
theorem Th10:
  for a,b,c,d being object st a <> b holds (a,b) --> (c,d) = (b,a) --> (d,c)
proof
  let a,b,c,d be object;
  assume A1: a <> b;
  A2: dom (a,b) --> (c,d) = {a,b} by FUNCT_4:62;
  then A3: dom (a,b) --> (c,d) = dom (b,a) --> (d,c) by FUNCT_4:62;
  for x being object st x in dom (a,b) --> (c,d)
    holds ((a,b) --> (c,d)).x = ((b,a) --> (d,c)).x
  proof
    let x be object;
    assume x in dom (a,b) --> (c,d);
    then per cases by A2, TARSKI:def 2;
    suppose A4: x = a;
      hence ((a,b) --> (c,d)).x = c by A1, FUNCT_4:63
        .= ((b,a) --> (d,c)).x by A4, FUNCT_4:63;
    end;
    suppose A5: x = b;
      hence ((a,b) --> (c,d)).x = d by FUNCT_4:63
        .= ((b,a) --> (d,c)).x by A1, A5, FUNCT_4:63;
    end;
  end;
  hence thesis by A3, FUNCT_1:2;
end;
