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;
