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
  C is I_Lattice implies C is \/-distributive
proof
  assume
A1: C is I_Lattice;
  now
    let X,a;
    set Y = {a"/\"a9: a9 in X}, b = "\/"(X,C), c = "\/"(Y,C), Z = {b9: a"/\"
    b9 [= c};
    X is_less_than a=>c
    proof
      let b9;
      assume b9 in X;
      then a"/\"b9 in Y;
      then a"/\"b9 [= c by Th38;
      then
A2:   b9 in Z;
      a=>c = "\/"(Z,C) by A1,Th51;
      hence thesis by A2,Th38;
    end;
    then b [= a=>c by Def21;
    then
A3: a"/\"b [= a"/\"(a=>c) by LATTICES:9;
    a"/\" (a=>c) [= c by A1,FILTER_0:def 7;
    hence a"/\"b [= c by A3,LATTICES:7;
  end;
  hence thesis by Th33;
end;
