reserve a,b,c for set;

theorem
  for X being set, O being Subset-Family of bool X ex B being
  Subset-Family of X st B = UniCl FinMeetCl union O & TopStruct(#X,B#) is
TopSpace & (for o being Subset-Family of X st o in O holds TopStruct(#X,B#) is
TopExtension of TopStruct(#X,o#)) & for T being TopSpace st the carrier of T =
  X & for o being Subset-Family of X st o in O holds T is TopExtension of
  TopStruct(#X,o#) holds T is TopExtension of TopStruct(#X,B#)
proof
  reconsider e = {} bool {}, tE = {{}{}} as Subset-Family of {};
  reconsider E = {} bool bool {} as empty Subset-Family of bool {};
  let X be set;
  let O be Subset-Family of bool X;
  reconsider B = UniCl FinMeetCl union O as Subset-Family of X;
  take B;
  thus B = UniCl FinMeetCl union O;
  reconsider dT = TopStruct(#{},tE#) as discrete TopStruct by TDLAT_3:def 1
,ZFMISC_1:1;
A1: {{},{}} = tE by ENUMSET1:29;
A2: FinMeetCl tE = {{},{}} by YELLOW_9:11;
A3: now
    assume
A4: X is empty;
    hence
A5: union O = e or union O = tE by ZFMISC_1:1,33;
    thus FinMeetCl union E = the topology of dT by YELLOW_9:17;
    hence TopStruct(#X,B#) is TopSpace by A2,A1,A4,A5,YELLOW_9:11;
  end;
  hence TopStruct(#X,B#) is TopSpace by CANTOR_1:15;
  reconsider TT = TopStruct(#X,B#) as TopSpace by A3,CANTOR_1:15;
  hereby
    let o be Subset-Family of X;
    set S = TopStruct(#X,o#);
A6: FinMeetCl union O c= B by CANTOR_1:1;
    assume o in O;
    then
A7: o c= union O by ZFMISC_1:74;
    union O c= FinMeetCl union O by CANTOR_1:4;
    then o c= FinMeetCl union O by A7;
    then the topology of S c= the topology of TT by A6;
    hence TopStruct(#X,B#) is TopExtension of S by YELLOW_9:def 5;
  end;
  let T be TopSpace such that
A8: the carrier of T = X and
A9: for o being Subset-Family of X st o in O holds T is TopExtension of
  TopStruct(#X,o#);
  thus the carrier of T = the carrier of TopStruct(#X,B#) by A8;
A10: X <> {} implies T is non empty by A8;
  now
    let a;
    assume
A11: a in O;
    then reconsider o = a as Subset-Family of X;
    T is TopExtension of TopStruct(#X,o#) by A9,A11;
    hence a c= the topology of T by YELLOW_9:def 5;
  end;
  then union O c= the topology of T by ZFMISC_1:76;
  then FinMeetCl union O c= FinMeetCl the topology of T by A8,CANTOR_1:14;
  then
A12: B c= UniCl FinMeetCl the topology of T by A8,CANTOR_1:9;
  X in the topology of T by A8,PRE_TOPC:def 1;
  hence thesis by A12,A10,CANTOR_1:7;
end;
