 reserve x,y,z,t for object,X,Y,Z,W for set;
 reserve R,S,T for Relation;

theorem Th38:
  (S * R = S & R * (R \ id dom R) = {} implies S * (R \ id dom R) = {}) &
  (R * S = S & (R \ id dom R) * R = {} implies (R \ id dom R) * S = {})
proof
  thus S * R = S & R * (R \ id dom R) = {} implies S * (R \ id dom R) = {}
  proof
    assume S * R = S & R * (R \ id dom R) = {};
    then S * (R \ id dom R) = S * {} by RELAT_1:36
      .= {};
    hence thesis;
  end;
  assume R * S = S & (R \ id dom R) * R = {};
  then (R \ id dom R) * S = {} * S by RELAT_1:36
    .= {};
  hence thesis;
end;
