
theorem Th21:
  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 holds ex Ai being Subset of J.i st Ai <> [#](J.i) & xi in Ai &
  proj(J,i)"Ai in G & Ai is open
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;
  assume proj(J,i)"({xi}) c= union G;
  then consider A being set such that
A2: A in product_prebasis J and
A3: A in G and
A4: proj(J,i)"({xi}) c= A by A1,Th20;
  A <> [#](product J)
  proof
    reconsider G1 = {A} as finite Subset of F by A3,ZFMISC_1:31;
    assume A = [#](product J);
    then union G1 = [#](product J) by ZFMISC_1:25;
    hence contradiction by A1;
  end;
  then consider Ai being Subset of J.i such that
A5: Ai <> [#](J.i) and
A6: xi in Ai and
A7: Ai is open and
A8: A=proj(J,i)"Ai by A2,A4,Th17;
  take Ai;
  thus Ai <> [#](J.i) by A5;
  thus xi in Ai by A6;
  thus proj(J,i)"Ai in G by A3,A8;
  thus thesis by A7;
end;
