let G be Group; :: thesis: for f being FinSequence of G holds Product (((Rev f) " ) ^ f) = 1_ G
let f be FinSequence of G; :: thesis: Product (((Rev f) " ) ^ f) = 1_ G
A1: len f = len (Rev f) by FINSEQ_5:def 3;
A2: len (Rev ((Rev f) " )) = len ((Rev ((Rev f) " )) " ) by Def4;
then A3: dom (Rev ((Rev f) " )) = dom ((Rev ((Rev f) " )) " ) by FINSEQ_3:31;
A4: len ((Rev f) " ) = len (Rev ((Rev f) " )) by FINSEQ_5:def 3;
A5: len (Rev f) = len ((Rev f) " ) by Def4;
then A6: dom (Rev f) = dom ((Rev f) " ) by FINSEQ_3:31;
for i being Nat st 1 <= i & i <= len f holds
f . i = ((Rev ((Rev f) " )) " ) . i

proof

let i be Nat; :: thesis: ( 1 <= i & i <= len f implies f . i = ((Rev ((Rev f) " )) " ) . i )
assume that
A7: 1 <= i and
A8: i <= len f ; :: thesis: f . i = ((Rev ((Rev f) " )) " ) . i
((len f) - i) + 1 = ((len f) -' i) + 1 by A8, XREAL_1:235;
then reconsider j = ((len f) - i) + 1 as Element of NAT ;
A9: i + j = (len f) + 1 ;
i in NAT by ORDINAL1:def 13;
then A10: i in Seg (len f) by A7, A8;
then A11: i in dom ((Rev ((Rev f) " )) " ) by A1, A5, A4, A2, FINSEQ_1:def 3;
i - 1 >= 0 by A7, XREAL_1:50;
then (len f) + 0 <= (len f) + (i - 1) by XREAL_1:9;
then A12: (len f) - (i - 1) <= ((len f) + (i - 1)) - (i - 1) by XREAL_1:15;
(len f) - i = (len f) -' i by A8, XREAL_1:235;
then ((len f) - i) + 1 >= 0 + 1 by XREAL_1:8;
then j in Seg (len f) by A12;
then A13: j in dom ((Rev f) " ) by A1, A5, FINSEQ_1:def 3;
A14: j + i = (len f) + 1 ;
A15: i in dom f by A10, FINSEQ_1:def 3;
then f . i = f /. i by PARTFUN1:def 8
.= (Rev f) /. j by A15, A9, FINSEQ_5:69
.= (((Rev f) /. j) " ) "
.= (((Rev f) " ) /. j) " by A6, A13, Def4
.= ((Rev ((Rev f) " )) /. i) " by A1, A5, A13, A14, FINSEQ_5:69
.= ((Rev ((Rev f) " )) " ) /. i by A3, A11, Def4
.= ((Rev ((Rev f) " )) " ) . i by A11, PARTFUN1:def 8 ;
hence f . i = ((Rev ((Rev f) " )) " ) . i ; :: thesis: verum

end;

then (Rev ((Rev f) " )) " = f by A1, A5, A4, A2, FINSEQ_1:18;
hence Product (((Rev f) " ) ^ f) = 1_ G by Th24; :: thesis: verum