let G be Group; :: thesis: for F being FinSequence of the carrier of G holds F |^ ((len F) |-> (@ 0 )) = (len F) |-> (1_ G)
let F be FinSequence of the carrier of G; :: thesis: F |^ ((len F) |-> (@ 0 )) = (len F) |-> (1_ G)
defpred S₁[ FinSequence of the carrier of G] means $1 |^ ((len $1) |-> (@ 0 )) = (len $1) |-> (1_ G);
A1: for F being FinSequence of the carrier of G
for a being Element of G st S₁[F] holds
S₁[F ^ <*a*>]

proof

let F be FinSequence of the carrier of G; :: thesis: for a being Element of G st S₁[F] holds
S₁[F ^ <*a*>]
let a be Element of G; :: thesis: ( S₁[F] implies S₁[F ^ <*a*>] )
set A = F ^ <*a*>;
assume A2: S₁[F] ; :: thesis: S₁[F ^ <*a*>]
A3: len <*a*> = 1 by FINSEQ_1:57;
A4: ( len <*(@ 0 )*> = 1 & len ((len F) |-> (@ 0 )) = len F ) by FINSEQ_1:57, FINSEQ_1:def 18;
A5: len (F ^ <*a*>) = (len F) + (len <*a*>) by FINSEQ_1:35;
hence (F ^ <*a*>) |^ ((len (F ^ <*a*>)) |-> (@ 0 )) = (F ^ <*a*>) |^ (((len F) |-> (@ 0 )) ^ <*(@ 0 )*>) by A3, FINSEQ_2:74
.= (F |^ ((len F) |-> (@ 0 ))) ^ (<*a*> |^ <*(@ 0 )*>) by A3, A4, Th25
.= ((len F) |-> (1_ G)) ^ <*(a |^ 0 )*> by A2, Th28
.= ((len F) |-> (1_ G)) ^ <*(1_ G)*> by GROUP_1:43
.= (len (F ^ <*a*>)) |-> (1_ G) by A5, A3, FINSEQ_2:74 ;
:: thesis: verum

end;

A6: S₁[ <*> the carrier of G]

proof

set A = <*> the carrier of G;
thus (<*> the carrier of G) |^ ((len (<*> the carrier of G)) |-> (@ 0 )) = (<*> the carrier of G) |^ (0 |-> (@ 0 ))
.= (<*> the carrier of G) |^ (<*> INT )
.= {} by Th27
.= (len (<*> the carrier of G)) |-> (1_ G) ; :: thesis: verum

end;

for F being FinSequence of the carrier of G holds S₁[F] from FINSEQ_2:sch 2(A6, A1);
hence F |^ ((len F) |-> (@ 0 )) = (len F) |-> (1_ G) ; :: thesis: verum