
theorem Th8:
for F be FinSequence of COMPLEX, Fi be FinSequence of REAL st Fi=Im F
  holds Sum(Fi) = Im Sum(F)
proof
defpred P[Nat] means
  for F be FinSequence of COMPLEX,
      Fi be FinSequence of REAL
      st len F = $1 & Fi=Im F
    holds Sum(Fi) = Im Sum(F);
A1: P[0]
  proof
    let F be FinSequence of COMPLEX,
        Fi be FinSequence of REAL;
    assume A2: len F = 0 & Fi=Im F;
    then dom Fi = dom F by COMSEQ_3:def 4
          .= Seg len F by FINSEQ_1:def 3; then
  A3:len Fi = 0 by A2,FINSEQ_1:def 3;
    thus Im (Sum (F)) = Im (0) by A2,Lm2,FINSEQ_1:20
                     .= Sum Fi by A3,COMPLEX1:4,FINSEQ_1:20,RVSUM_1:72;
  end;
A4:now let k be Nat;
     assume A5:P[k];
     now let F be FinSequence of COMPLEX,
             Fi be FinSequence of REAL;
       assume A6:len F = k+1 & Fi=Im F;
       reconsider F1= F|k as FinSequence of COMPLEX;
       A7: len F1 = k by A6,FINSEQ_1:59,NAT_1:11;
       reconsider F1i= Im 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 Im (Sum(F)) = Im (Sum(F1)+ x ) by Lm3
                       .= Im (Sum(F1)) + Im x by COMPLEX1:8
                       .= Sum (F1i) + Im x by A5,A7
                       .= Sum(F1i^<* Im x *> ) by RVSUM_1:74
                       .= Sum(Fi) by A6,A8,Th6;
     end;
     hence P[k+1];
   end;
A9:for k be Nat holds P[k] from NAT_1:sch 2(A1,A4);
let F be FinSequence of COMPLEX,
    Fi be FinSequence of REAL;
assume A10: Fi=Im F;
len F is Element of NAT;
hence Sum(Fi) = Im Sum(F) by A9,A10;
end;
