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 Th23:
  a in L|O iff ex b st a in b.O & b in L
  proof
    hereby
      assume
      a in L|O; then
      consider b being object such that
A1:   [b,a] in O & b in L by RELAT_1:def 13;
      reconsider b as Element of X by A1;
      take b;
      thus a in b.O by A1,RELAT_1:169;
      thus b in L by A1;
    end;
    given b such that
A2: a in b.O & b in L;
    [b,a] in O by A2,RELAT_1:169;
    hence a in L|O by A2,RELAT_1:def 13;
  end;
