defpred S₁[ Nat] means for F being FinSequence of COMPLEX
for Fr being FinSequence of REAL st len F = $1 & Fr = Re F holds
Sum Fr = Re (Sum F);
A1: S₁[ 0 ]

proof

let F be FinSequence of COMPLEX ; :: thesis: for Fr being FinSequence of REAL st len F = 0 & Fr = Re F holds
Sum Fr = Re (Sum F)
let Fr be FinSequence of REAL ; :: thesis: ( len F = 0 & Fr = Re F implies Sum Fr = Re (Sum F) )
assume A2: ( len F = 0 & Fr = Re F ) ; :: thesis: Sum Fr = Re (Sum F)
then dom Fr = dom F by COMSEQ_3:def 3
.= Seg (len F) by FINSEQ_1:def 3 ;
then A3: len Fr = 0 by A2, FINSEQ_1:def 3;
thus Re (Sum F) = Re 0 by A2, Lm2, FINSEQ_1:20
.= Sum Fr by A3, COMPLEX1:4, FINSEQ_1:20, RVSUM_1:72 ; :: thesis: verum

end;

A4: now :: thesis: for k being Nat st S₁[k] holds
S₁[k + 1]

let k be Nat; :: thesis: ( S₁[k] implies S₁[k + 1] )
assume A5: S₁[k] ; :: thesis: S₁[k + 1]

now :: thesis: for F being FinSequence of COMPLEX
for Fr being FinSequence of REAL st len F = k + 1 & Fr = Re F holds
Re (Sum F) = Sum Fr

let F be FinSequence of COMPLEX ; :: thesis: for Fr being FinSequence of REAL st len F = k + 1 & Fr = Re F holds
Re (Sum F) = Sum Fr
let Fr be FinSequence of REAL ; :: thesis: ( len F = k + 1 & Fr = Re F implies Re (Sum F) = Sum Fr )
assume A6: ( len F = k + 1 & Fr = Re F ) ; :: thesis: Re (Sum F) = Sum Fr
reconsider F1 = F | k as FinSequence of COMPLEX ;
A7: len F1 = k by A6, FINSEQ_1:59, NAT_1:11;
reconsider F1r = Re F1 as FinSequence of REAL ;
reconsider x = F . (k + 1) as Element of COMPLEX by XCMPLX_0:def 2;
A8: F = F1 ^ <*x*> by A6, FINSEQ_3:55;
hence Re (Sum F) = Re ((Sum F1) + x) by Lm3
.= (Re (Sum F1)) + (Re x) by COMPLEX1:8
.= (Sum F1r) + (Re x) by A5, A7
.= Sum (F1r ^ <*(Re x)*>) by RVSUM_1:74
.= Sum Fr by A6, A8, Th5 ;
:: thesis: verum

end;

hence S₁[k + 1] ; :: thesis: verum

end;

A9: for k being Nat holds S₁[k] from NAT_1:sch 2(A1, A4);
let F be FinSequence of COMPLEX ; :: thesis: for Fr being FinSequence of REAL st Fr = Re F holds
Sum Fr = Re (Sum F)
let Fr be FinSequence of REAL ; :: thesis: ( Fr = Re F implies Sum Fr = Re (Sum F) )
assume A10: Fr = Re F ; :: thesis: Sum Fr = Re (Sum F)
len F is Element of NAT ;
hence Sum Fr = Re (Sum F) by A9, A10; :: thesis: verum