reserve x, y for set,
  T for TopStruct,
  GX for TopSpace,
  P, Q, M, N for Subset of T,
  F, G for Subset-Family of T,
  W, Z for Subset-Family of GX,
  A for SubSpace of T;

theorem Th20:
  W is open & W is finite implies meet W is open
proof
  assume that
A1: W is open and
A2: W is finite;
  consider p being FinSequence such that
A3: rng p = W by A2,FINSEQ_1:52;
  consider n being Nat such that
A4: dom p = Seg n by FINSEQ_1:def 2;
  defpred X[Nat] means for Z st Z = p.:(Seg $1) & $1 <= n & 1 <= n holds meet
  Z is open;
A5: for k be Nat holds X[k] implies X[k+1]
  proof
    let k be Nat;
    assume
A6: for Z st Z = p.:(Seg k) & k <= n & 1 <= n holds meet Z is open;
    let Z such that
A7: Z = p.:(Seg(k+1));
    assume that
A8: k+1 <= n and
A9: 1 <= n;
A10: now
      reconsider G2 = Im(p,k+1) as Subset-Family of GX by A3,Th2,RELAT_1:111;
      reconsider G1 = p.:(Seg k) as Subset-Family of GX by A3,Th2,RELAT_1:111;
      assume
A11:  0 < k;
      k+1 <= n+1 by A8,NAT_1:12;
      then k <= n by XREAL_1:6;
      then Seg k c= dom p by A4,FINSEQ_1:5;
      then
A12:  G1 <> {} by A11,RELAT_1:119;
      k+1 <= n+1 by A8,NAT_1:12;
      then k <= n by XREAL_1:6;
      then
A13:  meet G1 is open by A6,A9;
      0 <= k & 0+1 = 1 by NAT_1:2;
      then 1 <= k+1 by XREAL_1:7;
      then
A14:  k+1 in dom p by A4,A8,FINSEQ_1:1;
      then G2 = {p.(k+1)} by FUNCT_1:59;
      then
A15:  meet G2 = p.(k+1) by SETFAM_1:10;
      {k+1} c= dom p by A14,ZFMISC_1:31;
      then
A16:  G2 <> {} by RELAT_1:119;
      p.(k+1) in W by A3,A14,FUNCT_1:def 3;
      then
A17:  meet G2 is open by A1,A15;
      p.:(Seg(k+1)) = p.:(Seg k \/ {k+1}) by FINSEQ_1:9
        .= p.:(Seg k) \/ p.:{k+1} by RELAT_1:120;
      then meet Z = meet G1 /\ meet G2 by A7,A12,A16,SETFAM_1:9;
      hence thesis by A13,A17,TOPS_1:11;
    end;
    now
      assume
A18:  k=0;
      then
A19:  1 in dom p by A4,A8,FINSEQ_1:1;
      then Im(p,1) = {p.1} by FUNCT_1:59;
      then meet Z = p.1 by A7,A18,FINSEQ_1:2,SETFAM_1:10;
      then meet Z in W by A3,A19,FUNCT_1:def 3;
      hence thesis by A1;
    end;
    hence thesis by A10,NAT_1:3;
  end;
A20: X[0]
  proof
    let Z;
    assume that
A21: Z = p.:(Seg 0) and
    0 <= n;
A22: {} in the topology of GX by PRE_TOPC:1;
    meet Z = {} by A21,SETFAM_1:1;
    hence thesis by A22;
  end;
A23: for k be Nat holds X[k] from NAT_1:sch 2(A20,A5);
A24: now
    assume
A25: 1 <= n;
    W = p.:(Seg n) by A3,A4,RELAT_1:113;
    hence thesis by A23,A25;
  end;
A26: now
    assume n = 0;
    then Seg n = {};
    then W = p.:{} by A3,A4,RELAT_1:113;
    then
A27: meet W = {} by SETFAM_1:1;
    {} in the topology of GX by PRE_TOPC:1;
    hence thesis by A27;
  end;
  now
    assume n <> 0;
    then
A28: 0 < n by NAT_1:3;
    1 = 0+1;
    hence 1 <= n by A28,NAT_1:13;
  end;
  hence thesis by A24,A26;
end;
