reserve x, X, Y for set;

theorem Th5:
  for R being Relation for X, Y being set holds X c= Y implies R
  |_2 X c= R |_2 Y
proof
  let R be Relation, X,Y be set;
  assume
A1: X c= Y;
  then X|`R c= Y|`R by RELAT_1:100;
  then
A2: X|`R|X c= Y|`R|X by RELAT_1:76;
  Y|`R|X c= Y|`R|Y by A1,RELAT_1:75;
  then X|`R|X c= Y|`R|Y by A2;
  then R|_2 X c= Y|`R|Y by WELLORD1:10;
  hence thesis by WELLORD1:10;
end;
