reserve A,X,X1,X2,Y,Y1,Y2 for set, a,b,c,d,x,y,z for object;
reserve P,P1,P2,Q,R,S for Relation;

theorem Th44:
  R*(id X) c= R & (id X)*R c= R
proof
  thus [x,y] in R*id X implies [x,y] in R
  proof
    assume [x,y] in R*id X;
    then ex z st [x,z] in R & [z,y] in id X by Def6;
    hence thesis by Def8;
  end;
  thus [x,y] in (id X)*R implies [x,y] in R
  proof
    assume [x,y] in (id X)*R;
    then ex z st [x,z] in id X & [z,y] in R by Def6;
    hence thesis by Def8;
  end;
end;
