
theorem Th32:
  for I be non empty finite set,
  G be Group,
  H be Subgroup of G,
  x be (the carrier of G)-valued total I -defined Function,
  x0 be (the carrier of H)-valued total I -defined Function
  st x=x0
  holds Product x = Product x0
  proof
    let I be non empty finite set,
    G be Group,
    H be Subgroup of G,
    x be (the carrier of G)-valued total I -defined Function,
    x0 be (the carrier of H)-valued total I -defined Function;
    assume A1:x=x0;
    consider f being FinSequence of G such that
    A2: Product x = Product f & f = x*canFS(I) by Def1;
    consider g being FinSequence of the carrier of H such that
    A3: Product x0 = Product g & g = x0*canFS(I) by Def1;
    thus thesis by A2,A1,A3,Th31;
  end;
