
theorem Th59:
  for T1,T2 being non empty TopSpace
  for B1 being Basis of T1, B2 being Basis of T2
  for T being Refinement of T1, T2
  holds B1 \/ B2 \/ INTERSECTION(B1,B2) is Basis of T
proof
  let T1,T2 be non empty TopSpace;
  let B1 be Basis of T1, B2 be Basis of T2;
  let T be Refinement of T1,T2;
  set BB = B1 \/ B2 \/ INTERSECTION(B1,B2);
  reconsider B = (the topology of T1) \/ the topology of T2 as prebasis of T
  by Def6;
A1: FinMeetCl B is Basis of T by Th23;
A2: (the carrier of T1) \/ the carrier of T2 = the carrier of T by Def6;
A3: B1 c= the topology of T1 by TOPS_2:64;
A4: B2 c= the topology of T2 by TOPS_2:64;
A5: the topology of T1 c= B by XBOOLE_1:7;
A6: the topology of T2 c= B by XBOOLE_1:7;
A7: B1 c= B by A3,A5;
A8: B2 c= B by A4,A6;
A9: B c= FinMeetCl B by CANTOR_1:4;
  then
A10: B1 c= FinMeetCl B by A7;
A11: B2 c= FinMeetCl B by A8,A9;
A12: B1 \/ B2 c= B by A3,A4,XBOOLE_1:13;
A13: INTERSECTION(B1,B2) c= FinMeetCl B by A10,A11,CANTOR_1:13;
  B1 \/ B2 c= FinMeetCl B by A9,A12;
  then
A14: BB c= FinMeetCl B by A13,XBOOLE_1:8;
A15: FinMeetCl B c= the topology of T by A1,TOPS_2:64;
  reconsider BB as Subset-Family of T by A14,XBOOLE_1:1;
  now
    let A be Subset of T such that
A16: A is open;
    let p be Point of T;
    assume p in A;
    then consider a being Subset of T such that
A17: a in FinMeetCl B and
A18: p in a and
A19: a c= A by A1,A16,Th31;
    consider Y being Subset-Family of T such that
A20: Y c= B and
A21: Y is finite and
A22: a = Intersect Y by A17,CANTOR_1:def 3;
    reconsider Y1 = Y /\ the topology of T1 as Subset-Family of T1;
    reconsider Y2 = Y /\ the topology of T2 as Subset-Family of T2;
A23: Y = B /\ Y by A20,XBOOLE_1:28
      .= Y1 \/ Y2 by XBOOLE_1:23;
    the carrier of T1 c= the carrier of T1;
    then reconsider cT1 = the carrier of T1 as Subset of T1;
    the carrier of T2 c= the carrier of T2;
    then reconsider cT2 = the carrier of T2 as Subset of T2;
A24: cT1 in the topology of T1 by PRE_TOPC:def 1;
A25: cT2 in the topology of T2 by PRE_TOPC:def 1;
A26: cT1 is open by A24;
A27: cT2 is open by A25;
    thus ex a being Subset of T st a in BB & p in a & a c= A
    proof per cases by A2,XBOOLE_0:def 3;
      suppose
A28:    Y1 \/ Y2 = {} & p in cT1;
        then consider b1 being Subset of T1 such that
A29:    b1 in B1 and
A30:    p in b1 and b1 c= cT1 by A26,Th31;
A31:    b1 in B1 \/ B2 by A29,XBOOLE_0:def 3;
A32:    a = the carrier of T by A22,A23,A28,SETFAM_1:def 9;
        b1 in BB by A31,XBOOLE_0:def 3;
        hence thesis by A19,A30,A32,XBOOLE_1:1;
      end;
      suppose
A33:    Y1 \/ Y2 = {} & p in cT2;
        then consider b2 being Subset of T2 such that
A34:    b2 in B2 and
A35:    p in b2 and b2 c= cT2 by A27,Th31;
A36:    b2 in B1 \/ B2 by A34,XBOOLE_0:def 3;
A37:    a = the carrier of T by A22,A23,A33,SETFAM_1:def 9;
        b2 in BB by A36,XBOOLE_0:def 3;
        hence thesis by A19,A35,A37,XBOOLE_1:1;
      end;
      suppose
A38:    Y1 = {} & Y2 <> {} & Y <> {};
        then
A39:    a = meet Y2 by A22,A23,SETFAM_1:def 9
          .= Intersect Y2 by A38,SETFAM_1:def 9;
        Y2 c= the topology of T2 by XBOOLE_1:17;
        then a in FinMeetCl the topology of T2 by A21,A39,CANTOR_1:def 3;
        then
A40:    a in the topology of T2 by CANTOR_1:5;
        the topology of T2 = UniCl B2 by Th22;
        then consider Z being Subset-Family of T2 such that
A41:    Z c= B2 and
A42:    a = union Z by A40,CANTOR_1:def 1;
        consider z being set such that
A43:    p in z and
A44:    z in Z by A18,A42,TARSKI:def 4;
        z in B1 \/ B2 by A41,A44,XBOOLE_0:def 3;
        then
A45:    z in BB by XBOOLE_0:def 3;
        z c= a by A42,A44,ZFMISC_1:74;
        hence thesis by A19,A43,A45,XBOOLE_1:1;
      end;
      suppose
A46:    Y1 <> {} & Y2 = {} & Y <> {};
        then
A47:    a = meet Y1 by A22,A23,SETFAM_1:def 9
          .= Intersect Y1 by A46,SETFAM_1:def 9;
        Y1 c= the topology of T1 by XBOOLE_1:17;
        then a in FinMeetCl the topology of T1 by A21,A47,CANTOR_1:def 3;
        then
A48:    a in the topology of T1 by CANTOR_1:5;
        the topology of T1 = UniCl B1 by Th22;
        then consider Z being Subset-Family of T1 such that
A49:    Z c= B1 and
A50:    a = union Z by A48,CANTOR_1:def 1;
        consider z being set such that
A51:    p in z and
A52:    z in Z by A18,A50,TARSKI:def 4;
        z in B1 \/ B2 by A49,A52,XBOOLE_0:def 3;
        then
A53:    z in BB by XBOOLE_0:def 3;
        z c= a by A50,A52,ZFMISC_1:74;
        hence thesis by A19,A51,A53,XBOOLE_1:1;
      end;
      suppose
A54:    Y1 <> {} & Y2 <> {} & Y <> {};
        then
A55:    a = meet Y by A22,SETFAM_1:def 9
          .= (meet Y1)/\meet Y2 by A23,A54,SETFAM_1:9
          .= (meet Y1)/\Intersect Y2 by A54,SETFAM_1:def 9
          .= (Intersect Y1)/\Intersect Y2 by A54,SETFAM_1:def 9;
A56:    Y1 c= the topology of T1 by XBOOLE_1:17;
A57:    Y2 c= the topology of T2 by XBOOLE_1:17;
A58:    Intersect Y1 in FinMeetCl the topology of T1 by A21,A56,CANTOR_1:def 3;
A59:    Intersect Y2 in FinMeetCl the topology of T2 by A21,A57,CANTOR_1:def 3;
A60:    Intersect Y1 in the topology of T1 by A58,CANTOR_1:5;
A61:    the topology of T1 = UniCl B1 by Th22;
A62:    Intersect Y2 in the topology of T2 by A59,CANTOR_1:5;
A63:    the topology of T2 = UniCl B2 by Th22;
        consider Z1 being Subset-Family of T1 such that
A64:    Z1 c= B1 and
A65:    Intersect Y1 = union Z1 by A60,A61,CANTOR_1:def 1;
        p in Intersect Y1 by A18,A55,XBOOLE_0:def 4;
        then consider z1 being set such that
A66:    p in z1 and
A67:    z1 in Z1 by A65,TARSKI:def 4;
        consider Z2 being Subset-Family of T2 such that
A68:    Z2 c= B2 and
A69:    Intersect Y2 = union Z2 by A62,A63,CANTOR_1:def 1;
        p in Intersect Y2 by A18,A55,XBOOLE_0:def 4;
        then consider z2 being set such that
A70:    p in z2 and
A71:    z2 in Z2 by A69,TARSKI:def 4;
A72:    z1 /\ z2 in INTERSECTION(B1,B2) by A64,A67,A68,A71,SETFAM_1:def 5;
A73:    z1 c= union Z1 by A67,ZFMISC_1:74;
A74:    z2 c= union Z2 by A71,ZFMISC_1:74;
A75:    z1 /\ z2 in BB by A72,XBOOLE_0:def 3;
A76:    z1 /\ z2 c= a by A55,A65,A69,A73,A74,XBOOLE_1:27;
        p in z1 /\ z2 by A66,A70,XBOOLE_0:def 4;
        hence thesis by A19,A75,A76,XBOOLE_1:1;
      end;
    end;
  end;
  hence thesis by A14,A15,Th32,XBOOLE_1:1;
end;
