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)*>;
A1: dom F = dom (Im F) by COMSEQ_3:def 4;
A2: Seg (len F) = dom F by FINSEQ_1:def 3;
A3: len <*(Im x)*> = 1 by FINSEQ_1:39;
A4: dom (Im (F ^ <*x*>)) = dom (F ^ <*x*>) by COMSEQ_3:def 4
.= Seg ((len F) + ()) by FINSEQ_1:def 7
.= Seg ((len (Im F)) + ()) by
.= Seg ((len (Im F)) + (len <*(Im x)*>)) by
.= dom ((Im F) ^ <*(Im x)*>) by FINSEQ_1:def 7 ;
now :: thesis: for k being Nat st k in dom (Im (F ^ <*x*>)) holds
(Im (F ^ <*x*>)) . k = ((Im F) ^ <*(Im x)*>) . k
let k be Nat; :: thesis: ( k in dom (Im (F ^ <*x*>)) implies (Im (F ^ <*x*>)) . k = ((Im F) ^ <*(Im x)*>) . k )
assume A5: 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 A6: k in Seg (len (F ^ <*x*>)) by FINSEQ_1:def 3;
then k in Seg ((len F) + ()) by FINSEQ_1:22;
then A7: ( 1 <= k & k <= (len F) + () ) by FINSEQ_1:1;
now :: thesis: (Im (F ^ <*x*>)) . k = ((Im F) ^ <*(Im x)*>) . k
per cases ( k = (len F) + 1 or k <> (len F) + 1 ) ;
suppose A8: k = (len F) + 1 ; :: thesis: (Im (F ^ <*x*>)) . k = ((Im F) ^ <*(Im x)*>) . k
thus (Im (F ^ <*x*>)) . k = Im ((F ^ <*x*>) . k) by
.= Im x by
.= ((Im F) ^ <*(Im x)*>) . ((len (Im F)) + 1) by FINSEQ_1:42
.= ((Im F) ^ <*(Im x)*>) . k by ; :: thesis: verum
end;
suppose A9: k <> (len F) + 1 ; :: thesis: (Im (F ^ <*x*>)) . k = ((Im F) ^ <*(Im x)*>) . k
k <= (len F) + 1 by ;
then k < (len F) + 1 by ;
then ( 1 <= k & k <= len F ) by ;
then k in Seg (len F) ;
then A10: k in dom F by FINSEQ_1:def 3;
then A11: k in dom (Im F) by COMSEQ_3:def 4;
thus (Im (F ^ <*x*>)) . k = Im ((F ^ <*x*>) . k) by
.= Im (F . k) by
.= (Im F) . k by
.= ((Im F) ^ <*(Im x)*>) . k by ; :: 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 ; :: thesis: verum