
theorem Th22:
  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
st (for i being Element of I holds J.i is compact) & (for G being finite Subset
  of F holds not [#](product J) c= union G) ex xi being Element of J.i st for G
  being finite Subset of F holds not proj(J,i)"({xi}) c= union G
proof
  defpred P[set] means not contradiction;
  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 that
A1: for i being Element of I holds J.i is compact and
A2: for G being finite Subset of F holds not [#](product J) c= union G;
  deffunc F(set) = proj(J,i)"$1;
  defpred P[object,object] means
   ex A being set st A = $2 &
   $1 in A & proj(J,i)"A in F & for V being Subset
  of J.i st V = $2 holds V is open;
  assume
A3: for xi being Element of J.i ex G being finite Subset of F st proj(J,
  i)"({xi}) c= union G;
A4: for xi being object st xi in the carrier of J.i
  ex Ai being object st Ai in bool the carrier of J.i & P[xi, Ai]
  proof
    let xi be object;
    assume xi in the carrier of J.i;
    then reconsider xi9=xi as Element of J.i;
    consider G being finite Subset of F such that
A5: proj(J,i)"({xi9}) c= union G by A3;
    consider Ai being Subset of J.i such that
    Ai <> [#](J.i) and
A6: xi in Ai and
A7: proj(J,i)"Ai in G and
A8: Ai is open by A2,A5,Th21;
    take Ai;
    thus Ai in bool the carrier of J.i;
    take Ai;
    thus Ai = Ai;
    thus xi in Ai by A6;
    thus proj(J,i)"Ai in F by A7;
    let V be Subset of J.i;
    assume V = Ai;
    hence thesis by A8;
  end;
  consider h being Function such that
A9: dom h = the carrier of J.i and
A10: rng h c= bool the carrier of J.i and
A11: for xi being object st xi in the carrier of J.i holds P[xi, h.xi] from
  FUNCT_1:sch 6(A4);
  reconsider bGip = rng h as Subset-Family of (J.i) by A10;
  reconsider bGip as Subset-Family of J.i;
A12: [#](J.i) c= union bGip
  proof
    let x be object;
    assume
A13:   x in [#](J.i);
    then P[x,h.x] by A11;
    then consider A being set such that
A14:   A = h.x &
      x in A & proj(J,i)"A in F & for V being Subset
        of J.i st V = h.x holds V is open;
     x in h.x & h.x in bGip by A9,FUNCT_1:def 3,A14,A13;
    hence thesis by TARSKI:def 4;
  end;
  for P being Subset of J.i holds P in bGip implies P is open
  proof
    let P be Subset of J.i;
    assume
   P in bGip;
    then consider x being object such that
A15:   x in dom h & P = h.x by FUNCT_1:def 3;
     P[x,h.x] by A9,A11,A15;
    hence thesis by A15;
  end;
  then
A16: bGip is open by TOPS_2:def 1;
  J.i is compact by A1;
  then consider Gip being Subset-Family of J.i such that
A17: Gip c= bGip and
A18: [#](J.i) c= union Gip and
A19: Gip is finite by A12,A16,Th14;
  reconsider Gip as non empty finite Subset-Family of J.i by A18,A19,ZFMISC_1:2
;
  set Gp={F(Ai) where Ai is Element of Gip: P[Ai]};
A20: Gp c= F
  proof
    let A be object;
    assume A in Gp;
    then consider Ai being Element of Gip such that
A21: A= proj(J,i)"Ai;
  Ai in Gip;
    then consider x being object such that
A22:   x in dom h & h.x = Ai by A17,FUNCT_1:def 3;
    P[x,h.x] by A9,A11,A22;
   hence thesis by A21,A22;
  end;
  Gp is finite from PRE_CIRC:sch 1;
  then reconsider Gp as finite Subset of F by A20;
  [#](product J) c= union Gp by A18,Th18;
  hence contradiction by A2;
end;
