reserve A,B,X,X1,Y,Y1,Y2,Z for set, a,x,y,z for object;
reserve P,R for Relation of X,Y;

theorem
  X c= X1 implies R|X1 = R
proof
  assume
A1: X c= X1;
  now
    let x,y be object;
    now
      assume
A2:   [x,y] in R;
      then x in X by ZFMISC_1:87;
      hence [x,y] in R|X1 by A1,A2,RELAT_1:def 11;
    end;
    hence [x,y] in R|X1 iff [x,y] in R by RELAT_1:def 11;
  end;
  hence thesis;
end;
