reserve a,b,p,x,x9,x1,x19,x2,y,y9,y1,y19,y2,z,z9,z1,z2 for object,
   X,X9,Y,Y9,Z,Z9 for set;
reserve A,D,D9 for non empty set;
reserve f,g,h for Function;
reserve A,B for set;
reserve x,y,i,j,k for object;

theorem
  for f being PartFunc of X,Y, x,y st x in X & y in Y holds f+*(x .--> y
  ) is PartFunc of X,Y
proof
  let f be PartFunc of X,Y, x,y;
  assume that
A1: x in X and
A2: y in Y;
A3: {x} c= X by A1,ZFMISC_1:31;
  {y} c= Y by A2,ZFMISC_1:31;
  then rng(x .--> y) c= Y by FUNCOP_1:8;
  then
A4: rng f \/ rng(x .--> y) c= Y by XBOOLE_1:8;
  rng(f+*(x .--> y)) c= rng f \/ rng(x .--> y) by Th17;
  then
A5: rng(f+*(x .--> y)) c= Y by A4;
  dom(f+*(x .--> y)) = dom f \/ dom(x .--> y) by Def1
    .= dom f \/ {x};
  hence thesis by A3,A5,RELSET_1:4,XBOOLE_1:8;
end;
