let F be FinSequence of COMPLEX ; :: thesis: for x being Element of COMPLEX holds Re (F ^ <*x*>) = (Re F) ^ <*(Re x)*>
let x be Element of COMPLEX ; :: thesis: Re (F ^ <*x*>) = (Re F) ^ <*(Re x)*>
set F1 = Re (F ^ <*x*>);
set F2 = (Re F) ^ <*(Re x)*>;
A1: dom F = dom (Re F) by COMSEQ_3:def 3;
A2: Seg (len F) = dom F by FINSEQ_1:def 3;
A3: len <*(Re x)*> = 1 by FINSEQ_1:39;
A4: dom (Re (F ^ <*x*>)) = dom (F ^ <*x*>) by COMSEQ_3:def 3
.= Seg ((len F) + (len <*x*>)) by FINSEQ_1:def 7
.= Seg ((len (Re F)) + (len <*x*>)) by A1, A2, FINSEQ_1:def 3
.= Seg ((len (Re F)) + (len <*(Re x)*>)) by A3, FINSEQ_1:39
.= dom ((Re F) ^ <*(Re x)*>) by FINSEQ_1:def 7 ;

now

let k be Nat; :: thesis: ( k in dom (Re (F ^ <*x*>)) implies (Re (F ^ <*x*>)) . k = ((Re F) ^ <*(Re x)*>) . k )
assume A5: k in dom (Re (F ^ <*x*>)) ; :: thesis: (Re (F ^ <*x*>)) . k = ((Re F) ^ <*(Re x)*>) . k
then k in dom (F ^ <*x*>) by COMSEQ_3:def 3;
then A6: k in Seg (len (F ^ <*x*>)) by FINSEQ_1:def 3;
then k in Seg ((len F) + (len <*x*>)) by FINSEQ_1:22;
then A7: ( 1 <= k & k <= (len F) + (len <*x*>) ) by FINSEQ_1:1;

now

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

suppose A8: k = (len F) + 1 ; :: thesis: (Re (F ^ <*x*>)) . k = ((Re F) ^ <*(Re x)*>) . k

thus (Re (F ^ <*x*>)) . k = Re ((F ^ <*x*>) . k) by A5, COMSEQ_3:def 3
.= Re x by A8, FINSEQ_1:42
.= ((Re F) ^ <*(Re x)*>) . ((len (Re F)) + 1) by FINSEQ_1:42
.= ((Re F) ^ <*(Re x)*>) . k by A8, A1, A2, FINSEQ_1:def 3 ; :: thesis: verum

end;

suppose A9: k <> (len F) + 1 ; :: thesis: (Re (F ^ <*x*>)) . k = ((Re F) ^ <*(Re x)*>) . k

k <= (len F) + 1 by A7, FINSEQ_1:39;
then k < (len F) + 1 by A9, XXREAL_0:1;
then ( 1 <= k & k <= len F ) by A6, FINSEQ_1:1, NAT_1:13;
then k in Seg (len F) by FINSEQ_1:1;
then A10: k in dom F by FINSEQ_1:def 3;
then A11: k in dom (Re F) by COMSEQ_3:def 3;
thus (Re (F ^ <*x*>)) . k = Re ((F ^ <*x*>) . k) by A5, COMSEQ_3:def 3
.= Re (F . k) by A10, FINSEQ_1:def 7
.= (Re F) . k by A11, COMSEQ_3:def 3
.= ((Re F) ^ <*(Re x)*>) . k by A11, FINSEQ_1:def 7 ; :: thesis: verum

end;

end;

end;

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

end;

hence Re (F ^ <*x*>) = (Re F) ^ <*(Re x)*> by A4, FINSEQ_1:13; :: thesis: verum