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
  Y c= Y1 implies Y1|`R = R
proof
  assume
A1: Y c= Y1;
  now
    let x,y be object;
    now
      assume
A2:   [x,y] in R;
      then y in Y by ZFMISC_1:87;
      hence [x,y] in Y1|`R by A1,A2,RELAT_1:def 12;
    end;
    hence [x,y] in Y1|`R iff [x,y] in R by RELAT_1:def 12;
  end;
  hence thesis;
end;
