 reserve x,y,z,t for object,X,Y,Z,W for set;
 reserve R,S,T for Relation;

theorem
  (R * (R \ id dom R) = {} implies id(dom CL R) * (R \ id dom CL R) = {}) &
  ((R \ id dom R) * R = {} implies
    (R \ id dom CL R) * id dom CL R = {})
proof
  thus R * (R \ id dom R) = {} implies id (dom CL R) * (R \ id dom CL R) = {}
  proof
A1: id dom CL R c= R by Th34;
    assume
A2: R * (R \ id dom R) = {};
    R \ id dom R = R \ CL R by XBOOLE_1:47
      .= R \ id dom CL R by Th28;
    hence thesis by A2,A1,RELAT_1:30,XBOOLE_1:3;
  end;
A3: id dom CL R c= R by Th34;
    assume
A4: (R \ id dom R) * R = {};
    R \ id dom R = R \ CL R by XBOOLE_1:47
      .= R \ id dom CL R by Th28;
    hence thesis by A4,A3,RELAT_1:29,XBOOLE_1:3;
end;
