reserve X for set,
  x,y,z for Element of BooleLatt X,
  s for set;
reserve y for Element of BooleLatt X;
reserve L for Lattice,
  p,q for Element of L;
reserve A for RelStr,
  a,b,c for Element of A;
reserve A for non empty RelStr,
  a,b,c,c9 for Element of A;
reserve V for with_suprema antisymmetric RelStr,
  u1,u2,u3,u4 for Element of V;
reserve N for with_infima antisymmetric RelStr,
  n1,n2,n3,n4 for Element of N;
reserve K for with_suprema with_infima reflexive antisymmetric RelStr,
  k1,k2,k3 for Element of K;
reserve p9,q9 for Element of LattPOSet L;
reserve C for complete Lattice,
  a,a9,b,b9,c,d for Element of C,
  X,Y for set;

theorem Th34:
  a = "/\"(X,C) iff a is_less_than X & for b st b is_less_than X holds b [= a
proof
  set Y = {b: b is_less_than X};
A1: a = "/\"(X,C) iff Y is_less_than a &
  for c st Y is_less_than c holds a [= c by Def21;
  thus a = "/\"(X,C) implies a is_less_than X &
  for b st b is_less_than X holds b [= a
  proof
    assume
A2: a = "/\"(X,C);
    then
A3: Y is_less_than a by Def21;
    thus a is_less_than X
    proof
      let b such that
A4:   b in X;
      Y is_less_than b
      proof
        let c;
        assume c in Y;
        then ex d being Element of C st c = d & d is_less_than X;
        hence thesis by A4;
      end;
      hence thesis by A2,Def21;
    end;
    let b;
    assume b is_less_than X;
    then b in Y;
    hence thesis by A3;
  end;
  assume that
A5: a is_less_than X and
A6: for b st b is_less_than X holds b [= a;
A7: Y is_less_than a
  proof
    let b;
    assume b in Y;
    then ex c st b = c & c is_less_than X;
    hence thesis by A6;
  end;
  a in Y by A5;
  hence thesis by A1,A7;
end;
