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Theorem usgfisALT 32755
Description: An undirected simple graph of finite order (i.e. with a finite number of vertices) is of finite size, i.e. it has also only a finite number of edges, analogous to usgrafis 24633. Remark: The proof of this theorem is very long compared with usgrafis 24633, because the theorem brfi1ind 12537 to perform the finite induction is taylored for binary relations, so that the theorem itself and the used lemmas must be transformed accordingly. Maybe a variant of brfi1ind 12537 could be provided, which is better suitable for this theorem. (Contributed by Alexander van der Vekens, 6-Jan-2018.) (Revised by AV, 15-Jan-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
usgfisALT  |-  ( G  e. FinUSGrph  ->  ( Edges  `  G )  e.  Fin )

Proof of Theorem usgfisALT
Dummy variables  e 
f  v  w  y  p  q  n are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fusgraimpclALT2 32736 . 2  |-  ( G  e. FinUSGrph  ->  ( G  e. USGrph  /\  ( 1st `  G
)  e.  Fin )
)
2 relusgra 24553 . . . . . . 7  |-  Rel USGrph
3 1st2nd 6845 . . . . . . 7  |-  ( ( Rel USGrph  /\  G  e. USGrph  )  ->  G  =  <. ( 1st `  G ) ,  ( 2nd `  G
) >. )
42, 3mpan 670 . . . . . 6  |-  ( G  e. USGrph  ->  G  =  <. ( 1st `  G ) ,  ( 2nd `  G
) >. )
5 eleq1 2529 . . . . . . . 8  |-  ( G  =  <. ( 1st `  G
) ,  ( 2nd `  G ) >.  ->  ( G  e. USGrph  <->  <. ( 1st `  G
) ,  ( 2nd `  G ) >.  e. USGrph  )
)
6 df-br 4457 . . . . . . . 8  |-  ( ( 1st `  G ) USGrph 
( 2nd `  G
)  <->  <. ( 1st `  G
) ,  ( 2nd `  G ) >.  e. USGrph  )
75, 6syl6bbr 263 . . . . . . 7  |-  ( G  =  <. ( 1st `  G
) ,  ( 2nd `  G ) >.  ->  ( G  e. USGrph  <->  ( 1st `  G
) USGrph  ( 2nd `  G
) ) )
8 mptresid 5338 . . . . . . . . . 10  |-  ( q  e.  { p  e.  ( Edges  `  <. v ,  e >. )  |  n  e/  p }  |->  q )  =  (  _I  |`  { p  e.  ( Edges  `  <. v ,  e
>. )  |  n  e/  p } )
9 fvex 5882 . . . . . . . . . . 11  |-  ( Edges  `  <. v ,  e >. )  e.  _V
109mptrabex 6145 . . . . . . . . . 10  |-  ( q  e.  { p  e.  ( Edges  `  <. v ,  e >. )  |  n  e/  p }  |->  q )  e.  _V
118, 10eqeltrri 2542 . . . . . . . . 9  |-  (  _I  |`  { p  e.  ( Edges  `  <. v ,  e
>. )  |  n  e/  p } )  e. 
_V
12 eleq1 2529 . . . . . . . . . 10  |-  ( e  =  ( 2nd `  G
)  ->  ( e  e.  Fin  <->  ( 2nd `  G
)  e.  Fin )
)
1312adantl 466 . . . . . . . . 9  |-  ( ( v  =  ( 1st `  G )  /\  e  =  ( 2nd `  G
) )  ->  (
e  e.  Fin  <->  ( 2nd `  G )  e.  Fin ) )
14 eleq1 2529 . . . . . . . . . 10  |-  ( e  =  f  ->  (
e  e.  Fin  <->  f  e.  Fin ) )
1514adantl 466 . . . . . . . . 9  |-  ( ( v  =  w  /\  e  =  f )  ->  ( e  e.  Fin  <->  f  e.  Fin ) )
16 df-br 4457 . . . . . . . . . 10  |-  ( v USGrph 
e  <->  <. v ,  e
>.  e. USGrph  )
17 vex 3112 . . . . . . . . . . . . 13  |-  v  e. 
_V
18 vex 3112 . . . . . . . . . . . . 13  |-  e  e. 
_V
1917, 18op1st 6807 . . . . . . . . . . . 12  |-  ( 1st `  <. v ,  e
>. )  =  v
2019eqcomi 2470 . . . . . . . . . . 11  |-  v  =  ( 1st `  <. v ,  e >. )
21 eqid 2457 . . . . . . . . . . 11  |-  ( Edges  `  <. v ,  e >. )  =  ( Edges  `  <. v ,  e >. )
22 eqid 2457 . . . . . . . . . . 11  |-  { p  e.  ( Edges  `  <. v ,  e >. )  |  n  e/  p }  =  { p  e.  ( Edges  `  <. v ,  e >. )  |  n  e/  p }
2320, 21, 22usgresvm1ALT 32752 . . . . . . . . . 10  |-  ( (
<. v ,  e >.  e. USGrph  /\  n  e.  v )  ->  ( v  \  { n } ) USGrph 
(  _I  |`  { p  e.  ( Edges  `  <. v ,  e >. )  |  n  e/  p } ) )
2416, 23sylanb 472 . . . . . . . . 9  |-  ( ( v USGrph  e  /\  n  e.  v )  ->  (
v  \  { n } ) USGrph  (  _I  |`  { p  e.  ( Edges  `  <. v ,  e >. )  |  n  e/  p } ) )
25 eleq1 2529 . . . . . . . . . 10  |-  ( f  =  (  _I  |`  { p  e.  ( Edges  `  <. v ,  e >. )  |  n  e/  p } )  ->  (
f  e.  Fin  <->  (  _I  |` 
{ p  e.  ( Edges  `  <. v ,  e
>. )  |  n  e/  p } )  e. 
Fin ) )
2625adantl 466 . . . . . . . . 9  |-  ( ( w  =  ( v 
\  { n }
)  /\  f  =  (  _I  |`  { p  e.  ( Edges  `  <. v ,  e >. )  |  n  e/  p } ) )  -> 
( f  e.  Fin  <->  (  _I  |`  { p  e.  ( Edges  `  <. v ,  e >. )  |  n  e/  p } )  e.  Fin ) )
27 usgrafisbaseALT 32745 . . . . . . . . 9  |-  ( ( v USGrph  e  /\  ( # `
 v )  =  0 )  ->  e  e.  Fin )
28 residfi 32621 . . . . . . . . . . . 12  |-  ( (  _I  |`  { p  e.  ( Edges  `  <. v ,  e >. )  |  n  e/  p } )  e.  Fin  <->  {
p  e.  ( Edges  `  <. v ,  e >. )  |  n  e/  p }  e.  Fin )
29 simpr1 1002 . . . . . . . . . . . . . . . . 17  |-  ( ( ( y  +  1 )  e.  NN0  /\  ( v USGrph  e  /\  ( # `
 v )  =  ( y  +  1 )  /\  n  e.  v ) )  -> 
v USGrph  e )
30 eleq1 2529 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( y  +  1 )  =  ( # `  v
)  ->  ( (
y  +  1 )  e.  NN0  <->  ( # `  v
)  e.  NN0 )
)
3130eqcoms 2469 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
# `  v )  =  ( y  +  1 )  ->  (
( y  +  1 )  e.  NN0  <->  ( # `  v
)  e.  NN0 )
)
32 hashclb 12433 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( v  e.  _V  ->  (
v  e.  Fin  <->  ( # `  v
)  e.  NN0 )
)
3332bicomd 201 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( v  e.  _V  ->  (
( # `  v )  e.  NN0  <->  v  e.  Fin ) )
3417, 33ax-mp 5 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
# `  v )  e.  NN0  <->  v  e.  Fin )
3534biimpi 194 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
# `  v )  e.  NN0  ->  v  e.  Fin )
3631, 35syl6bi 228 . . . . . . . . . . . . . . . . . . 19  |-  ( (
# `  v )  =  ( y  +  1 )  ->  (
( y  +  1 )  e.  NN0  ->  v  e.  Fin ) )
37363ad2ant2 1018 . . . . . . . . . . . . . . . . . 18  |-  ( ( v USGrph  e  /\  ( # `
 v )  =  ( y  +  1 )  /\  n  e.  v )  ->  (
( y  +  1 )  e.  NN0  ->  v  e.  Fin ) )
3837impcom 430 . . . . . . . . . . . . . . . . 17  |-  ( ( ( y  +  1 )  e.  NN0  /\  ( v USGrph  e  /\  ( # `
 v )  =  ( y  +  1 )  /\  n  e.  v ) )  -> 
v  e.  Fin )
3929, 38jca 532 . . . . . . . . . . . . . . . 16  |-  ( ( ( y  +  1 )  e.  NN0  /\  ( v USGrph  e  /\  ( # `
 v )  =  ( y  +  1 )  /\  n  e.  v ) )  -> 
( v USGrph  e  /\  v  e.  Fin )
)
40 usgrav 24556 . . . . . . . . . . . . . . . . . . 19  |-  ( v USGrph 
e  ->  ( v  e.  _V  /\  e  e. 
_V ) )
41 df-br 4457 . . . . . . . . . . . . . . . . . . . 20  |-  ( v FinUSGrph  e 
<-> 
<. v ,  e >.  e. FinUSGrph  )
42 isfusgra0 32730 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( v  e.  _V  /\  e  e.  _V )  ->  ( v FinUSGrph  e  <->  ( v USGrph  e  /\  v  e.  Fin ) ) )
4341, 42syl5bbr 259 . . . . . . . . . . . . . . . . . . 19  |-  ( ( v  e.  _V  /\  e  e.  _V )  ->  ( <. v ,  e
>.  e. FinUSGrph 
<->  ( v USGrph  e  /\  v  e.  Fin )
) )
4440, 43syl 16 . . . . . . . . . . . . . . . . . 18  |-  ( v USGrph 
e  ->  ( <. v ,  e >.  e. FinUSGrph  <->  ( v USGrph  e  /\  v  e.  Fin ) ) )
45443ad2ant1 1017 . . . . . . . . . . . . . . . . 17  |-  ( ( v USGrph  e  /\  ( # `
 v )  =  ( y  +  1 )  /\  n  e.  v )  ->  ( <. v ,  e >.  e. FinUSGrph  <->  ( v USGrph  e  /\  v  e.  Fin ) ) )
4645adantl 466 . . . . . . . . . . . . . . . 16  |-  ( ( ( y  +  1 )  e.  NN0  /\  ( v USGrph  e  /\  ( # `
 v )  =  ( y  +  1 )  /\  n  e.  v ) )  -> 
( <. v ,  e
>.  e. FinUSGrph 
<->  ( v USGrph  e  /\  v  e.  Fin )
) )
4739, 46mpbird 232 . . . . . . . . . . . . . . 15  |-  ( ( ( y  +  1 )  e.  NN0  /\  ( v USGrph  e  /\  ( # `
 v )  =  ( y  +  1 )  /\  n  e.  v ) )  ->  <. v ,  e >.  e. FinUSGrph  )
48 eqidd 2458 . . . . . . . . . . . . . . . . . . . . 21  |-  ( v USGrph 
e  ->  ( 1st ` 
<. v ,  e >.
)  =  ( 1st `  <. v ,  e
>. ) )
4948, 19syl6req 2515 . . . . . . . . . . . . . . . . . . . 20  |-  ( v USGrph 
e  ->  v  =  ( 1st `  <. v ,  e >. )
)
5049eleq2d 2527 . . . . . . . . . . . . . . . . . . 19  |-  ( v USGrph 
e  ->  ( n  e.  v  <->  n  e.  ( 1st `  <. v ,  e
>. ) ) )
5150biimpd 207 . . . . . . . . . . . . . . . . . 18  |-  ( v USGrph 
e  ->  ( n  e.  v  ->  n  e.  ( 1st `  <. v ,  e >. )
) )
5251a1d 25 . . . . . . . . . . . . . . . . 17  |-  ( v USGrph 
e  ->  ( ( # `
 v )  =  ( y  +  1 )  ->  ( n  e.  v  ->  n  e.  ( 1st `  <. v ,  e >. )
) ) )
53523imp 1190 . . . . . . . . . . . . . . . 16  |-  ( ( v USGrph  e  /\  ( # `
 v )  =  ( y  +  1 )  /\  n  e.  v )  ->  n  e.  ( 1st `  <. v ,  e >. )
)
5453adantl 466 . . . . . . . . . . . . . . 15  |-  ( ( ( y  +  1 )  e.  NN0  /\  ( v USGrph  e  /\  ( # `
 v )  =  ( y  +  1 )  /\  n  e.  v ) )  ->  n  e.  ( 1st ` 
<. v ,  e >.
) )
5547, 54jca 532 . . . . . . . . . . . . . 14  |-  ( ( ( y  +  1 )  e.  NN0  /\  ( v USGrph  e  /\  ( # `
 v )  =  ( y  +  1 )  /\  n  e.  v ) )  -> 
( <. v ,  e
>.  e. FinUSGrph  /\  n  e.  ( 1st `  <. v ,  e >. )
) )
56 eqid 2457 . . . . . . . . . . . . . . 15  |-  ( 1st `  <. v ,  e
>. )  =  ( 1st `  <. v ,  e
>. )
57 df-ov 6299 . . . . . . . . . . . . . . 15  |-  ( v Edges 
e )  =  ( Edges  `  <. v ,  e
>. )
5857eqcomi 2470 . . . . . . . . . . . . . . . 16  |-  ( Edges  `  <. v ,  e >. )  =  ( v Edges  e
)
59 rabeq 3103 . . . . . . . . . . . . . . . 16  |-  ( ( Edges  `  <. v ,  e
>. )  =  (
v Edges  e )  ->  { p  e.  ( Edges  ` 
<. v ,  e >.
)  |  n  e/  p }  =  {
p  e.  ( v Edges 
e )  |  n  e/  p } )
6058, 59ax-mp 5 . . . . . . . . . . . . . . 15  |-  { p  e.  ( Edges  `  <. v ,  e >. )  |  n  e/  p }  =  { p  e.  ( v Edges  e )  |  n  e/  p }
6156, 57, 60usgfisALTlem2 32754 . . . . . . . . . . . . . 14  |-  ( (
<. v ,  e >.  e. FinUSGrph 
/\  n  e.  ( 1st `  <. v ,  e >. )
)  ->  ( (
v Edges  e )  e. 
Fin 
<->  { p  e.  ( Edges  `  <. v ,  e
>. )  |  n  e/  p }  e.  Fin ) )
6255, 61syl 16 . . . . . . . . . . . . 13  |-  ( ( ( y  +  1 )  e.  NN0  /\  ( v USGrph  e  /\  ( # `
 v )  =  ( y  +  1 )  /\  n  e.  v ) )  -> 
( ( v Edges  e
)  e.  Fin  <->  { p  e.  ( Edges  `  <. v ,  e >. )  |  n  e/  p }  e.  Fin )
)
6362biimprd 223 . . . . . . . . . . . 12  |-  ( ( ( y  +  1 )  e.  NN0  /\  ( v USGrph  e  /\  ( # `
 v )  =  ( y  +  1 )  /\  n  e.  v ) )  -> 
( { p  e.  ( Edges  `  <. v ,  e >. )  |  n  e/  p }  e.  Fin  ->  ( v Edges  e
)  e.  Fin )
)
6428, 63syl5bi 217 . . . . . . . . . . 11  |-  ( ( ( y  +  1 )  e.  NN0  /\  ( v USGrph  e  /\  ( # `
 v )  =  ( y  +  1 )  /\  n  e.  v ) )  -> 
( (  _I  |`  { p  e.  ( Edges  `  <. v ,  e >. )  |  n  e/  p } )  e.  Fin  ->  ( v Edges  e )  e.  Fin ) )
6564imp 429 . . . . . . . . . 10  |-  ( ( ( ( y  +  1 )  e.  NN0  /\  ( v USGrph  e  /\  ( # `  v )  =  ( y  +  1 )  /\  n  e.  v ) )  /\  (  _I  |`  { p  e.  ( Edges  `  <. v ,  e >. )  |  n  e/  p } )  e.  Fin )  ->  ( v Edges  e
)  e.  Fin )
66 usgedgffibi 32739 . . . . . . . . . . . . 13  |-  ( v USGrph 
e  ->  ( e  e.  Fin  <->  ( v Edges  e
)  e.  Fin )
)
67663ad2ant1 1017 . . . . . . . . . . . 12  |-  ( ( v USGrph  e  /\  ( # `
 v )  =  ( y  +  1 )  /\  n  e.  v )  ->  (
e  e.  Fin  <->  ( v Edges  e )  e.  Fin ) )
6867adantl 466 . . . . . . . . . . 11  |-  ( ( ( y  +  1 )  e.  NN0  /\  ( v USGrph  e  /\  ( # `
 v )  =  ( y  +  1 )  /\  n  e.  v ) )  -> 
( e  e.  Fin  <->  (
v Edges  e )  e. 
Fin ) )
6968adantr 465 . . . . . . . . . 10  |-  ( ( ( ( y  +  1 )  e.  NN0  /\  ( v USGrph  e  /\  ( # `  v )  =  ( y  +  1 )  /\  n  e.  v ) )  /\  (  _I  |`  { p  e.  ( Edges  `  <. v ,  e >. )  |  n  e/  p } )  e.  Fin )  ->  ( e  e. 
Fin 
<->  ( v Edges  e )  e.  Fin ) )
7065, 69mpbird 232 . . . . . . . . 9  |-  ( ( ( ( y  +  1 )  e.  NN0  /\  ( v USGrph  e  /\  ( # `  v )  =  ( y  +  1 )  /\  n  e.  v ) )  /\  (  _I  |`  { p  e.  ( Edges  `  <. v ,  e >. )  |  n  e/  p } )  e.  Fin )  ->  e  e.  Fin )
712, 11, 13, 15, 24, 26, 27, 70brfi1ind 12537 . . . . . . . 8  |-  ( ( ( 1st `  G
) USGrph  ( 2nd `  G
)  /\  ( 1st `  G )  e.  Fin )  ->  ( 2nd `  G
)  e.  Fin )
7271ex 434 . . . . . . 7  |-  ( ( 1st `  G ) USGrph 
( 2nd `  G
)  ->  ( ( 1st `  G )  e. 
Fin  ->  ( 2nd `  G
)  e.  Fin )
)
737, 72syl6bi 228 . . . . . 6  |-  ( G  =  <. ( 1st `  G
) ,  ( 2nd `  G ) >.  ->  ( G  e. USGrph  ->  ( ( 1st `  G )  e.  Fin  ->  ( 2nd `  G )  e. 
Fin ) ) )
744, 73mpcom 36 . . . . 5  |-  ( G  e. USGrph  ->  ( ( 1st `  G )  e.  Fin  ->  ( 2nd `  G
)  e.  Fin )
)
7574imp 429 . . . 4  |-  ( ( G  e. USGrph  /\  ( 1st `  G )  e. 
Fin )  ->  ( 2nd `  G )  e. 
Fin )
76 rnfi 7823 . . . 4  |-  ( ( 2nd `  G )  e.  Fin  ->  ran  ( 2nd `  G )  e.  Fin )
7775, 76syl 16 . . 3  |-  ( ( G  e. USGrph  /\  ( 1st `  G )  e. 
Fin )  ->  ran  ( 2nd `  G )  e.  Fin )
78 edgval 24557 . . . . 5  |-  ( G  e. USGrph  ->  ( Edges  `  G )  =  ran  ( 2nd `  G ) )
7978eleq1d 2526 . . . 4  |-  ( G  e. USGrph  ->  ( ( Edges  `  G
)  e.  Fin  <->  ran  ( 2nd `  G )  e.  Fin ) )
8079adantr 465 . . 3  |-  ( ( G  e. USGrph  /\  ( 1st `  G )  e. 
Fin )  ->  (
( Edges  `  G )  e. 
Fin 
<->  ran  ( 2nd `  G
)  e.  Fin )
)
8177, 80mpbird 232 . 2  |-  ( ( G  e. USGrph  /\  ( 1st `  G )  e. 
Fin )  ->  ( Edges  `  G )  e.  Fin )
821, 81syl 16 1  |-  ( G  e. FinUSGrph  ->  ( Edges  `  G )  e.  Fin )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819    e/ wnel 2653   {crab 2811   _Vcvv 3109    \ cdif 3468   {csn 4032   <.cop 4038   class class class wbr 4456    |-> cmpt 4515    _I cid 4799   ran crn 5009    |` cres 5010   Rel wrel 5013   ` cfv 5594  (class class class)co 6296   1stc1st 6797   2ndc2nd 6798   Fincfn 7535   1c1 9510    + caddc 9512   NN0cn0 10816   #chash 12408   USGrph cusg 24548   Edges cedg 24549   FinUSGrph cfusg 32726
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-2o 7149  df-oadd 7152  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-card 8337  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-n0 10817  df-z 10886  df-uz 11107  df-fz 11698  df-hash 12409  df-uhgra 24510  df-umgra 24531  df-uslgra 24550  df-usgra 24551  df-edg 24554  df-gord 32691  df-gsiz 32692  df-fusg 32727
This theorem is referenced by: (None)
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