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

theorem Th14:
  id(X \/ Y) = id(X) \/ id(Y) & id(X /\ Y) = id(X) /\ id(Y) &
  id(X \ Y) = id(X) \ id(Y)
proof
  thus id(X \/ Y) = id(X) \/ id(Y)
  proof
    let x,y be object;
    thus [x,y] in id(X \/ Y) implies [x,y] in id(X) \/ id(Y)
    proof
      assume
A1:   [x,y] in id(X \/ Y);
      then x in X \/ Y by RELAT_1:def 10;
      then
A2:   x in X or x in Y by XBOOLE_0:def 3;
      x = y by A1,RELAT_1:def 10;
      then [x,y] in id(X) or [x,y] in id(Y) by A2,RELAT_1:def 10;
      hence thesis by XBOOLE_0:def 3;
    end;
    assume [x,y] in id(X) \/ id(Y); then
A3: [x,y] in id(X) or [x,y] in id(Y) by XBOOLE_0:def 3;
    then x in X or x in Y by RELAT_1:def 10;
    then
A4: x in X \/ Y by XBOOLE_0:def 3;
    x = y by A3,RELAT_1:def 10;
    hence thesis by A4,RELAT_1:def 10;
  end;
  thus id(X /\ Y) = id(X) /\ id(Y)
  proof
    let x,y be object;
    thus [x,y] in id(X /\ Y) implies [x,y] in id(X) /\ id(Y)
    proof
      assume
A5:   [x,y] in id(X /\ Y); then
A6:   x in X /\ Y by RELAT_1:def 10;
A7:   x = y by A5,RELAT_1:def 10;
      x in Y by A6,XBOOLE_0:def 4; then
A8:   [x,y] in id(Y) by A7,RELAT_1:def 10;
      x in X by A6,XBOOLE_0:def 4;
      then [x,y] in id(X) by A7,RELAT_1:def 10;
      hence thesis by A8,XBOOLE_0:def 4;
    end;
    assume
A9: [x,y] in id(X) /\ id Y; then
A10: [x,y] in id(X) by XBOOLE_0:def 4; then
A11: x = y by RELAT_1:def 10;
    [x,y] in id(Y) by A9,XBOOLE_0:def 4; then
A12: x in Y by RELAT_1:def 10;
    x in X by A10,RELAT_1:def 10;
    then x in X /\ Y by A12,XBOOLE_0:def 4;
    hence thesis by A11,RELAT_1:def 10;
  end;
    let x,y be object;
    thus [x,y] in id(X \ Y) implies [x,y] in id(X) \ id(Y)
    proof
      assume
A13:  [x,y] in id(X \ Y); then
A14:  x in X \ Y by RELAT_1:def 10;
      then not x in Y by XBOOLE_0:def 5; then
A15:  not [x,y] in id(Y) by RELAT_1:def 10;
      x = y by A13,RELAT_1:def 10;
      then [x,y] in id(X) by A14,RELAT_1:def 10;
      hence thesis by A15,XBOOLE_0:def 5;
    end;
    assume
A16: [x,y] in id(X) \ id(Y); then
A17: x = y by RELAT_1:def 10;
    not [x,y] in id(Y) by A16,XBOOLE_0:def 5; then
A18: not (x in Y & x = y) by RELAT_1:def 10;
    x in X by A16,RELAT_1:def 10;
    then x in X \ Y by A16,A18,RELAT_1:def 10,XBOOLE_0:def 5;
    hence thesis by A17,RELAT_1:def 10;
end;
