
theorem Th19:
  for I being non empty set, J being TopStruct-yielding non-Empty
  ManySortedSet of I, i being Element of I, xi being Element of J.i, G being
Subset of product_prebasis J st proj(J,i)"({xi}) c= union G & (for A being set
st A in product_prebasis J & A in G holds not proj(J,i)"({xi}) c= A) holds [#](
  product J) c= union G
proof
  let I be non empty set, J be TopStruct-yielding non-Empty ManySortedSet of I,
  i be Element of I, xi be Element of J.i, G be Subset of product_prebasis J;
  assume that
A1: proj(J,i)"({xi}) c= union G and
A2: for A being set st A in product_prebasis J & A in G holds not proj(J
  ,i)"({xi}) c= A;
  let f be object;
  assume f in [#](product J);
  then reconsider f9=f as Element of product J;
  set g = f9+*(i,xi);
A3: g in proj(J,i)"({xi}) by Th11;
  then consider Ag being set such that
A4: g in Ag and
A5: Ag in G by A1,TARSKI:def 4;
  consider i2 being Element of I, Ai2 being Subset of J.i2 such that
  Ai2 is open and
A6: proj(J,i2)"Ai2 = Ag by A5,Th16;
A7: Ai2 <> [#](J.i2)
  proof
    assume Ai2 = [#](J.i2);
    then proj(J,i2)"Ai2 = [#] product J by Th10
      .= the carrier of product J;
    hence contradiction by A2,A5,A6;
  end;
A8: not proj(J,i)"({xi}) c= proj(J,i2)"Ai2 by A2,A5,A6;
  i<>i2
  proof
    assume
A9: i = i2;
    then reconsider Ai29=Ai2 as Subset of J.i;
    proj(J,i)"({xi}) /\ proj(J,i)"Ai29 <> {} by A3,A4,A6,A9,XBOOLE_0:def 4;
    then
A10: proj(J,i)"({xi}) meets proj(J,i)"Ai29;
    not xi in Ai2 by A8,A7,A9,Th12;
    hence contradiction by A10,Th9;
  end;
  then f in proj(J,i2)"Ai2 by A4,A6,Th13;
  hence thesis by A5,A6,TARSKI:def 4;
end;
