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 Th45:
  dom R c= X implies (id X)*R = R
proof
  assume
A1: dom R c= X;
  R c= (id X)*R
  proof
    let x,y;
    assume
A2: [x,y] in R;
    then x in dom R by XTUPLE_0:def 12;
    then [x,x] in id X by A1,Def8;
    hence thesis by A2,Def6;
  end;
  hence thesis by Th44;
end;
