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 Th11:
  for a,b being Function of Y,BOOLEAN st a '<' b holds
  'not' b '<' 'not' a
proof
  let a,b being Function of Y,BOOLEAN such that
A1: a '<' b;
  let x be Element of Y;
  assume
A2: ('not' b).x= TRUE;
  b.x = ('not' 'not' b).x .= 'not' TRUE by A2,MARGREL1:def 19
    .= FALSE;
  then a.x <> TRUE by A1;
  then a.x = FALSE by XBOOLEAN:def 3;
  hence ('not' a).x = 'not' FALSE by MARGREL1:def 19
    .=TRUE;
end;
