reserve A for QC-alphabet;

theorem Th1:
  for x,y being set, f being Function holds Im(f+*(x .--> y),x) = { y}
proof
  let x,y be set, f be Function;
  now
    let u be object;
    thus u in (f+*(x .--> y)).:{x} implies u = y
    proof
      assume u in (f+*(x .--> y)).:{x};
      then consider z being object such that
      z in dom(f+*(x .--> y)) and
A1:   z in {x} and
A2:   u = (f+*(x .--> y)).z by FUNCT_1:def 6;
      z in dom(x .--> y) by A1;
      then u = (x .--> y).z by A2,FUNCT_4:13;
      hence thesis by A1,FUNCOP_1:7;
    end;
A3: x in {x} by TARSKI:def 1;
    then
A4: x in dom(x .--> y);
    then
A5: x in dom(f+*(x .--> y)) by FUNCT_4:12;
    (x .--> y).x = y by A3,FUNCOP_1:7;
    then y = (f+*(x .--> y)).x by A4,FUNCT_4:13;
    hence u = y implies u in (f+*(x .--> y)).:{x} by A3,A5,FUNCT_1:def 6;
  end;
  hence thesis by TARSKI:def 1;
end;
