
theorem Th20:
  for I being non empty set, J being TopStruct-yielding non-Empty
ManySortedSet of I, i being Element of I, F being Subset of product_prebasis J
  holds (for G being finite Subset of F holds not [#](product J) c= union G)
implies for xi being Element of J.i, G being finite Subset of F st proj(J,i)"({
xi}) c= union G ex A being set st A in product_prebasis J & A in G & proj(J,i)"
  ({xi}) c= A
proof
  let I be non empty set, J be TopStruct-yielding non-Empty ManySortedSet of I,
  i be Element of I, F be Subset of product_prebasis J;
  assume
A1: for G being finite Subset of F holds not [#](product J) c= union G;
  let xi be Element of J.i, G be finite Subset of F;
  reconsider G9=G as Subset of product_prebasis J by XBOOLE_1:1;
  assume
A2: proj(J,i)"({xi}) c= union G;
  assume for A being set st A in product_prebasis J & A in G holds not proj(J
  ,i)"({xi}) c= A;
  then [#](product J) c= union G9 by A2,Th19;
  hence contradiction by A1;
end;
