reserve p,q,x,x1,x2,y,y1,y2,z,z1,z2 for set;
reserve A,B,V,X,X1,X2,Y,Y1,Y2,Z for set;
reserve C,C1,C2,D,D1,D2 for non empty set;

theorem Th5:
  for f,g being Function st Y c= rng g & g"Y c= rng f holds Y c= rng(g*f)
proof
  let f,g be Function such that
A1: Y c= rng g and
A2: g"Y c= rng f;
  let y be object;
  assume
A3: y in Y;
  then consider z being object such that
A4: z in dom g & y = g.z by A1,FUNCT_1:def 3;
  z in g"Y by A3,A4,FUNCT_1:def 7;
  then consider x being object such that
A5: x in dom f & z = f.x by A2,FUNCT_1:def 3;
  x in dom(g*f) & (g*f).x = y by A4,A5,FUNCT_1:11,13;
  hence thesis by FUNCT_1:def 3;
end;
