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Theorem ps-2b 33435
Description: Variation of projective geometry axiom ps-2 33431. (Contributed by NM, 3-Jul-2012.)
Hypotheses
Ref Expression
ps-2b.l  |-  .<_  =  ( le `  K )
ps-2b.j  |-  .\/  =  ( join `  K )
ps-2b.m  |-  ./\  =  ( meet `  K )
ps-2b.z  |-  .0.  =  ( 0. `  K )
ps-2b.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
ps-2b  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( -.  P  .<_  ( Q  .\/  R )  /\  S  =/= 
T  /\  ( S  .<_  ( P  .\/  Q
)  /\  T  .<_  ( Q  .\/  R ) ) ) )  -> 
( ( P  .\/  R )  ./\  ( S  .\/  T ) )  =/= 
.0.  )

Proof of Theorem ps-2b
Dummy variable  u is distinct from all other variables.
StepHypRef Expression
1 simp11 1018 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( -.  P  .<_  ( Q  .\/  R )  /\  S  =/= 
T  /\  ( S  .<_  ( P  .\/  Q
)  /\  T  .<_  ( Q  .\/  R ) ) ) )  ->  K  e.  HL )
2 simp12 1019 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( -.  P  .<_  ( Q  .\/  R )  /\  S  =/= 
T  /\  ( S  .<_  ( P  .\/  Q
)  /\  T  .<_  ( Q  .\/  R ) ) ) )  ->  P  e.  A )
3 simp13 1020 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( -.  P  .<_  ( Q  .\/  R )  /\  S  =/= 
T  /\  ( S  .<_  ( P  .\/  Q
)  /\  T  .<_  ( Q  .\/  R ) ) ) )  ->  Q  e.  A )
4 simp21 1021 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( -.  P  .<_  ( Q  .\/  R )  /\  S  =/= 
T  /\  ( S  .<_  ( P  .\/  Q
)  /\  T  .<_  ( Q  .\/  R ) ) ) )  ->  R  e.  A )
52, 3, 43jca 1168 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( -.  P  .<_  ( Q  .\/  R )  /\  S  =/= 
T  /\  ( S  .<_  ( P  .\/  Q
)  /\  T  .<_  ( Q  .\/  R ) ) ) )  -> 
( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )
6 simp22 1022 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( -.  P  .<_  ( Q  .\/  R )  /\  S  =/= 
T  /\  ( S  .<_  ( P  .\/  Q
)  /\  T  .<_  ( Q  .\/  R ) ) ) )  ->  S  e.  A )
7 simp23 1023 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( -.  P  .<_  ( Q  .\/  R )  /\  S  =/= 
T  /\  ( S  .<_  ( P  .\/  Q
)  /\  T  .<_  ( Q  .\/  R ) ) ) )  ->  T  e.  A )
86, 7jca 532 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( -.  P  .<_  ( Q  .\/  R )  /\  S  =/= 
T  /\  ( S  .<_  ( P  .\/  Q
)  /\  T  .<_  ( Q  .\/  R ) ) ) )  -> 
( S  e.  A  /\  T  e.  A
) )
9 simp31 1024 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( -.  P  .<_  ( Q  .\/  R )  /\  S  =/= 
T  /\  ( S  .<_  ( P  .\/  Q
)  /\  T  .<_  ( Q  .\/  R ) ) ) )  ->  -.  P  .<_  ( Q 
.\/  R ) )
10 simp32 1025 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( -.  P  .<_  ( Q  .\/  R )  /\  S  =/= 
T  /\  ( S  .<_  ( P  .\/  Q
)  /\  T  .<_  ( Q  .\/  R ) ) ) )  ->  S  =/=  T )
119, 10jca 532 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( -.  P  .<_  ( Q  .\/  R )  /\  S  =/= 
T  /\  ( S  .<_  ( P  .\/  Q
)  /\  T  .<_  ( Q  .\/  R ) ) ) )  -> 
( -.  P  .<_  ( Q  .\/  R )  /\  S  =/=  T
) )
12 simp33 1026 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( -.  P  .<_  ( Q  .\/  R )  /\  S  =/= 
T  /\  ( S  .<_  ( P  .\/  Q
)  /\  T  .<_  ( Q  .\/  R ) ) ) )  -> 
( S  .<_  ( P 
.\/  Q )  /\  T  .<_  ( Q  .\/  R ) ) )
13 ps-2b.l . . . 4  |-  .<_  =  ( le `  K )
14 ps-2b.j . . . 4  |-  .\/  =  ( join `  K )
15 ps-2b.a . . . 4  |-  A  =  ( Atoms `  K )
1613, 14, 15ps-2 33431 . . 3  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( S  e.  A  /\  T  e.  A ) )  /\  ( ( -.  P  .<_  ( Q  .\/  R
)  /\  S  =/=  T )  /\  ( S 
.<_  ( P  .\/  Q
)  /\  T  .<_  ( Q  .\/  R ) ) ) )  ->  E. u  e.  A  ( u  .<_  ( P 
.\/  R )  /\  u  .<_  ( S  .\/  T ) ) )
171, 5, 8, 11, 12, 16syl32anc 1227 . 2  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( -.  P  .<_  ( Q  .\/  R )  /\  S  =/= 
T  /\  ( S  .<_  ( P  .\/  Q
)  /\  T  .<_  ( Q  .\/  R ) ) ) )  ->  E. u  e.  A  ( u  .<_  ( P 
.\/  R )  /\  u  .<_  ( S  .\/  T ) ) )
18 simp111 1117 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A )  /\  ( -.  P  .<_  ( Q  .\/  R )  /\  S  =/=  T  /\  ( S  .<_  ( P 
.\/  Q )  /\  T  .<_  ( Q  .\/  R ) ) ) )  /\  u  e.  A  /\  ( u  .<_  ( P 
.\/  R )  /\  u  .<_  ( S  .\/  T ) ) )  ->  K  e.  HL )
19 hlatl 33314 . . . . 5  |-  ( K  e.  HL  ->  K  e.  AtLat )
2018, 19syl 16 . . . 4  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A )  /\  ( -.  P  .<_  ( Q  .\/  R )  /\  S  =/=  T  /\  ( S  .<_  ( P 
.\/  Q )  /\  T  .<_  ( Q  .\/  R ) ) ) )  /\  u  e.  A  /\  ( u  .<_  ( P 
.\/  R )  /\  u  .<_  ( S  .\/  T ) ) )  ->  K  e.  AtLat )
21 hllat 33317 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  Lat )
2218, 21syl 16 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A )  /\  ( -.  P  .<_  ( Q  .\/  R )  /\  S  =/=  T  /\  ( S  .<_  ( P 
.\/  Q )  /\  T  .<_  ( Q  .\/  R ) ) ) )  /\  u  e.  A  /\  ( u  .<_  ( P 
.\/  R )  /\  u  .<_  ( S  .\/  T ) ) )  ->  K  e.  Lat )
23 simp112 1118 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A )  /\  ( -.  P  .<_  ( Q  .\/  R )  /\  S  =/=  T  /\  ( S  .<_  ( P 
.\/  Q )  /\  T  .<_  ( Q  .\/  R ) ) ) )  /\  u  e.  A  /\  ( u  .<_  ( P 
.\/  R )  /\  u  .<_  ( S  .\/  T ) ) )  ->  P  e.  A )
24 simp121 1120 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A )  /\  ( -.  P  .<_  ( Q  .\/  R )  /\  S  =/=  T  /\  ( S  .<_  ( P 
.\/  Q )  /\  T  .<_  ( Q  .\/  R ) ) ) )  /\  u  e.  A  /\  ( u  .<_  ( P 
.\/  R )  /\  u  .<_  ( S  .\/  T ) ) )  ->  R  e.  A )
25 eqid 2451 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
2625, 14, 15hlatjcl 33320 . . . . . 6  |-  ( ( K  e.  HL  /\  P  e.  A  /\  R  e.  A )  ->  ( P  .\/  R
)  e.  ( Base `  K ) )
2718, 23, 24, 26syl3anc 1219 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A )  /\  ( -.  P  .<_  ( Q  .\/  R )  /\  S  =/=  T  /\  ( S  .<_  ( P 
.\/  Q )  /\  T  .<_  ( Q  .\/  R ) ) ) )  /\  u  e.  A  /\  ( u  .<_  ( P 
.\/  R )  /\  u  .<_  ( S  .\/  T ) ) )  -> 
( P  .\/  R
)  e.  ( Base `  K ) )
28 simp122 1121 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A )  /\  ( -.  P  .<_  ( Q  .\/  R )  /\  S  =/=  T  /\  ( S  .<_  ( P 
.\/  Q )  /\  T  .<_  ( Q  .\/  R ) ) ) )  /\  u  e.  A  /\  ( u  .<_  ( P 
.\/  R )  /\  u  .<_  ( S  .\/  T ) ) )  ->  S  e.  A )
29 simp123 1122 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A )  /\  ( -.  P  .<_  ( Q  .\/  R )  /\  S  =/=  T  /\  ( S  .<_  ( P 
.\/  Q )  /\  T  .<_  ( Q  .\/  R ) ) ) )  /\  u  e.  A  /\  ( u  .<_  ( P 
.\/  R )  /\  u  .<_  ( S  .\/  T ) ) )  ->  T  e.  A )
3025, 14, 15hlatjcl 33320 . . . . . 6  |-  ( ( K  e.  HL  /\  S  e.  A  /\  T  e.  A )  ->  ( S  .\/  T
)  e.  ( Base `  K ) )
3118, 28, 29, 30syl3anc 1219 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A )  /\  ( -.  P  .<_  ( Q  .\/  R )  /\  S  =/=  T  /\  ( S  .<_  ( P 
.\/  Q )  /\  T  .<_  ( Q  .\/  R ) ) ) )  /\  u  e.  A  /\  ( u  .<_  ( P 
.\/  R )  /\  u  .<_  ( S  .\/  T ) ) )  -> 
( S  .\/  T
)  e.  ( Base `  K ) )
32 ps-2b.m . . . . . 6  |-  ./\  =  ( meet `  K )
3325, 32latmcl 15333 . . . . 5  |-  ( ( K  e.  Lat  /\  ( P  .\/  R )  e.  ( Base `  K
)  /\  ( S  .\/  T )  e.  (
Base `  K )
)  ->  ( ( P  .\/  R )  ./\  ( S  .\/  T ) )  e.  ( Base `  K ) )
3422, 27, 31, 33syl3anc 1219 . . . 4  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A )  /\  ( -.  P  .<_  ( Q  .\/  R )  /\  S  =/=  T  /\  ( S  .<_  ( P 
.\/  Q )  /\  T  .<_  ( Q  .\/  R ) ) ) )  /\  u  e.  A  /\  ( u  .<_  ( P 
.\/  R )  /\  u  .<_  ( S  .\/  T ) ) )  -> 
( ( P  .\/  R )  ./\  ( S  .\/  T ) )  e.  ( Base `  K
) )
35 simp2 989 . . . 4  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A )  /\  ( -.  P  .<_  ( Q  .\/  R )  /\  S  =/=  T  /\  ( S  .<_  ( P 
.\/  Q )  /\  T  .<_  ( Q  .\/  R ) ) ) )  /\  u  e.  A  /\  ( u  .<_  ( P 
.\/  R )  /\  u  .<_  ( S  .\/  T ) ) )  ->  u  e.  A )
36 simp3 990 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A )  /\  ( -.  P  .<_  ( Q  .\/  R )  /\  S  =/=  T  /\  ( S  .<_  ( P 
.\/  Q )  /\  T  .<_  ( Q  .\/  R ) ) ) )  /\  u  e.  A  /\  ( u  .<_  ( P 
.\/  R )  /\  u  .<_  ( S  .\/  T ) ) )  -> 
( u  .<_  ( P 
.\/  R )  /\  u  .<_  ( S  .\/  T ) ) )
3725, 15atbase 33243 . . . . . . 7  |-  ( u  e.  A  ->  u  e.  ( Base `  K
) )
3835, 37syl 16 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A )  /\  ( -.  P  .<_  ( Q  .\/  R )  /\  S  =/=  T  /\  ( S  .<_  ( P 
.\/  Q )  /\  T  .<_  ( Q  .\/  R ) ) ) )  /\  u  e.  A  /\  ( u  .<_  ( P 
.\/  R )  /\  u  .<_  ( S  .\/  T ) ) )  ->  u  e.  ( Base `  K ) )
3925, 13, 32latlem12 15359 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( u  e.  ( Base `  K )  /\  ( P  .\/  R )  e.  ( Base `  K
)  /\  ( S  .\/  T )  e.  (
Base `  K )
) )  ->  (
( u  .<_  ( P 
.\/  R )  /\  u  .<_  ( S  .\/  T ) )  <->  u  .<_  ( ( P  .\/  R
)  ./\  ( S  .\/  T ) ) ) )
4022, 38, 27, 31, 39syl13anc 1221 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A )  /\  ( -.  P  .<_  ( Q  .\/  R )  /\  S  =/=  T  /\  ( S  .<_  ( P 
.\/  Q )  /\  T  .<_  ( Q  .\/  R ) ) ) )  /\  u  e.  A  /\  ( u  .<_  ( P 
.\/  R )  /\  u  .<_  ( S  .\/  T ) ) )  -> 
( ( u  .<_  ( P  .\/  R )  /\  u  .<_  ( S 
.\/  T ) )  <-> 
u  .<_  ( ( P 
.\/  R )  ./\  ( S  .\/  T ) ) ) )
4136, 40mpbid 210 . . . 4  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A )  /\  ( -.  P  .<_  ( Q  .\/  R )  /\  S  =/=  T  /\  ( S  .<_  ( P 
.\/  Q )  /\  T  .<_  ( Q  .\/  R ) ) ) )  /\  u  e.  A  /\  ( u  .<_  ( P 
.\/  R )  /\  u  .<_  ( S  .\/  T ) ) )  ->  u  .<_  ( ( P 
.\/  R )  ./\  ( S  .\/  T ) ) )
42 ps-2b.z . . . . 5  |-  .0.  =  ( 0. `  K )
4325, 13, 42, 15atlen0 33264 . . . 4  |-  ( ( ( K  e.  AtLat  /\  ( ( P  .\/  R )  ./\  ( S  .\/  T ) )  e.  ( Base `  K
)  /\  u  e.  A )  /\  u  .<_  ( ( P  .\/  R )  ./\  ( S  .\/  T ) ) )  ->  ( ( P 
.\/  R )  ./\  ( S  .\/  T ) )  =/=  .0.  )
4420, 34, 35, 41, 43syl31anc 1222 . . 3  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A )  /\  ( -.  P  .<_  ( Q  .\/  R )  /\  S  =/=  T  /\  ( S  .<_  ( P 
.\/  Q )  /\  T  .<_  ( Q  .\/  R ) ) ) )  /\  u  e.  A  /\  ( u  .<_  ( P 
.\/  R )  /\  u  .<_  ( S  .\/  T ) ) )  -> 
( ( P  .\/  R )  ./\  ( S  .\/  T ) )  =/= 
.0.  )
4544rexlimdv3a 2942 . 2  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( -.  P  .<_  ( Q  .\/  R )  /\  S  =/= 
T  /\  ( S  .<_  ( P  .\/  Q
)  /\  T  .<_  ( Q  .\/  R ) ) ) )  -> 
( E. u  e.  A  ( u  .<_  ( P  .\/  R )  /\  u  .<_  ( S 
.\/  T ) )  ->  ( ( P 
.\/  R )  ./\  ( S  .\/  T ) )  =/=  .0.  )
)
4617, 45mpd 15 1  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( -.  P  .<_  ( Q  .\/  R )  /\  S  =/= 
T  /\  ( S  .<_  ( P  .\/  Q
)  /\  T  .<_  ( Q  .\/  R ) ) ) )  -> 
( ( P  .\/  R )  ./\  ( S  .\/  T ) )  =/= 
.0.  )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2644   E.wrex 2796   class class class wbr 4393   ` cfv 5519  (class class class)co 6193   Basecbs 14285   lecple 14356   joincjn 15225   meetcmee 15226   0.cp0 15318   Latclat 15326   Atomscatm 33217   AtLatcal 33218   HLchlt 33304
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-id 4737  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-riota 6154  df-ov 6196  df-oprab 6197  df-poset 15227  df-plt 15239  df-lub 15255  df-glb 15256  df-join 15257  df-meet 15258  df-p0 15320  df-lat 15327  df-clat 15389  df-oposet 33130  df-ol 33132  df-oml 33133  df-covers 33220  df-ats 33221  df-atl 33252  df-cvlat 33276  df-hlat 33305
This theorem is referenced by:  ps-2c  33481
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