
theorem
  for T1,T2 being non empty TopSpace, T be Refinement of T1, T2
  for B1 being prebasis of T1, B2 being prebasis of T2
  holds B1 \/ B2 \/ {the carrier of T1, the carrier of T2} is prebasis of T
proof
  let T1,T2 be non empty TopSpace, T be Refinement of T1,T2;
  let B1 be prebasis of T1, B2 be prebasis of T2;
  reconsider B = (the topology of T1) \/ (the topology of T2) as prebasis of T
  by Def6;
  set cT1 = the carrier of T1, cT2 = the carrier of T2;
  reconsider C1 = B1 \/ {cT1} as prebasis of T1 by Th29;
  reconsider C2 = B2 \/ {cT2} as prebasis of T2 by Th29;
A1: B1 c= the topology of T1 by TOPS_2:64;
A2: C1 c= the topology of T1 by TOPS_2:64;
A3: B2 c= the topology of T2 by TOPS_2:64;
A4: C2 c= the topology of T2 by TOPS_2:64;
A5: cT1 in the topology of T1 by PRE_TOPC:def 1;
A6: cT2 in the topology of T2 by PRE_TOPC:def 1;
A7: B1 \/ B2 c= B by A1,A3,XBOOLE_1:13;
A8: B c= the topology of T by TOPS_2:64;
A9: cT1 in B by A5,XBOOLE_0:def 3;
A10: cT2 in B by A6,XBOOLE_0:def 3;
A11: B1 \/ B2 c= the topology of T by A7,A8;
A12: {cT1,cT2} c= the topology of T by A8,A9,A10,ZFMISC_1:32;
A13: {cT1,cT2} c= B by A9,A10,ZFMISC_1:32;
  B1 \/ B2 \/ {cT1,cT2} c= the topology of T by A11,A12,XBOOLE_1:8;
  then reconsider BB = B1 \/ B2 \/
  {cT1, cT2} as Subset-Family of T by XBOOLE_1:1;
A14: the topology of T1 c= B by XBOOLE_1:7;
A15: the topology of T2 c= B by XBOOLE_1:7;
A16: C1 c= B by A2,A14;
  C2 c= B by A4,A15;
  then reconsider D1 = C1, D2 = C2 as Subset-Family of T by A16,XBOOLE_1:1;
  reconsider D1, D2 as Subset-Family of T;
  reconsider D1, D2 as Subset-Family of T;
A17: UniCl FinMeetCl BB = UniCl FinMeetCl FinMeetCl BB by CANTOR_1:11
    .= UniCl FinMeetCl UniCl FinMeetCl BB by Th21;
A18: FinMeetCl B is Basis of T by Th23;
A19: FinMeetCl C1 is Basis of T1 by Th23;
A20: FinMeetCl C2 is Basis of T2 by Th23;
A21: UniCl FinMeetCl B = the topology of T by A18,Th22;
A22: UniCl FinMeetCl C1 = the topology of T1 by A19,Th22;
A23: UniCl FinMeetCl C2 = the topology of T2 by A20,Th22;
A24: B1 c= B1 \/ B2 by XBOOLE_1:7;
A25: B2 c= B1 \/ B2 by XBOOLE_1:7;
A26: {cT1} c= {cT1,cT2} by ZFMISC_1:7;
A27: {cT2} c= {cT1, cT2 } by ZFMISC_1:7;
A28: D1 c= BB by A24,A26,XBOOLE_1:13;
A29: D2 c= BB by A25,A27,XBOOLE_1:13;
  BB c= B by A7,A13,XBOOLE_1:8;
  then
A30: FinMeetCl BB c= FinMeetCl B by CANTOR_1:14;
A31: FinMeetCl D1 c= FinMeetCl BB by A28,CANTOR_1:14;
A32: FinMeetCl D2 c= FinMeetCl BB by A29,CANTOR_1:14;
A33: UniCl FinMeetCl BB c= the topology of T by A21,A30,CANTOR_1:9;
A34: cT1 in {cT1} by TARSKI:def 1;
A35: cT2 in {cT2} by TARSKI:def 1;
A36: cT1 in C1 by A34,XBOOLE_0:def 3;
A37: cT2 in C2 by A35,XBOOLE_0:def 3;
A38: FinMeetCl D1 = {the carrier of T} \/ FinMeetCl C1 by A36,Th20;
A39: FinMeetCl D2 = {the carrier of T} \/ FinMeetCl C2 by A37,Th20;
A40: FinMeetCl C1 c= FinMeetCl D1 by A38,XBOOLE_1:7;
A41: FinMeetCl C2 c= FinMeetCl D2 by A39,XBOOLE_1:7;
A42: FinMeetCl C1 c= FinMeetCl BB by A31,A40;
A43: FinMeetCl C2 c= FinMeetCl BB by A32,A41;
A44: the topology of T1 c= UniCl FinMeetCl BB by A22,A42,Th19;
  the topology of T2 c= UniCl FinMeetCl BB by A23,A43,Th19;
  then B c= UniCl FinMeetCl BB by A44,XBOOLE_1:8;
  then FinMeetCl B c= FinMeetCl UniCl FinMeetCl BB by CANTOR_1:14;
  then the topology of T c= UniCl FinMeetCl BB by A17,A21,CANTOR_1:9;
  then the topology of T = UniCl FinMeetCl BB by A33;
  then FinMeetCl BB is Basis of T by Th22;
  hence thesis by Th23;
end;
