theorem
  for S being non empty product-like set holds S = product product" S
proof
  let S be non empty product-like set;
  thus S c= product product" S by Th74;
  let x be object;
  assume x in product product" S;
  then consider g being Function such that
A1: x = g and
A2: dom g = dom product" S and
A3: for z being object st z in dom product" S holds g.z in (product" S).z
  by Def5;
  consider p being Function such that
A4: S = product p by Def13;
  set s = the Element of S;
A5: dom g = DOM S by A2,Def12
    .= dom s by Lm2
    .= dom p by A4,Th9;
  for z being object st z in dom p holds g.z in p.z
  proof
    let z be object;
    assume
A6: z in dom p;
    then g.z in (product" S).z by A2,A3,A5;
    then g.z in pi(S,z) by A2,A5,A6,Def12;
    then ex f being Function st f in S & g.z = f.z by Def6;
    hence thesis by A4,A6,Th9;
  end;
  hence thesis by A1,A4,A5,Th9;
end;
