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 Th34:
  'not' Ex(Ex(a,A,G),B,G) '<' 'not' All(Ex(a,B,G),A,G)
proof
  let z be Element of Y;
  assume ('not' Ex(Ex(a,A,G),B,G)).z=TRUE;
  then
A1: 'not' (Ex(Ex(a,A,G),B,G)).z=TRUE by MARGREL1:def 19;
A2: now
    assume ex x being Element of Y st x in EqClass(z,CompF(B,G)) & (Ex(a,A,G)
    ).x=TRUE;
    then (B_SUP(Ex(a,A,G),CompF(B,G))).z = TRUE by BVFUNC_1:def 17;
    then (Ex(Ex(a,A,G),B,G)).z=TRUE by BVFUNC_2:def 10;
    hence contradiction by A1;
  end;
A3: Ex(a,A,G) = B_SUP(a,CompF(A,G)) by BVFUNC_2:def 10;
A4: for x being Element of Y st x in EqClass(z,CompF(B,G)) holds for y being
  Element of Y st y in EqClass(x,CompF(A,G)) holds a.y<>TRUE
  proof
    let x be Element of Y;
    assume x in EqClass(z,CompF(B,G));
    then (Ex(a,A,G)).x<>TRUE by A2;
    hence thesis by A3,BVFUNC_1:def 17;
  end;
  for x being Element of Y st x in EqClass(z,CompF(B,G)) holds (a).x<> TRUE
  proof
    let x be Element of Y;
A5: x in EqClass(x,CompF(A,G)) by EQREL_1:def 6;
    assume x in EqClass(z,CompF(B,G));
    hence thesis by A4,A5;
  end;
  then (B_SUP(a,CompF(B,G))).z = FALSE by BVFUNC_1:def 17;
  then z in EqClass(z,CompF(A,G)) & (Ex(a,B,G)).z=FALSE by BVFUNC_2:def 10
,EQREL_1:def 6;
  then (B_INF(Ex(a,B,G),CompF(A,G))).z = FALSE by BVFUNC_1:def 16;
  then (All(Ex(a,B,G),A,G)).z=FALSE by BVFUNC_2:def 9;
  then 'not' (All(Ex(a,B,G),A,G)).z=TRUE;
  hence thesis by MARGREL1:def 19;
end;
