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

theorem Th35:
  (id X c= R & id(X) * (R \ id X) = {} implies R|X = id X) &
  (id X c= R & (R \ id X) * id X = {} implies X|`R = id X)
proof
  thus id X c= R & id(X) * (R \ id X) = {} implies R|X = id X
  proof
    assume that
A1: id(X) c= R and
A2: id(X) * (R \ id(X)) = {};
    R|X = id(X) * R by RELAT_1:65
      .= id(X) * (R \/ id(X)) by A1,XBOOLE_1:12
      .= id(X) * ((R \ id(X)) \/ id(X)) by XBOOLE_1:39
      .= {} \/ id(X) * id(X) by A2,RELAT_1:32
      .= id(X) by Th12;
    hence thesis;
  end;
    assume that
A3: id(X) c= R and
A4: (R \ id X) * id X = {};
    X|`R = R * id X by RELAT_1:92
      .= (R \/ id X) * id X by A3,XBOOLE_1:12
      .= ((R \ id X) \/ id X) * id X by XBOOLE_1:39
      .= {} \/ id(X) * id X by A4,Th6
      .= id X by Th12;
    hence thesis;
end;
