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
  a = "\/"({b: b [= a}, C) & a = "/\"({c: a [= c}, C)
proof
  set X = {b: b [= a}, Y = {c: a [= c};
A1: a in X;
A2: a in Y;
  X is_less_than a
  proof
    let b;
    assume b in X;
    then ex c st b = c & c [= a;
    hence thesis;
  end;
  hence a = "\/"(X,C) by A1,Th40;
  a is_less_than Y
  proof
    let b;
    assume b in Y;
    then ex c st b = c & a [= c;
    hence thesis;
  end;
  hence thesis by A2,Th41;
end;
