reserve s for set,
  i,j for natural Number,
  k for Nat,
  x,x1,x2,x3 for Real,
  r,r1,r2,r3,r4 for Real,
  F,F1,F2,F3 for real-valued FinSequence,
  R,R1,R2 for Element of i-tuples_on REAL;

theorem Th97:
  for F1, F2 being complex-valued FinSequence holds
  Product (F1^F2) = Product F1 * Product F2
proof
  let F1, F2 be complex-valued FinSequence;
A1: rng (F1^F2) c= COMPLEX by VALUED_0:def 1;
  rng F1 c= COMPLEX & rng F2 c= COMPLEX by VALUED_0:def 1;
  then reconsider FF = F1^F2, f1 = F1, f2 = F2 as FinSequence of COMPLEX
  by A1,FINSEQ_1:def 4;
  thus Product (F1^F2) = multcomplex $$ FF by Def13
    .= multcomplex.(Product f1,Product f2) by FINSOP_1:5
    .= Product F1 * Product F2 by BINOP_2:def 5;
end;
