theorem
  Y1|`R is Relation of X,Y1
proof
  now
    let x,y be object;
    assume [x,y] in Y1|`R;
    then y in Y1 & x in X by RELAT_1:def 12,ZFMISC_1:87;
    hence [x,y] in [:X,Y1:] by ZFMISC_1:87;
  end;
  hence thesis by RELAT_1:def 3;
end;
