theorem Th73:
  for S being non empty functional set,
  i being set st i in dom product" S holds
  (product" S).i = the set of all f.i where f is Element of S
proof
  let S be non empty functional set, i be set;
  assume
A1: i in dom product" S;
  hereby
    let x be object;
    assume x in (product" S).i;
    then x in pi(S,i) by A1,Def12;
    then ex f being Function st f in S & x = f.i by Def6;
    hence x in the set of all f.i where f is Element of S;
  end;
  let x be object;
  assume x in the set of all f.i where f is Element of S;
  then ex f being Element of S st x = f.i;
  then x in pi(S,i) by Def6;
  hence thesis by A1,Def12;
end;
