
theorem Th6:
  for I be non empty set,
      G be Group,
      H be Subgroup of G,
      x be finite-support Function of I,G,
      y be finite-support Function of I,H
  st x = y
  holds Product x = Product y
  proof
    let I be non empty set,
        G be Group,
        H be Subgroup of G,
        x be finite-support Function of I,G,
        y be finite-support Function of I,H;
    assume
    A1: x = y; then
    A2: support x = support y by Th3;
    reconsider fx = (x|support(x)) * canFS(support x)
        as FinSequence of G by FINSEQ_2:32;
    reconsider fy = (y|support(y)) * canFS(support y)
        as FinSequence of H by FINSEQ_2:32;
    thus Product x = Product fx by GROUP_17:def 1
                  .= Product fy by A1,A2,GROUP_19:45
                  .= Product y by GROUP_17:def 1;
  end;
