
theorem Th55:
  for I being non empty set
  for J being TopStruct-yielding non-Empty ManySortedSet of I
  for f being one-to-one I-valued Function
  st f" is non-empty & dom f c= bool product Carrier J
  holds product_basis_selector(J,f) is non-empty
proof
  let I be non empty set;
  let J be TopStruct-yielding non-Empty ManySortedSet of I;
  let f be one-to-one I-valued Function;
  assume A1: f" is non-empty & dom f c= bool product Carrier J;
  assume product_basis_selector(J,f) is non non-empty;
  then consider x being object such that
    A2: x in dom product_basis_selector(J,f) and
    A3: product_basis_selector(J,f).x is empty by FUNCT_1:def 9;
  A4: x in rng f by A2;
  then reconsider i = x as Element of I;
  proj(J,i).:(f".i) is empty by A3, A2, Th54;
  then dom proj(J,i) misses f".i by RELAT_1:118;
  then dom proj(Carrier J,i) misses f".i by WAYBEL18:def 4;
  then A5: product Carrier J misses f".i by CARD_3:def 16;
  A6: rng(f") c= bool product Carrier J by A1, FUNCT_1:33;
  i in dom(f") by A4, FUNCT_1:33;
  then f".i in rng(f") by FUNCT_1:3;
  hence contradiction by A1,A5, A6, XBOOLE_1:67;
end;
