let F be FinSequence of COMPLEX ; :: thesis: for x being Element of COMPLEX holds Im (F ^ <*x*>) = (Im F) ^ <*(Im x)*>
let x be Element of COMPLEX ; :: thesis: Im (F ^ <*x*>) = (Im F) ^ <*(Im x)*>
set F1 = Im (F ^ <*x*>);
set F2 = (Im F) ^ <*(Im x)*>;
A0: dom F = dom (Im F) by COMSEQ_3:def 4;
A1: Seg (len F) = dom F by FINSEQ_1:def 3;
A2: len <*(Im x)*> = 1 by FINSEQ_1:56;
P1: dom (Im (F ^ <*x*>)) = dom (F ^ <*x*>) by COMSEQ_3:def 4
.= Seg ((len F) + (len <*x*>)) by FINSEQ_1:def 7
.= Seg ((len (Im F)) + (len <*x*>)) by A0, A1, FINSEQ_1:def 3
.= Seg ((len (Im F)) + (len <*(Im x)*>)) by A2, FINSEQ_1:56
.= dom ((Im F) ^ <*(Im x)*>) by FINSEQ_1:def 7 ;

now

let k be Nat; :: thesis: ( k in dom (Im (F ^ <*x*>)) implies (Im (F ^ <*x*>)) . k = ((Im F) ^ <*(Im x)*>) . k )
assume P20: k in dom (Im (F ^ <*x*>)) ; :: thesis: (Im (F ^ <*x*>)) . k = ((Im F) ^ <*(Im x)*>) . k
then k in dom (F ^ <*x*>) by COMSEQ_3:def 4;
then P21: k in Seg (len (F ^ <*x*>)) by FINSEQ_1:def 3;
then k in Seg ((len F) + (len <*x*>)) by FINSEQ_1:35;
then P22: ( 1 <= k & k <= (len F) + (len <*x*>) ) by FINSEQ_1:3;

now

per cases ( k = (len F) + 1 or k <> (len F) + 1 ) ;

suppose C1: k = (len F) + 1 ; :: thesis: (Im (F ^ <*x*>)) . k = ((Im F) ^ <*(Im x)*>) . k

thus (Im (F ^ <*x*>)) . k = Im ((F ^ <*x*>) . k) by P20, COMSEQ_3:def 4
.= Im x by FINSEQ_1:59, C1
.= ((Im F) ^ <*(Im x)*>) . ((len (Im F)) + 1) by FINSEQ_1:59
.= ((Im F) ^ <*(Im x)*>) . k by C1, A0, A1, FINSEQ_1:def 3 ; :: thesis: verum

end;

suppose C2: k <> (len F) + 1 ; :: thesis: (Im (F ^ <*x*>)) . k = ((Im F) ^ <*(Im x)*>) . k

k <= (len F) + 1 by P22, FINSEQ_1:56;
then k < (len F) + 1 by C2, XXREAL_0:1;
then ( 1 <= k & k <= len F ) by P21, FINSEQ_1:3, NAT_1:13;
then k in Seg (len F) by FINSEQ_1:3;
then C20: k in dom F by FINSEQ_1:def 3;
then C21: k in dom (Im F) by COMSEQ_3:def 4;
thus (Im (F ^ <*x*>)) . k = Im ((F ^ <*x*>) . k) by P20, COMSEQ_3:def 4
.= Im (F . k) by C20, FINSEQ_1:def 7
.= (Im F) . k by C21, COMSEQ_3:def 4
.= ((Im F) ^ <*(Im x)*>) . k by C21, FINSEQ_1:def 7 ; :: thesis: verum

end;

end;

end;

hence (Im (F ^ <*x*>)) . k = ((Im F) ^ <*(Im x)*>) . k ; :: thesis: verum

end;

hence Im (F ^ <*x*>) = (Im F) ^ <*(Im x)*> by FINSEQ_1:17, P1; :: thesis: verum