reserve Y for non empty set,
  a for Function of Y,BOOLEAN,
  G for Subset of PARTITIONS(Y),
  P,Q for a_partition of Y;

theorem Th12:
  for a,b being Function of Y,BOOLEAN, G being Subset of PARTITIONS(Y),
      P being a_partition of Y st a '<' b holds All(a,P,G) '<' All(b,P,G)
proof
  let a,b be Function of Y,BOOLEAN, G be Subset of PARTITIONS(Y), P be
  a_partition of Y such that
A1: a '<' b;
  let x be Element of Y;
  assume
A2: All(a,P,G).x= TRUE;
A3: All(a,P,G) = B_INF(a,CompF(P,G)) by BVFUNC_2:def 9;
A4: for y being Element of Y st y in EqClass(x,CompF(P,G)) holds b.y=TRUE
  proof
    let y be Element of Y;
    assume y in EqClass(x,CompF(P,G));
    then a.y=TRUE by A3,A2,BVFUNC_1:def 16;
    hence thesis by A1;
  end;
  All(b,P,G) = B_INF(b,CompF(P,G)) by BVFUNC_2:def 9;
  hence thesis by A4,BVFUNC_1:def 16;
end;
