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 Th27:
  All(All('not' a,A,G),B,G) '<' 'not' All(All(a,B,G),A,G)
proof
  let z be Element of Y;
A1: z in EqClass(z,CompF(A,G)) by EQREL_1:def 6;
A2: z in EqClass(z,CompF(B,G)) by EQREL_1:def 6;
  assume
A3: (All(All('not' a,A,G),B,G)).z=TRUE;
  now
    assume not (for x being Element of Y st x in EqClass(z,CompF(B,G)) holds
    (All('not' a,A,G)).x=TRUE);
    then (B_INF(All('not' a,A,G),CompF(B,G))).z = FALSE by BVFUNC_1:def 16;
    hence contradiction by A3,BVFUNC_2:def 9;
  end;
  then All('not' a,A,G) = B_INF('not' a,CompF(A,G)) & (All('not' a,A,G)).z=
  TRUE by A2,BVFUNC_2:def 9;
  then ('not' a).z=TRUE by A1,BVFUNC_1:def 16;
  then 'not' (a).z=TRUE by MARGREL1:def 19;
  then (B_INF(a,CompF(B,G))).z = FALSE by A2,BVFUNC_1:def 16;
  then (All(a,B,G)).z=FALSE by BVFUNC_2:def 9;
  then (B_INF(All(a,B,G),CompF(A,G))).z = FALSE by A1,BVFUNC_1:def 16;
  then (All(All(a,B,G),A,G)).z=FALSE by BVFUNC_2:def 9;
  then 'not' (All(All(a,B,G),A,G)).z=TRUE;
  hence thesis by MARGREL1:def 19;
end;
