reserve n for Element of NAT,
  p,q,r,s for Element of HP-WFF;

theorem Th12:
  for a,b,c,d being set st a <> b holds
  ((a,b) --> (c,d)) * ((a,b)--> (b,a)) = (a,b) --> (d,c)
proof
  let a,b,c,d be set such that
A1: a <> b;
  set f = ((a,b) --> (c,d))*((a,b) --> (b,a));
A2: dom((a,b) --> (b,a)) = {a,b} by FUNCT_4:62;
  b in {a,b} by TARSKI:def 2;
  then
A3: f.b = ((a,b) --> (c,d)).(((a,b) --> (b,a)).b) by A2,FUNCT_1:13
    .= ((a,b) --> (c,d)).a by FUNCT_4:63
    .= c by A1,FUNCT_4:63;
  a in {a,b} by TARSKI:def 2;
  then
A4: f.a = ((a,b) --> (c,d)).(((a,b) --> (b,a)).a) by A2,FUNCT_1:13
    .= ((a,b) --> (c,d)).b by A1,FUNCT_4:63
    .= d by FUNCT_4:63;
  rng((a,b) --> (b,a)) = {a,b} by A1,FUNCT_4:64
    .= dom((a,b) --> (c,d)) by FUNCT_4:62;
  hence thesis by A4,A3,FUNCT_4:66,A2,RELAT_1:27;
end;
