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 Th33:
  C is \/-distributive iff for X, a holds a"/\""\/"(X,C) [= "\/"({a"/\"
b: b in X}, C)
proof
  thus C is \/-distributive implies for X, a holds
  a"/\""\/"(X,C) [= "\/"({a"/\"b: b in X}, C)
  proof
    assume
A1: for X for a,b,c st X is_less_than a &
    (for d st X is_less_than d holds a [= d) &
    {b"/\"a9: a9 in X} is_less_than c &
    for d st {b"/\"b9: b9 in X} is_less_than d holds c [= d holds b"/\"a [= c;
    let X, a;
    set Y = {a"/\"b: b in X};
A2: X is_less_than "\/"(X,C) by Def21;
A3: for d st X is_less_than d holds "\/"(X,C) [= d by Def21;
A4: Y is_less_than "\/"(Y,C) by Def21;
    for d st Y is_less_than d holds "\/"(Y,C) [= d by Def21;
    hence thesis by A1,A2,A3,A4;
  end;
  assume
A5: for X, a holds a"/\""\/"(X,C) [= "\/"({a"/\"b: b in X}, C);
  let X;
  let a,b,c;
  assume
A6: X is_less_than a & (for d st X is_less_than d holds a [= d) &
  {b"/\"a9: a9 in X} is_less_than c &
  for d st {b"/\"b9: b9 in X} is_less_than d holds c [= d;
  then
A7: a = "\/"(X,C) by Def21;
  c = "\/"({b"/\"a9: a9 in X}, C) by A6,Def21;
  hence b"/\"a [= c by A5,A7;
end;
