reserve L for Lattice,
  p,q,r for Element of L,
  p9,q9,r9 for Element of L.:,
  x, y for set;
reserve I,J for Ideal of L,
  F for Filter of L;
reserve D for non empty Subset of L,
  D9 for non empty Subset of L.:;
reserve D1,D2 for non empty Subset of L,
  D19,D29 for non empty Subset of L.:;
reserve B for B_Lattice,
  IB,JB for Ideal of B,
  a,b for Element of B;
reserve a9 for Element of (B qua Lattice).:;
reserve P for non empty ClosedSubset of L,
  o1,o2 for BinOp of P;

theorem
  L is implicative & p [= q implies latt (L,[#p,q#]) is implicative
proof
  assume
A1: L is implicative;
  set P = [#p,q#], K = latt (L,P);
  assume
A2: p [= q;
  let a9,b9 be Element of latt (L,P);
  reconsider a = a9, b = b9 as Element of L by Th68;
  set c = a => b;
A3: carr(K) = P by Th72;
  then p [= a by A2,Th62;
  then
A4: p"\/"(c"/\"a) = (p"\/"c)"/\"a by A1,LATTICES:def 12;
A5: a"/\"c [= b by A1,FILTER_0:def 7;
  p [= b by A2,A3,Th62;
  then p"\/"(a"/\"c) [= b by A5,FILTER_0:6;
  then
A6: (p"\/"(a"/\"c))"/\"q [= b by FILTER_0:2;
  set d = (c"\/"p)"/\"q;
  p [= c"\/"p by LATTICES:5;
  then d [= q & p [= d by A2,FILTER_0:7,LATTICES:6;
  then reconsider d9 = d as Element of K by A2,A3,Th62;
  take d9;
  (p"\/"c)"/\"a"/\"q = a"/\"d & a"/\"d = a9 "/\"d9 by Th73,LATTICES:def 7;
  hence a9"/\"d9 [= b9 by A4,A6,Th74;
  let e9 be Element of K;
  reconsider e = e9, ae = a9"/\"e9 as Element of L by Th68;
  e [= q by A2,A3,Th62;
  then
A7: e = e"/\"q by LATTICES:4;
  assume a9"/\"e9 [= b9;
  then ae [= b by Th74;
  then a"/\"e [= b by Th73;
  then
A8: e [= c by A1,FILTER_0:def 7;
  p [= e by A2,A3,Th62;
  then e = e"\/"p;
  then e [= c"\/"p by A8,FILTER_0:1;
  then e [= d by A7,LATTICES:9;
  hence e9 [= d9 by Th74;
end;
