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;
reserve F,F1,F2 for filtering Operation of X;
reserve i for Element of NAT;
reserve X for ConstructorDB, A for FinSequence of the Constrs of X,
  x for Element of X;
reserve Y for ref-finite ConstructorDB,
  B for FinSequence of the Constrs of Y,
  y for Element of Y;

theorem
  for c1,c2 being Element of X holds
  A = <*c1,c2*> implies ATLEAST A = (c1 occur) AND (c2 occur)
  proof
    let c1,c2 be Element of X;
    assume A1: A = <*c1,c2*>; then
A2: rng A = {c1,c2} by FINSEQ_2:127;
A3: {x occur: x in rng A} = {c1 occur, c2 occur}
    proof
      thus {x occur: x in rng A} c= {c1 occur, c2 occur}
      proof
        let z be object; assume z in {x occur: x in rng A}; then
        consider x such that
A4:     z = x occur & x in rng A;
        x = c1 or x = c2 by A2,A4,TARSKI:def 2;
        hence thesis by A4,TARSKI:def 2;
      end;
      let z be object; assume z in {c1 occur,c2 occur}; then
A5:   z = c1 occur or z = c2 occur by TARSKI:def 2;
      c1 in rng A & c2 in rng A by A2,TARSKI:def 2;
      hence thesis by A5;
    end;
    thus ATLEAST A = meet {x occur: x in rng A} by A1,Th78
    .= (c1 occur) AND (c2 occur) by A3,SETFAM_1:11;
  end;
