
theorem Th17:
  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, A being set
st A in product_prebasis J & proj(J,i)"({xi}) c= A holds A = [#](product J) or
ex Ai being Subset of J.i st Ai <> [#](J.i) & xi in Ai & Ai is open & A=proj(J,
  i)"Ai
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, A be set;
  assume A in product_prebasis J;
  then consider i1 being Element of I, Ai1 being Subset of J.i1 such that
A1: Ai1 is open and
A2: proj(J,i1)"Ai1 = A by Th16;
  assume
A3: proj(J,i)"({xi}) c= A;
  assume not A = [#](product J);
  then
A4: Ai1 <> [#](J.i1) by A2,Th10;
  then reconsider Ai=Ai1 as Subset of J.i by A3,A2,Th12;
  take Ai;
  thus Ai <> [#](J.i) by A3,A2,A4,Th12;
  thus xi in Ai by A3,A2,A4,Th12;
  J.i = J.i1 by A3,A2,A4,Th12;
  hence Ai is open by A1;
  thus thesis by A3,A2,A4,Th12;
end;
