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Theorem dalem-cly 33597
Description: Lemma for dalem9 33598. Center of perspectivity  C is not in plane  Y (when  Y and  Z are different planes). (Contributed by NM, 13-Aug-2012.)
Hypotheses
Ref Expression
dalema.ph  |-  ( ph  <->  ( ( ( K  e.  HL  /\  C  e.  ( Base `  K
) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A
) )  /\  ( Y  e.  O  /\  Z  e.  O )  /\  ( ( -.  C  .<_  ( P  .\/  Q
)  /\  -.  C  .<_  ( Q  .\/  R
)  /\  -.  C  .<_  ( R  .\/  P
) )  /\  ( -.  C  .<_  ( S 
.\/  T )  /\  -.  C  .<_  ( T 
.\/  U )  /\  -.  C  .<_  ( U 
.\/  S ) )  /\  ( C  .<_  ( P  .\/  S )  /\  C  .<_  ( Q 
.\/  T )  /\  C  .<_  ( R  .\/  U ) ) ) ) )
dalemc.l  |-  .<_  =  ( le `  K )
dalemc.j  |-  .\/  =  ( join `  K )
dalemc.a  |-  A  =  ( Atoms `  K )
dalem-cly.o  |-  O  =  ( LPlanes `  K )
dalem-cly.y  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
dalem-cly.z  |-  Z  =  ( ( S  .\/  T )  .\/  U )
Assertion
Ref Expression
dalem-cly  |-  ( (
ph  /\  Y  =/=  Z )  ->  -.  C  .<_  Y )

Proof of Theorem dalem-cly
StepHypRef Expression
1 dalema.ph . . . . . . 7  |-  ( ph  <->  ( ( ( K  e.  HL  /\  C  e.  ( Base `  K
) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A
) )  /\  ( Y  e.  O  /\  Z  e.  O )  /\  ( ( -.  C  .<_  ( P  .\/  Q
)  /\  -.  C  .<_  ( Q  .\/  R
)  /\  -.  C  .<_  ( R  .\/  P
) )  /\  ( -.  C  .<_  ( S 
.\/  T )  /\  -.  C  .<_  ( T 
.\/  U )  /\  -.  C  .<_  ( U 
.\/  S ) )  /\  ( C  .<_  ( P  .\/  S )  /\  C  .<_  ( Q 
.\/  T )  /\  C  .<_  ( R  .\/  U ) ) ) ) )
21dalemkelat 33550 . . . . . 6  |-  ( ph  ->  K  e.  Lat )
3 dalemc.a . . . . . . 7  |-  A  =  ( Atoms `  K )
41, 3dalemceb 33564 . . . . . 6  |-  ( ph  ->  C  e.  ( Base `  K ) )
5 dalem-cly.o . . . . . . 7  |-  O  =  ( LPlanes `  K )
61, 5dalemyeb 33575 . . . . . 6  |-  ( ph  ->  Y  e.  ( Base `  K ) )
7 eqid 2450 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
8 dalemc.l . . . . . . 7  |-  .<_  =  ( le `  K )
9 dalemc.j . . . . . . 7  |-  .\/  =  ( join `  K )
107, 8, 9latleeqj1 15321 . . . . . 6  |-  ( ( K  e.  Lat  /\  C  e.  ( Base `  K )  /\  Y  e.  ( Base `  K
) )  ->  ( C  .<_  Y  <->  ( C  .\/  Y )  =  Y ) )
112, 4, 6, 10syl3anc 1219 . . . . 5  |-  ( ph  ->  ( C  .<_  Y  <->  ( C  .\/  Y )  =  Y ) )
121dalemclpjs 33560 . . . . . . . . . . . . 13  |-  ( ph  ->  C  .<_  ( P  .\/  S ) )
131dalemkehl 33549 . . . . . . . . . . . . . 14  |-  ( ph  ->  K  e.  HL )
14 dalem-cly.y . . . . . . . . . . . . . . 15  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
151, 8, 9, 3, 5, 14dalemcea 33586 . . . . . . . . . . . . . 14  |-  ( ph  ->  C  e.  A )
161dalemsea 33555 . . . . . . . . . . . . . 14  |-  ( ph  ->  S  e.  A )
171dalempea 33552 . . . . . . . . . . . . . 14  |-  ( ph  ->  P  e.  A )
181dalemqea 33553 . . . . . . . . . . . . . . 15  |-  ( ph  ->  Q  e.  A )
191dalem-clpjq 33563 . . . . . . . . . . . . . . 15  |-  ( ph  ->  -.  C  .<_  ( P 
.\/  Q ) )
208, 9, 3atnlej1 33305 . . . . . . . . . . . . . . 15  |-  ( ( K  e.  HL  /\  ( C  e.  A  /\  P  e.  A  /\  Q  e.  A
)  /\  -.  C  .<_  ( P  .\/  Q
) )  ->  C  =/=  P )
2113, 15, 17, 18, 19, 20syl131anc 1232 . . . . . . . . . . . . . 14  |-  ( ph  ->  C  =/=  P )
228, 9, 3hlatexch1 33321 . . . . . . . . . . . . . 14  |-  ( ( K  e.  HL  /\  ( C  e.  A  /\  S  e.  A  /\  P  e.  A
)  /\  C  =/=  P )  ->  ( C  .<_  ( P  .\/  S
)  ->  S  .<_  ( P  .\/  C ) ) )
2313, 15, 16, 17, 21, 22syl131anc 1232 . . . . . . . . . . . . 13  |-  ( ph  ->  ( C  .<_  ( P 
.\/  S )  ->  S  .<_  ( P  .\/  C ) ) )
2412, 23mpd 15 . . . . . . . . . . . 12  |-  ( ph  ->  S  .<_  ( P  .\/  C ) )
259, 3hlatjcom 33294 . . . . . . . . . . . . 13  |-  ( ( K  e.  HL  /\  C  e.  A  /\  P  e.  A )  ->  ( C  .\/  P
)  =  ( P 
.\/  C ) )
2613, 15, 17, 25syl3anc 1219 . . . . . . . . . . . 12  |-  ( ph  ->  ( C  .\/  P
)  =  ( P 
.\/  C ) )
2724, 26breqtrrd 4402 . . . . . . . . . . 11  |-  ( ph  ->  S  .<_  ( C  .\/  P ) )
281dalemclqjt 33561 . . . . . . . . . . . . 13  |-  ( ph  ->  C  .<_  ( Q  .\/  T ) )
291dalemtea 33556 . . . . . . . . . . . . . 14  |-  ( ph  ->  T  e.  A )
301dalemrea 33554 . . . . . . . . . . . . . . 15  |-  ( ph  ->  R  e.  A )
31 simp312 1136 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( K  e.  HL  /\  C  e.  ( Base `  K
) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A
) )  /\  ( Y  e.  O  /\  Z  e.  O )  /\  ( ( -.  C  .<_  ( P  .\/  Q
)  /\  -.  C  .<_  ( Q  .\/  R
)  /\  -.  C  .<_  ( R  .\/  P
) )  /\  ( -.  C  .<_  ( S 
.\/  T )  /\  -.  C  .<_  ( T 
.\/  U )  /\  -.  C  .<_  ( U 
.\/  S ) )  /\  ( C  .<_  ( P  .\/  S )  /\  C  .<_  ( Q 
.\/  T )  /\  C  .<_  ( R  .\/  U ) ) ) )  ->  -.  C  .<_  ( Q  .\/  R ) )
321, 31sylbi 195 . . . . . . . . . . . . . . 15  |-  ( ph  ->  -.  C  .<_  ( Q 
.\/  R ) )
338, 9, 3atnlej1 33305 . . . . . . . . . . . . . . 15  |-  ( ( K  e.  HL  /\  ( C  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  -.  C  .<_  ( Q  .\/  R
) )  ->  C  =/=  Q )
3413, 15, 18, 30, 32, 33syl131anc 1232 . . . . . . . . . . . . . 14  |-  ( ph  ->  C  =/=  Q )
358, 9, 3hlatexch1 33321 . . . . . . . . . . . . . 14  |-  ( ( K  e.  HL  /\  ( C  e.  A  /\  T  e.  A  /\  Q  e.  A
)  /\  C  =/=  Q )  ->  ( C  .<_  ( Q  .\/  T
)  ->  T  .<_  ( Q  .\/  C ) ) )
3613, 15, 29, 18, 34, 35syl131anc 1232 . . . . . . . . . . . . 13  |-  ( ph  ->  ( C  .<_  ( Q 
.\/  T )  ->  T  .<_  ( Q  .\/  C ) ) )
3728, 36mpd 15 . . . . . . . . . . . 12  |-  ( ph  ->  T  .<_  ( Q  .\/  C ) )
389, 3hlatjcom 33294 . . . . . . . . . . . . 13  |-  ( ( K  e.  HL  /\  C  e.  A  /\  Q  e.  A )  ->  ( C  .\/  Q
)  =  ( Q 
.\/  C ) )
3913, 15, 18, 38syl3anc 1219 . . . . . . . . . . . 12  |-  ( ph  ->  ( C  .\/  Q
)  =  ( Q 
.\/  C ) )
4037, 39breqtrrd 4402 . . . . . . . . . . 11  |-  ( ph  ->  T  .<_  ( C  .\/  Q ) )
411, 3dalemseb 33568 . . . . . . . . . . . 12  |-  ( ph  ->  S  e.  ( Base `  K ) )
427, 9, 3hlatjcl 33293 . . . . . . . . . . . . 13  |-  ( ( K  e.  HL  /\  C  e.  A  /\  P  e.  A )  ->  ( C  .\/  P
)  e.  ( Base `  K ) )
4313, 15, 17, 42syl3anc 1219 . . . . . . . . . . . 12  |-  ( ph  ->  ( C  .\/  P
)  e.  ( Base `  K ) )
441, 3dalemteb 33569 . . . . . . . . . . . 12  |-  ( ph  ->  T  e.  ( Base `  K ) )
457, 9, 3hlatjcl 33293 . . . . . . . . . . . . 13  |-  ( ( K  e.  HL  /\  C  e.  A  /\  Q  e.  A )  ->  ( C  .\/  Q
)  e.  ( Base `  K ) )
4613, 15, 18, 45syl3anc 1219 . . . . . . . . . . . 12  |-  ( ph  ->  ( C  .\/  Q
)  e.  ( Base `  K ) )
477, 8, 9latjlej12 15325 . . . . . . . . . . . 12  |-  ( ( K  e.  Lat  /\  ( S  e.  ( Base `  K )  /\  ( C  .\/  P )  e.  ( Base `  K
) )  /\  ( T  e.  ( Base `  K )  /\  ( C  .\/  Q )  e.  ( Base `  K
) ) )  -> 
( ( S  .<_  ( C  .\/  P )  /\  T  .<_  ( C 
.\/  Q ) )  ->  ( S  .\/  T )  .<_  ( ( C  .\/  P )  .\/  ( C  .\/  Q ) ) ) )
482, 41, 43, 44, 46, 47syl122anc 1228 . . . . . . . . . . 11  |-  ( ph  ->  ( ( S  .<_  ( C  .\/  P )  /\  T  .<_  ( C 
.\/  Q ) )  ->  ( S  .\/  T )  .<_  ( ( C  .\/  P )  .\/  ( C  .\/  Q ) ) ) )
4927, 40, 48mp2and 679 . . . . . . . . . 10  |-  ( ph  ->  ( S  .\/  T
)  .<_  ( ( C 
.\/  P )  .\/  ( C  .\/  Q ) ) )
501, 3dalempeb 33565 . . . . . . . . . . 11  |-  ( ph  ->  P  e.  ( Base `  K ) )
511, 3dalemqeb 33566 . . . . . . . . . . 11  |-  ( ph  ->  Q  e.  ( Base `  K ) )
527, 9latjjdi 15361 . . . . . . . . . . 11  |-  ( ( K  e.  Lat  /\  ( C  e.  ( Base `  K )  /\  P  e.  ( Base `  K )  /\  Q  e.  ( Base `  K
) ) )  -> 
( C  .\/  ( P  .\/  Q ) )  =  ( ( C 
.\/  P )  .\/  ( C  .\/  Q ) ) )
532, 4, 50, 51, 52syl13anc 1221 . . . . . . . . . 10  |-  ( ph  ->  ( C  .\/  ( P  .\/  Q ) )  =  ( ( C 
.\/  P )  .\/  ( C  .\/  Q ) ) )
5449, 53breqtrrd 4402 . . . . . . . . 9  |-  ( ph  ->  ( S  .\/  T
)  .<_  ( C  .\/  ( P  .\/  Q ) ) )
551dalemclrju 33562 . . . . . . . . . . 11  |-  ( ph  ->  C  .<_  ( R  .\/  U ) )
561dalemuea 33557 . . . . . . . . . . . 12  |-  ( ph  ->  U  e.  A )
57 simp313 1137 . . . . . . . . . . . . . 14  |-  ( ( ( ( K  e.  HL  /\  C  e.  ( Base `  K
) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A
) )  /\  ( Y  e.  O  /\  Z  e.  O )  /\  ( ( -.  C  .<_  ( P  .\/  Q
)  /\  -.  C  .<_  ( Q  .\/  R
)  /\  -.  C  .<_  ( R  .\/  P
) )  /\  ( -.  C  .<_  ( S 
.\/  T )  /\  -.  C  .<_  ( T 
.\/  U )  /\  -.  C  .<_  ( U 
.\/  S ) )  /\  ( C  .<_  ( P  .\/  S )  /\  C  .<_  ( Q 
.\/  T )  /\  C  .<_  ( R  .\/  U ) ) ) )  ->  -.  C  .<_  ( R  .\/  P ) )
581, 57sylbi 195 . . . . . . . . . . . . 13  |-  ( ph  ->  -.  C  .<_  ( R 
.\/  P ) )
598, 9, 3atnlej1 33305 . . . . . . . . . . . . 13  |-  ( ( K  e.  HL  /\  ( C  e.  A  /\  R  e.  A  /\  P  e.  A
)  /\  -.  C  .<_  ( R  .\/  P
) )  ->  C  =/=  R )
6013, 15, 30, 17, 58, 59syl131anc 1232 . . . . . . . . . . . 12  |-  ( ph  ->  C  =/=  R )
618, 9, 3hlatexch1 33321 . . . . . . . . . . . 12  |-  ( ( K  e.  HL  /\  ( C  e.  A  /\  U  e.  A  /\  R  e.  A
)  /\  C  =/=  R )  ->  ( C  .<_  ( R  .\/  U
)  ->  U  .<_  ( R  .\/  C ) ) )
6213, 15, 56, 30, 60, 61syl131anc 1232 . . . . . . . . . . 11  |-  ( ph  ->  ( C  .<_  ( R 
.\/  U )  ->  U  .<_  ( R  .\/  C ) ) )
6355, 62mpd 15 . . . . . . . . . 10  |-  ( ph  ->  U  .<_  ( R  .\/  C ) )
649, 3hlatjcom 33294 . . . . . . . . . . 11  |-  ( ( K  e.  HL  /\  C  e.  A  /\  R  e.  A )  ->  ( C  .\/  R
)  =  ( R 
.\/  C ) )
6513, 15, 30, 64syl3anc 1219 . . . . . . . . . 10  |-  ( ph  ->  ( C  .\/  R
)  =  ( R 
.\/  C ) )
6663, 65breqtrrd 4402 . . . . . . . . 9  |-  ( ph  ->  U  .<_  ( C  .\/  R ) )
671, 9, 3dalemsjteb 33572 . . . . . . . . . 10  |-  ( ph  ->  ( S  .\/  T
)  e.  ( Base `  K ) )
681, 9, 3dalempjqeb 33571 . . . . . . . . . . 11  |-  ( ph  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
697, 9latjcl 15309 . . . . . . . . . . 11  |-  ( ( K  e.  Lat  /\  C  e.  ( Base `  K )  /\  ( P  .\/  Q )  e.  ( Base `  K
) )  ->  ( C  .\/  ( P  .\/  Q ) )  e.  (
Base `  K )
)
702, 4, 68, 69syl3anc 1219 . . . . . . . . . 10  |-  ( ph  ->  ( C  .\/  ( P  .\/  Q ) )  e.  ( Base `  K
) )
711, 3dalemueb 33570 . . . . . . . . . 10  |-  ( ph  ->  U  e.  ( Base `  K ) )
727, 9, 3hlatjcl 33293 . . . . . . . . . . 11  |-  ( ( K  e.  HL  /\  C  e.  A  /\  R  e.  A )  ->  ( C  .\/  R
)  e.  ( Base `  K ) )
7313, 15, 30, 72syl3anc 1219 . . . . . . . . . 10  |-  ( ph  ->  ( C  .\/  R
)  e.  ( Base `  K ) )
747, 8, 9latjlej12 15325 . . . . . . . . . 10  |-  ( ( K  e.  Lat  /\  ( ( S  .\/  T )  e.  ( Base `  K )  /\  ( C  .\/  ( P  .\/  Q ) )  e.  (
Base `  K )
)  /\  ( U  e.  ( Base `  K
)  /\  ( C  .\/  R )  e.  (
Base `  K )
) )  ->  (
( ( S  .\/  T )  .<_  ( C  .\/  ( P  .\/  Q
) )  /\  U  .<_  ( C  .\/  R
) )  ->  (
( S  .\/  T
)  .\/  U )  .<_  ( ( C  .\/  ( P  .\/  Q ) )  .\/  ( C 
.\/  R ) ) ) )
752, 67, 70, 71, 73, 74syl122anc 1228 . . . . . . . . 9  |-  ( ph  ->  ( ( ( S 
.\/  T )  .<_  ( C  .\/  ( P 
.\/  Q ) )  /\  U  .<_  ( C 
.\/  R ) )  ->  ( ( S 
.\/  T )  .\/  U )  .<_  ( ( C  .\/  ( P  .\/  Q ) )  .\/  ( C  .\/  R ) ) ) )
7654, 66, 75mp2and 679 . . . . . . . 8  |-  ( ph  ->  ( ( S  .\/  T )  .\/  U ) 
.<_  ( ( C  .\/  ( P  .\/  Q ) )  .\/  ( C 
.\/  R ) ) )
771, 3dalemreb 33567 . . . . . . . . 9  |-  ( ph  ->  R  e.  ( Base `  K ) )
787, 9latjjdi 15361 . . . . . . . . 9  |-  ( ( K  e.  Lat  /\  ( C  e.  ( Base `  K )  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  R  e.  ( Base `  K )
) )  ->  ( C  .\/  ( ( P 
.\/  Q )  .\/  R ) )  =  ( ( C  .\/  ( P  .\/  Q ) ) 
.\/  ( C  .\/  R ) ) )
792, 4, 68, 77, 78syl13anc 1221 . . . . . . . 8  |-  ( ph  ->  ( C  .\/  (
( P  .\/  Q
)  .\/  R )
)  =  ( ( C  .\/  ( P 
.\/  Q ) ) 
.\/  ( C  .\/  R ) ) )
8076, 79breqtrrd 4402 . . . . . . 7  |-  ( ph  ->  ( ( S  .\/  T )  .\/  U ) 
.<_  ( C  .\/  (
( P  .\/  Q
)  .\/  R )
) )
81 dalem-cly.z . . . . . . 7  |-  Z  =  ( ( S  .\/  T )  .\/  U )
8214oveq2i 6187 . . . . . . 7  |-  ( C 
.\/  Y )  =  ( C  .\/  (
( P  .\/  Q
)  .\/  R )
)
8380, 81, 823brtr4g 4408 . . . . . 6  |-  ( ph  ->  Z  .<_  ( C  .\/  Y ) )
84 breq2 4380 . . . . . 6  |-  ( ( C  .\/  Y )  =  Y  ->  ( Z  .<_  ( C  .\/  Y )  <->  Z  .<_  Y ) )
8583, 84syl5ibcom 220 . . . . 5  |-  ( ph  ->  ( ( C  .\/  Y )  =  Y  ->  Z  .<_  Y ) )
8611, 85sylbid 215 . . . 4  |-  ( ph  ->  ( C  .<_  Y  ->  Z  .<_  Y ) )
871dalemzeo 33559 . . . . . 6  |-  ( ph  ->  Z  e.  O )
881dalemyeo 33558 . . . . . 6  |-  ( ph  ->  Y  e.  O )
898, 5lplncmp 33488 . . . . . 6  |-  ( ( K  e.  HL  /\  Z  e.  O  /\  Y  e.  O )  ->  ( Z  .<_  Y  <->  Z  =  Y ) )
9013, 87, 88, 89syl3anc 1219 . . . . 5  |-  ( ph  ->  ( Z  .<_  Y  <->  Z  =  Y ) )
91 eqcom 2458 . . . . 5  |-  ( Z  =  Y  <->  Y  =  Z )
9290, 91syl6bb 261 . . . 4  |-  ( ph  ->  ( Z  .<_  Y  <->  Y  =  Z ) )
9386, 92sylibd 214 . . 3  |-  ( ph  ->  ( C  .<_  Y  ->  Y  =  Z )
)
9493necon3ad 2655 . 2  |-  ( ph  ->  ( Y  =/=  Z  ->  -.  C  .<_  Y ) )
9594imp 429 1  |-  ( (
ph  /\  Y  =/=  Z )  ->  -.  C  .<_  Y )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1757    =/= wne 2641   class class class wbr 4376   ` cfv 5502  (class class class)co 6176   Basecbs 14262   lecple 14333   joincjn 15202   Latclat 15303   Atomscatm 33190   HLchlt 33277   LPlanesclpl 33418
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 1709  ax-7 1729  ax-8 1759  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429  ax-rep 4487  ax-sep 4497  ax-nul 4505  ax-pow 4554  ax-pr 4615  ax-un 6458
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 1702  df-eu 2263  df-mo 2264  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-ne 2643  df-ral 2797  df-rex 2798  df-reu 2799  df-rab 2801  df-v 3056  df-sbc 3271  df-csb 3373  df-dif 3415  df-un 3417  df-in 3419  df-ss 3426  df-nul 3722  df-if 3876  df-pw 3946  df-sn 3962  df-pr 3964  df-op 3968  df-uni 4176  df-iun 4257  df-br 4377  df-opab 4435  df-mpt 4436  df-id 4720  df-xp 4930  df-rel 4931  df-cnv 4932  df-co 4933  df-dm 4934  df-rn 4935  df-res 4936  df-ima 4937  df-iota 5465  df-fun 5504  df-fn 5505  df-f 5506  df-f1 5507  df-fo 5508  df-f1o 5509  df-fv 5510  df-riota 6137  df-ov 6179  df-oprab 6180  df-poset 15204  df-plt 15216  df-lub 15232  df-glb 15233  df-join 15234  df-meet 15235  df-p0 15297  df-lat 15304  df-clat 15366  df-oposet 33103  df-ol 33105  df-oml 33106  df-covers 33193  df-ats 33194  df-atl 33225  df-cvlat 33249  df-hlat 33278  df-llines 33424  df-lplanes 33425
This theorem is referenced by:  dalem9  33598
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