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;
reserve z,z1,z2 for Element of COMPLEX;

theorem
  Product (i|->(z1*z2)) = (Product (i|->z1))*(Product (i|->z2))
proof
  reconsider i as Nat by TARSKI:1;
  reconsider zz = i|->(z1*z2) as FinSequence of COMPLEX by Lm5;
  Product (i|->(z1*z2)) = Product zz
    .= multcomplex$$(i|->multcomplex.(z1,z2)) by BINOP_2:def 5
    .= multcomplex.(multcomplex$$(i|->z1),multcomplex$$(i|->z2)) by SETWOP_2:36
    .= (Product (i|->z1))*(Product (i|->z2)) by BINOP_2:def 5;
  hence thesis;
end;
