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Theorem ps-2b 34278
Description: Variation of projective geometry axiom ps-2 34274. (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 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 ) ) ) )  ->  K  e.  HL )
2 simp12 1027 . . . 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 1028 . . . 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 1029 . . . 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 1176 . . 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 1030 . . . 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 1031 . . . 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 1032 . . . 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 1033 . . . 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 1034 . . 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 34274 . . 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 1236 . 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 1125 . . . . 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 34157 . . . . 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 34160 . . . . . 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 1126 . . . . . 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 1128 . . . . . 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 2467 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
2625, 14, 15hlatjcl 34163 . . . . . 6  |-  ( ( K  e.  HL  /\  P  e.  A  /\  R  e.  A )  ->  ( P  .\/  R
)  e.  ( Base `  K ) )
2718, 23, 24, 26syl3anc 1228 . . . . 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 1129 . . . . . 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 1130 . . . . . 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 34163 . . . . . 6  |-  ( ( K  e.  HL  /\  S  e.  A  /\  T  e.  A )  ->  ( S  .\/  T
)  e.  ( Base `  K ) )
3118, 28, 29, 30syl3anc 1228 . . . . 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 15535 . . . . 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 1228 . . . 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 997 . . . 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 998 . . . . 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 34086 . . . . . . 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 15561 . . . . . 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 1230 . . . . 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 34107 . . . 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 1231 . . 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 2957 . 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 973    = wceq 1379    e. wcel 1767    =/= wne 2662   E.wrex 2815   class class class wbr 4447   ` cfv 5586  (class class class)co 6282   Basecbs 14486   lecple 14558   joincjn 15427   meetcmee 15428   0.cp0 15520   Latclat 15528   Atomscatm 34060   AtLatcal 34061   HLchlt 34147
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-poset 15429  df-plt 15441  df-lub 15457  df-glb 15458  df-join 15459  df-meet 15460  df-p0 15522  df-lat 15529  df-clat 15591  df-oposet 33973  df-ol 33975  df-oml 33976  df-covers 34063  df-ats 34064  df-atl 34095  df-cvlat 34119  df-hlat 34148
This theorem is referenced by:  ps-2c  34324
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