theorem Th74:
  for S being functional with_common_domain set holds S c= product product" S
proof
  let S be functional with_common_domain set;
  let f be object;
  assume
A1: f in S;
  then reconsider f as Element of S;
A2: dom f = DOM S by A1,Lm2
    .= dom product" S by Def12;
  for i being object st i in dom product" S holds f.i in (product" S).i
  proof
    let i be object;
    assume i in dom product" S;
    then (product" S).i = pi(S,i) by Def12;
    hence thesis by A1,Def6;
  end;
  hence thesis by A2,Th9;
end;
