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 Th13:
  'not' Ex(All(a,A,G),B,G) '<' Ex(Ex('not' a,B,G),A,G)
proof
A1: All(a,A,G) = B_INF(a,CompF(A,G)) by BVFUNC_2:def 9;
  let z be Element of Y;
A2: z in EqClass(z,CompF(B,G)) by EQREL_1:def 6;
  assume ('not' Ex(All(a,A,G),B,G)).z=TRUE;
  then
A3: 'not' (Ex(All(a,A,G),B,G)).z=TRUE by MARGREL1:def 19;
  Ex(All(a,A,G),B,G) = B_SUP(All(a,A,G),CompF(B,G)) by BVFUNC_2:def 10;
  then (All(a,A,G)).z<>TRUE by A3,A2,BVFUNC_1:def 17;
  then consider x1 being Element of Y such that
A4: x1 in EqClass(z,CompF(A,G)) and
A5: (a).x1<>TRUE by A1,BVFUNC_1:def 16;
  (a).x1=FALSE by A5,XBOOLEAN:def 3;
  then
A6: ('not' a).x1='not' FALSE by MARGREL1:def 19;
A7: Ex('not' a,B,G) = B_SUP('not' a,CompF(B,G)) by BVFUNC_2:def 10;
  x1 in EqClass(x1,CompF(B,G)) by EQREL_1:def 6;
  then (Ex('not' a,B,G)).x1=TRUE by A7,A6,BVFUNC_1:def 17;
  then (B_SUP(Ex('not' a,B,G),CompF(A,G))).z = TRUE by A4,BVFUNC_1:def 17;
  hence thesis by BVFUNC_2:def 10;
end;
