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;
reserve x for set;
reserve x for object;
reserve A1,A2,B1,B2 for non empty set,
  f for Function of A1,B1,
  g for Function of A2,B2,
  Y1 for non empty Subset of A1,
  Y2 for non empty Subset of A2;

theorem
 for f being Function, x,y being object st x <> y
 holds f+~(x,y)+~(x,z) = f+~(x,y)
proof let f be Function,x,y be object;
 assume x <> y;
  then not x in rng(f+~(x,y)) by Th100;
 hence f+~(x,y)+~(x,z) = f+~(x,y) by Th103;
end;
