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Theorem cdleme0e 34200
Description: Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 13-Jun-2012.)
Hypotheses
Ref Expression
cdleme0.l  |-  .<_  =  ( le `  K )
cdleme0.j  |-  .\/  =  ( join `  K )
cdleme0.m  |-  ./\  =  ( meet `  K )
cdleme0.a  |-  A  =  ( Atoms `  K )
cdleme0.h  |-  H  =  ( LHyp `  K
)
cdleme0.u  |-  U  =  ( ( P  .\/  Q )  ./\  W )
cdleme0c.3  |-  V  =  ( ( P  .\/  R )  ./\  W )
Assertion
Ref Expression
cdleme0e  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )  ->  U  =/=  V )

Proof of Theorem cdleme0e
StepHypRef Expression
1 cdleme0.u . . . . 5  |-  U  =  ( ( P  .\/  Q )  ./\  W )
2 cdleme0c.3 . . . . 5  |-  V  =  ( ( P  .\/  R )  ./\  W )
31, 2oveq12i 6213 . . . 4  |-  ( U 
./\  V )  =  ( ( ( P 
.\/  Q )  ./\  W )  ./\  ( ( P  .\/  R )  ./\  W ) )
4 simp1l 1012 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )  ->  K  e.  HL )
5 hlol 33345 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  OL )
64, 5syl 16 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )  ->  K  e.  OL )
7 simp21l 1105 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )  ->  P  e.  A )
8 simp22 1022 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )  ->  Q  e.  A )
9 eqid 2454 . . . . . . . 8  |-  ( Base `  K )  =  (
Base `  K )
10 cdleme0.j . . . . . . . 8  |-  .\/  =  ( join `  K )
11 cdleme0.a . . . . . . . 8  |-  A  =  ( Atoms `  K )
129, 10, 11hlatjcl 33350 . . . . . . 7  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
134, 7, 8, 12syl3anc 1219 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )  -> 
( P  .\/  Q
)  e.  ( Base `  K ) )
14 simp23l 1109 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )  ->  R  e.  A )
159, 10, 11hlatjcl 33350 . . . . . . 7  |-  ( ( K  e.  HL  /\  P  e.  A  /\  R  e.  A )  ->  ( P  .\/  R
)  e.  ( Base `  K ) )
164, 7, 14, 15syl3anc 1219 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )  -> 
( P  .\/  R
)  e.  ( Base `  K ) )
17 simp1r 1013 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )  ->  W  e.  H )
18 cdleme0.h . . . . . . . 8  |-  H  =  ( LHyp `  K
)
199, 18lhpbase 33981 . . . . . . 7  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
2017, 19syl 16 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )  ->  W  e.  ( Base `  K ) )
21 cdleme0.m . . . . . . 7  |-  ./\  =  ( meet `  K )
229, 21latmmdir 33219 . . . . . 6  |-  ( ( K  e.  OL  /\  ( ( P  .\/  Q )  e.  ( Base `  K )  /\  ( P  .\/  R )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
) )  ->  (
( ( P  .\/  Q )  ./\  ( P  .\/  R ) )  ./\  W )  =  ( ( ( P  .\/  Q
)  ./\  W )  ./\  ( ( P  .\/  R )  ./\  W )
) )
236, 13, 16, 20, 22syl13anc 1221 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )  -> 
( ( ( P 
.\/  Q )  ./\  ( P  .\/  R ) )  ./\  W )  =  ( ( ( P  .\/  Q ) 
./\  W )  ./\  ( ( P  .\/  R )  ./\  W )
) )
24 hllat 33347 . . . . . . . . 9  |-  ( K  e.  HL  ->  K  e.  Lat )
254, 24syl 16 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )  ->  K  e.  Lat )
269, 11atbase 33273 . . . . . . . . 9  |-  ( R  e.  A  ->  R  e.  ( Base `  K
) )
2714, 26syl 16 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )  ->  R  e.  ( Base `  K ) )
289, 11atbase 33273 . . . . . . . . 9  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
297, 28syl 16 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )  ->  P  e.  ( Base `  K ) )
309, 11atbase 33273 . . . . . . . . 9  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
318, 30syl 16 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )  ->  Q  e.  ( Base `  K ) )
32 simp3r 1017 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )  ->  -.  R  .<_  ( P 
.\/  Q ) )
33 cdleme0.l . . . . . . . . . 10  |-  .<_  =  ( le `  K )
349, 33, 10latnlej1r 15360 . . . . . . . . 9  |-  ( ( K  e.  Lat  /\  ( R  e.  ( Base `  K )  /\  P  e.  ( Base `  K )  /\  Q  e.  ( Base `  K
) )  /\  -.  R  .<_  ( P  .\/  Q ) )  ->  R  =/=  Q )
3534necomd 2723 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  ( R  e.  ( Base `  K )  /\  P  e.  ( Base `  K )  /\  Q  e.  ( Base `  K
) )  /\  -.  R  .<_  ( P  .\/  Q ) )  ->  Q  =/=  R )
3625, 27, 29, 31, 32, 35syl131anc 1232 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )  ->  Q  =/=  R )
37 simp3 990 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )  -> 
( P  =/=  Q  /\  -.  R  .<_  ( P 
.\/  Q ) ) )
3833, 10, 11hlatcon3 33434 . . . . . . . 8  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q
) ) )  ->  -.  P  .<_  ( Q 
.\/  R ) )
394, 7, 8, 14, 37, 38syl131anc 1232 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )  ->  -.  P  .<_  ( Q 
.\/  R ) )
4033, 10, 21, 112llnma2 33772 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  P  e.  A
)  /\  ( Q  =/=  R  /\  -.  P  .<_  ( Q  .\/  R
) ) )  -> 
( ( P  .\/  Q )  ./\  ( P  .\/  R ) )  =  P )
414, 8, 14, 7, 36, 39, 40syl132anc 1237 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )  -> 
( ( P  .\/  Q )  ./\  ( P  .\/  R ) )  =  P )
4241oveq1d 6216 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )  -> 
( ( ( P 
.\/  Q )  ./\  ( P  .\/  R ) )  ./\  W )  =  ( P  ./\  W ) )
4323, 42eqtr3d 2497 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )  -> 
( ( ( P 
.\/  Q )  ./\  W )  ./\  ( ( P  .\/  R )  ./\  W ) )  =  ( P  ./\  W )
)
443, 43syl5eq 2507 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )  -> 
( U  ./\  V
)  =  ( P 
./\  W ) )
45 simp1 988 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )  -> 
( K  e.  HL  /\  W  e.  H ) )
46 simp21 1021 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )  -> 
( P  e.  A  /\  -.  P  .<_  W ) )
47 eqid 2454 . . . . 5  |-  ( 0.
`  K )  =  ( 0. `  K
)
4833, 21, 47, 11, 18lhpmat 34013 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  -> 
( P  ./\  W
)  =  ( 0.
`  K ) )
4945, 46, 48syl2anc 661 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )  -> 
( P  ./\  W
)  =  ( 0.
`  K ) )
5044, 49eqtrd 2495 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )  -> 
( U  ./\  V
)  =  ( 0.
`  K ) )
51 hlatl 33344 . . . 4  |-  ( K  e.  HL  ->  K  e.  AtLat )
524, 51syl 16 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )  ->  K  e.  AtLat )
53 simp3l 1016 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )  ->  P  =/=  Q )
5433, 10, 21, 11, 18, 1lhpat2 34028 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q ) )  ->  U  e.  A
)
5545, 46, 8, 53, 54syl112anc 1223 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )  ->  U  e.  A )
569, 33, 10latnlej1l 15359 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( R  e.  ( Base `  K )  /\  P  e.  ( Base `  K )  /\  Q  e.  ( Base `  K
) )  /\  -.  R  .<_  ( P  .\/  Q ) )  ->  R  =/=  P )
5756necomd 2723 . . . . 5  |-  ( ( K  e.  Lat  /\  ( R  e.  ( Base `  K )  /\  P  e.  ( Base `  K )  /\  Q  e.  ( Base `  K
) )  /\  -.  R  .<_  ( P  .\/  Q ) )  ->  P  =/=  R )
5825, 27, 29, 31, 32, 57syl131anc 1232 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )  ->  P  =/=  R )
5933, 10, 21, 11, 18, 2lhpat2 34028 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R  e.  A  /\  P  =/=  R ) )  ->  V  e.  A
)
6045, 46, 14, 58, 59syl112anc 1223 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )  ->  V  e.  A )
6121, 47, 11atnem0 33302 . . 3  |-  ( ( K  e.  AtLat  /\  U  e.  A  /\  V  e.  A )  ->  ( U  =/=  V  <->  ( U  ./\ 
V )  =  ( 0. `  K ) ) )
6252, 55, 60, 61syl3anc 1219 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )  -> 
( U  =/=  V  <->  ( U  ./\  V )  =  ( 0. `  K ) ) )
6350, 62mpbird 232 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )  ->  U  =/=  V )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2648   class class class wbr 4401   ` cfv 5527  (class class class)co 6201   Basecbs 14293   lecple 14365   joincjn 15234   meetcmee 15235   0.cp0 15327   Latclat 15335   OLcol 33158   Atomscatm 33247   AtLatcal 33248   HLchlt 33334   LHypclh 33967
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 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483
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 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-riota 6162  df-ov 6204  df-oprab 6205  df-poset 15236  df-plt 15248  df-lub 15264  df-glb 15265  df-join 15266  df-meet 15267  df-p0 15329  df-p1 15330  df-lat 15336  df-clat 15398  df-oposet 33160  df-ol 33162  df-oml 33163  df-covers 33250  df-ats 33251  df-atl 33282  df-cvlat 33306  df-hlat 33335  df-lhyp 33971
This theorem is referenced by:  cdleme3fN  34216  cdleme3g  34217  cdleme11e  34246
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