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
  rng P2 c= X & P2*R = id dom P1 & R*P1 = id X implies P1 = P2
proof
  (P2*R)*P1 = P2*(R*P1) & id(dom P1)*P1 = P1 by Th30,Th45;
  hence thesis by Th47;
end;
