reserve X,Y,z,s for set, L,L1,L2,A,B for List of X, x for Element of X,
  O,O1,O2,O3 for Operation of X, a,b,y for Element of X, n,m for Nat;

theorem Th35:
  x.(O1\&O2) = (x.O1)\&O2
  proof
    per cases;
    suppose X = {}; then
      x.(O1\&O2) = {} & (x.O1)\&O2 = {};
      hence thesis;
    end;
    suppose
      X <> {}; then
      reconsider L = {x} as List of X by ZFMISC_1:31;
A1:   {a.O1\&O2: a in L} = {x.O1\&O2}
      proof
        thus {a.O1\&O2: a in L} c= {x.O1\&O2}
        proof
          let z be object; assume z in {a.O1\&O2: a in L}; then
          consider a such that
A2:       z = a.O1\&O2 & a in L;
          a = x by A2,TARSKI:def 1;
          hence thesis by A2,TARSKI:def 1;
        end;
        let z be object; assume z in {x.O1\&O2}; then
        z = x.O1\&O2 & x in L by TARSKI:def 1;
        hence thesis;
      end;
      thus x.(O1\&O2) = union {a.O1\&O2: a in L} by Def20
      .= x.O1\&O2 by A1,ZFMISC_1:25;
    end;
  end;
