reserve X,X1,X2,Y,Y1,Y2 for set, p,x,x1,x2,y,y1,y2,z,z1,z2 for object;
reserve f,g,g1,g2,h for Function,
  R,S for Relation;

theorem
  z in rng(g*f) implies z in rng g
proof
  assume z in rng(g*f);
  then consider x being object such that
A1: x in dom(g*f) and
A2: z = (g*f).x by Def3;
  f.x in dom g & (g*f).x = g.(f.x) by A1,Th11,Th12;
  hence thesis by A2,Def3;
end;
