reserve A for QC-alphabet;

theorem Th2:
  for K,L being set for x,y being set, f being Function holds (f+*(
  L --> y)).:K c= f.:K \/ {y}
proof
  let K,L be set, x,y be set, f be Function, z be object;
  assume z in (f+*(L --> y)).:K;
  then consider u being object such that
A1: u in dom(f+*(L --> y)) and
A2: u in K and
A3: z = (f+*(L --> y)).u by FUNCT_1:def 6;
A4: dom(L --> y) = L;
  now
    per cases;
    case
A5:   u in L;
      then z = (L --> y).u by A3,A4,FUNCT_4:13;
      then z = y by A5,FUNCOP_1:7;
      hence z in {y} by TARSKI:def 1;
    end;
    case
A6:   not u in L;
      then
A7:   z = f.u by A3,A4,FUNCT_4:11;
      u in dom f by A1,A4,A6,FUNCT_4:12;
      hence z in f.:K by A2,A7,FUNCT_1:def 6;
    end;
  end;
  hence thesis by XBOOLE_0:def 3;
end;
