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 x,y being object holds
  x in rng f implies y in rng(f+~(x,y))
proof let x,y be object;
  assume x in rng f;
  then consider z being object such that
A1: z in dom f and
A2: f.z = x by FUNCT_1:def 3;
A3: dom((x .--> y)*f) c= dom(f+~(x,y)) by Th10;
  x in dom(x.-->y) by FUNCOP_1:74;
  then
A4: z in dom((x .--> y)*f) by A1,A2,FUNCT_1:11;
  then (f+~(x,y)).z = ((x .--> y)*f).z by Th13
    .= (x .--> y).(f.z) by A1,FUNCT_1:13
    .= y by A2,FUNCOP_1:72;
  hence thesis by A4,A3,FUNCT_1:3;
end;
