reserve i, x, I for set,
  A, B, M for ManySortedSet of I,
  f, f1 for Function;

theorem Th2:
  for I being non empty set for A being ManySortedSet of I for B
  being ManySortedSubset of A holds rng B c= union rng bool A
proof
  let I be non empty set, A be ManySortedSet of I, B be ManySortedSubset of A;
  let a be object;
  assume a in rng B;
  then consider i being object such that
A1: i in I and
A2: a = B.i by PBOOLE:138;
  i in dom bool A & bool (A.i) = (bool A).i by A1,MBOOLEAN:def 1,PARTFUN1:def 2
;
  then
A3: bool (A.i) in rng bool A by FUNCT_1:def 3;
  B c= A by PBOOLE:def 18;
  then B in bool A by MBOOLEAN:18;
  then B.i in (bool A).i by A1;
  then a in bool (A.i) by A1,A2,MBOOLEAN:def 1;
  hence thesis by A3,TARSKI:def 4;
end;
