reserve Y for non empty set,
  a, b for Function of Y,BOOLEAN,
  G for Subset of PARTITIONS(Y),
  A, B for a_partition of Y;

theorem
  All('not' Ex(a,A,G),B,G) '<' 'not' Ex(All(a,B,G),A,G)
proof
  let z be Element of Y;
A1: All('not' Ex(a,A,G),B,G) = B_INF('not' Ex(a,A,G),CompF(B,G)) & z in
  EqClass( z,CompF(B,G)) by BVFUNC_2:def 9,EQREL_1:def 6;
  assume (All('not' Ex(a,A,G),B,G)).z=TRUE;
  then ('not' Ex(a,A,G)).z=TRUE by A1,BVFUNC_1:def 16;
  then
A2: Ex(a,A,G) = B_SUP(a,CompF(A,G)) & 'not' (Ex(a,A,G)).z=TRUE by
BVFUNC_2:def 10,MARGREL1:def 19;
A3: All(a,B,G) = B_INF(a,CompF(B,G)) by BVFUNC_2:def 9;
  for x being Element of Y st x in EqClass(z,CompF(A,G)) holds (All(a,B,G
  )).x<>TRUE
  proof
    let x be Element of Y;
    assume x in EqClass(z,CompF(A,G)); then
A4: (a).x<>TRUE by A2,BVFUNC_1:def 17;
    x in EqClass(x,CompF(B,G)) by EQREL_1:def 6;
    hence thesis by A3,A4,BVFUNC_1:def 16;
  end;
  then Ex(All(a,B,G),A,G) = B_SUP(All(a,B,G),CompF(A,G)) & (B_SUP(All(a,B,G),
  CompF( A,G))).z = FALSE by BVFUNC_1:def 17,BVFUNC_2:def 10;
  then 'not' (Ex(All(a,B,G),A,G)).z=TRUE;
  hence thesis by MARGREL1:def 19;
end;
