
theorem Th37:
  for F,G be non-empty non empty Function,
        h be non empty Function
  st dom F = dom G = dom h
   & for i be object st i in dom h holds
     ex hi be Function of F.i,G.i st hi = h.i & hi is bijective
  holds ProductMap(F,G,h) is bijective
  proof
    let F,G be non-empty non empty Function,
          h be non empty Function;
    assume that
    A1: dom F = dom G = dom h and
    A2: for i be object st i in dom h holds
        ex hi be Function of F.i,G.i st hi = h.i & hi is bijective;
    set p = ProductMap(F,G,h);
    now
      let i be object;
      assume i in dom h; then
      ex hi be Function of F.i,G.i
      st hi = h.i & hi is bijective by A2;
      hence ex hi be Function of F.i,G.i st hi = h.i & hi is onto;
    end; then
    A3: p is onto by A1,Th35;
    now
      let i be object;
      assume i in dom h; then
      ex hi be Function of F.i,G.i
      st hi = h.i & hi is bijective by A2;
      hence ex hi be Function of F.i,G.i
            st hi = h.i &  hi is one-to-one;
    end; then
    p is one-to-one by A1,Th36;
    hence thesis by A3;
  end;
