reserve X for set, R,R1,R2 for Relation;
reserve x,y,z for set;
reserve n,m,k for Nat;

theorem Th3:
  for X,Y being set holds (id X).:Y = X /\ Y
  proof let X,Y be set;
    thus (id X).:Y c= X /\ Y
    proof let x be object; assume x in (id X).:Y; then
      consider y be object such that
A1:   [y,x] in id X & y in Y by RELAT_1:def 13;
      x = y & y in X by A1,RELAT_1:def 10;
      hence thesis by A1,XBOOLE_0:def 4;
    end;
    let x be object; assume x in X /\ Y; then
A2: x in X & x in Y by XBOOLE_0:def 4; then
    [x,x] in id X by RELAT_1:def 10;
    hence thesis by A2,RELAT_1:def 13;
  end;
