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 Th3:
  x in L WHERE O iff x in L & x.O <> {}
  proof
    hereby
      assume x in L WHERE O; then
      consider y such that
A1:   x = y & ex a st y,a in O & y in L;
      thus x in L & x.O <> {} by A1,Th2;
    end;
    assume
A2: x in L & x.O <> {};
    set y = the Element of x.O;
    y in x.O by A2; then
    reconsider y as Element of X;
    x,y in O by A2,Th2;
    hence thesis by A2;
  end;
