
theorem Th36:
  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 one-to-one
  holds ProductMap(F,G,h) is one-to-one
  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 one-to-one;
    set p = ProductMap(F,G,h);
    A4: for i be object st i in dom h holds h.i is Function of F.i,G.i
    proof
      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 one-to-one by A2;
      hence thesis;
    end;
    for x1,x2 being object st x1 in product F & x2 in product F & p.x1 = p.x2
    holds x1 = x2
    proof
      let x1,x2 be object;
      assume
      A5: x1 in product F & x2 in product F & p.x1 = p.x2;
      thus x1 = x2
      proof
        reconsider x1, x2 as Element of product F by A5;
        A7: dom x2 = dom F by CARD_3:9;
        for i be object st i in dom x1 holds x1.i = x2.i
        proof
          let i be object;
          assume i in dom x1; then
          reconsider i as Element of dom h by A1,CARD_3:9;
          consider hi1 be Function of F.i,G.i such that
          A9: hi1 = h.i & (p.x1).i = hi1.(x1.i) by A1,A4,Def5;
          consider hi2 be Function of F.i,G.i such that
          A10: hi2 = h.i & (p.x2).i = hi2.(x2.i) by A1,A4,Def5;
          A14: ex hi be Function of F.i,G.i
               st hi = h.i & hi is one-to-one by A2;
          A15: x1.i in F.i by A1,CARD_3:9;
          x2.i in F.i by A1,CARD_3:9;
          hence thesis by A1,A5,A9,A10,A14,A15,FUNCT_2:19;
        end;
        hence thesis by A7,CARD_3:9,FUNCT_1:2;
      end;
    end;
    hence thesis by FUNCT_2:19;
  end;
