
theorem Th29:
  for X, Y being set, f, i, j being object st i <> j
  holds f in product((i,j) --> (X,Y)) iff
    ex x,y being object st x in X & y in Y & f = (i,j) --> (x,y)
proof
  let X,Y be set, f, i, j be object;
  assume A1: i <> j;
  thus f in product((i,j) --> (X,Y)) implies
    ex x,y being object st x in X & y in Y & f = (i,j) --> (x,y)
  proof
    assume f in product((i,j) --> (X,Y));
    then consider g being Function such that
      A2: g = f & dom g = dom (i,j) --> (X,Y) and
      A3: for z being object st z in dom (i,j) --> (X,Y)
        holds g.z in ((i,j) --> (X,Y)).z by CARD_3:def 5;
    take g.i, g.j;
    A4: dom (i,j) --> (X,Y) = {i,j} by FUNCT_4:62;
    then i in dom (i,j) --> (X,Y) by TARSKI:def 2;
    then g.i in ((i,j) --> (X,Y)).i by A3;
    hence g.i in X by A1, FUNCT_4:63;
    j in dom (i,j) --> (X,Y) by A4, TARSKI:def 2;
    then g.j in ((i,j) --> (X,Y)).j by A3;
    hence thesis by A2, A4, FUNCT_4:66,FUNCT_4:63;
  end;
  given x,y being object such that
    A5: x in X & y in Y & f = (i,j) --> (x,y);
  reconsider g = f as Function by A5;
  A6: dom g = {i,j} by A5, FUNCT_4:62
    .= dom (i,j) --> (X,Y) by FUNCT_4:62;
  for z being object st z in dom (i,j) --> (X,Y)
    holds g.z in ((i,j) --> (X,Y)).z
  proof
    let z be object;
    assume z in dom (i,j) --> (X,Y);
    then z in {i,j} by FUNCT_4:62;
    then per cases by TARSKI:def 2;
    suppose A7: z = i;
      then g.z = x by A1, A5, FUNCT_4:63;
      hence thesis by A1, A5, A7, FUNCT_4:63;
    end;
    suppose A8: z = j;
      then g.z = y by A5, FUNCT_4:63;
      hence thesis by A5, A8, FUNCT_4:63;
    end;
  end;
  hence thesis by A6, CARD_3:9;
end;
