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

theorem
  (id(dom CL R) * (R \ id(dom CL R)) = {} implies R|(dom CL R) = id dom CL R &
    R|(rng CL R) = id dom CL R) &
  ((R \ id rng CL R) * id(rng CL R) = {} implies
    (dom CL R)|`R = id dom CL R &
    (rng CL R)|`R = id rng CL R)
proof
  thus id(dom CL R) * (R \ id dom CL R) = {} implies
  R|(dom CL R) = id(dom CL R) & R|(rng CL R) = id dom CL R
  proof
    assume
A1: id(dom CL R) * (R \ id dom CL R) = {};
    id(dom CL R) c= R by Th34;
    then R|(dom CL R) = id dom CL R by A1,Th35;
    hence thesis by Th26;
  end;
    assume
A2: (R \ id rng CL R) * id rng CL R = {};
    id rng CL R c= R by Th34;
    then (rng CL R)|`R = id rng CL R by A2,Th35;
    then (dom CL R)|`R = id rng CL R by Th26;
    hence thesis by Th26;
end;
