
theorem Th38:
  for L being non empty RelStr, S be full non empty SubRelStr of L
  , X be set holds S|^X is full SubRelStr of L|^X
proof
  let L be non empty RelStr, S be full non empty SubRelStr of L, X be set;
  per cases;
  suppose
A1: X = {};
    then
A2: L|^X = RelStr (#{{}},id {{}}#) by YELLOW_1:27;
    S|^X = RelStr (#{{}}, id {{}}#) by A1,YELLOW_1:27;
    hence thesis by A2,YELLOW_6:6;
  end;
  suppose
    X <> {};
    then reconsider X as non empty set;
    for i be Element of X holds (X --> S).i is full SubRelStr of (X --> L).i;
    hence thesis by Th36;
  end;
end;
