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
  for a,b being Function of Y,BOOLEAN,
      G being Subset of PARTITIONS(Y), P being a_partition of Y st
    a '<' b holds Ex(a,P,G) '<' Ex(b,P,G)
proof
  let a,b be Function of Y,BOOLEAN, G be Subset of PARTITIONS(Y), P be
  a_partition of Y;
A1: Ex(b,P,G) = Ex('not' 'not' b,P,G) .= 'not' All('not' b,P,G) by BVFUNC_2:18;
  assume a '<' b;
  then 'not' b '<' 'not' a by Th11;
  then
A2: All('not' b,P,G) '<' All('not' a,P,G) by Th12;
  Ex(a,P,G) = Ex('not' 'not' a,P,G) .= 'not' All('not' a,P,G) by BVFUNC_2:18;
  hence thesis by A1,A2,Th11;
end;
