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

theorem Th13:
  for a,b,c,d being set, f being Function st a <> b & c in dom f &
  d in dom f holds f*((a,b) --> (c,d)) = (a,b) --> (f.c,f.d)
proof
  let a,b,c,d be set, f be Function such that
A1: a <> b and
A2: c in dom f & d in dom f;
A3: dom((a,b) --> (c,d)) = {a,b} by FUNCT_4:62;
  then a in dom((a,b) --> (c,d)) by TARSKI:def 2;
  then
A4: (f*((a,b) --> (c,d))).a = f.(((a,b) --> (c,d)).a) by FUNCT_1:13
    .= f.c by A1,FUNCT_4:63;
  b in dom((a,b) --> (c,d)) by A3,TARSKI:def 2;
  then
A5: (f*((a,b) --> (c,d))).b = f.(((a,b) --> (c,d)).b) by FUNCT_1:13
    .= f.d by FUNCT_4:63;
A6: rng((a,b) --> (c,d)) c= {c,d} by FUNCT_4:62;
  {c,d} c= dom f by A2,ZFMISC_1:32;
  then dom(f*((a,b) --> (c,d))) = {a,b} by A3,A6,RELAT_1:27,XBOOLE_1:1;
  hence thesis by A4,A5,FUNCT_4:66;
end;
