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;

theorem
  rng g c= rng(f +* g)
proof
  let y be object;
  assume y in rng g;
  then consider x being object such that
A1: x in dom g & y = g.x by FUNCT_1:def 3;
  x in dom(f +* g) & (f +* g).x = y by A1,Th12,Th13;
  hence thesis by FUNCT_1:def 3;
end;
