
theorem
  for T1,T2 being non empty TopSpace
  for B1 being prebasis of T1, B2 being prebasis of T2
  st union B1 = the carrier of T1 & union B2 = the carrier of T2 holds
  {[:a,b:] where a is Subset of T1, b is Subset of T2: a in B1 & b in B2}
  is prebasis of [:T1,T2:]
proof
  let T1,T2 be non empty TopSpace;
  let B1 be prebasis of T1, B2 be prebasis of T2 such that
A1: union B1 = the carrier of T1 and
A2: union B2 = the carrier of T2;
  set cT1 = the carrier of T1, cT2 = the carrier of T2;
  set BB1 = {[:the carrier of T1, b:] where b is Subset of T2: b in B2},
  BB2 = {[:a, the carrier of T2:] where a is Subset of T1: a in B1};
  set CC = {[:a,b:] where a is Subset of T1, b is Subset of T2:
  a in B1 & b in B2};
  set T = [:T1,T2:];
  reconsider BB=BB1 \/ BB2 as prebasis of [:T1,T2:] by Th41;
A3: FinMeetCl BB is Basis of T by Th23;
  CC c= bool the carrier of [:T1,T2:]
  proof
    let x be object;
    assume x in CC;
    then ex a being Subset of T1, b being Subset of T2
    st x = [:a,b:] & a in B1 & b in B2;
    hence thesis;
  end;
  then reconsider CC as Subset-Family of [:T1,T2:];
  reconsider CC as Subset-Family of [:T1,T2:];
  CC c= the topology of T
  proof
    let x be object;
    assume x in CC;
    then consider a being Subset of T1, b being Subset of T2 such that
A4: x = [:a,b:] and
A5: a in B1 and
A6: b in B2;
A7: B1 c= the topology of T1 by TOPS_2:64;
A8: B2 c= the topology of T2 by TOPS_2:64;
A9: a is open by A5,A7;
    b is open by A6,A8;
    then [:a,b:] is open by A9,BORSUK_1:6;
    hence thesis by A4;
  end;
  then UniCl CC c= UniCl the topology of T by CANTOR_1:9;
  then
A10: UniCl CC c= the topology of T by CANTOR_1:6;
  BB c= UniCl CC
  proof
    let x be object;
    assume
A11: x in BB;
    per cases by A11,XBOOLE_0:def 3;
    suppose x in BB1;
      then consider b being Subset of T2 such that
A12:  x = [:cT1,b:] and
A13:  b in B2;
      set Y = {[:a,b:] where a is Subset of T1: a in B1};
      Y c= bool the carrier of T
      proof
        let y be object;
        assume y in Y;
        then ex a being Subset of T1 st y = [:a,b:] & a in B1;
        hence thesis;
      end;
      then reconsider Y as Subset-Family of T;
      reconsider Y as Subset-Family of T;
A14:  Y c= CC
      proof
        let y be object;
        assume y in Y;
        then ex a being Subset of T1 st y = [:a,b:] & a in B1;
        hence thesis by A13;
      end;
      [:cT1,b:] = union Y
      proof
        hereby
          let z be object;
          assume z in [:cT1,b:];
          then consider p1, p2 being object such that
A15:      p1 in cT1 and
A16:      p2 in b and
A17:      [p1,p2] = z by ZFMISC_1:def 2;
          consider a being set such that
A18:      p1 in a and
A19:      a in B1 by A1,A15,TARSKI:def 4;
          reconsider a as Subset of T1 by A19;
A20:      [:a,b:] in Y by A19;
          z in [:a,b:] by A16,A17,A18,ZFMISC_1:def 2;
          hence z in union Y by A20,TARSKI:def 4;
        end;
        let z be object;
        assume z in union Y;
        then consider y being set such that
A21:    z in y and
A22:    y in Y by TARSKI:def 4;
        ex a being Subset of T1 st y = [:a,b:] & a in B1 by A22;
        then y c= [:cT1,b:] by ZFMISC_1:95;
        hence thesis by A21;
      end;
      hence thesis by A14,CANTOR_1:def 1,A12;
    end;
    suppose x in BB2;
      then consider a being Subset of T1 such that
A23:  x = [:a,cT2:] and
A24:  a in B1;
      set Y = {[:a,b:] where b is Subset of T2: b in B2};
      Y c= bool the carrier of T
      proof
        let y be object;
        assume y in Y;
        then ex b being Subset of T2 st y = [:a,b:] & b in B2;
        hence thesis;
      end;
      then reconsider Y as Subset-Family of T;
      reconsider Y as Subset-Family of T;
A25:  Y c= CC
      proof
        let y be object;
        assume y in Y;
        then ex b being Subset of T2 st y = [:a,b:] & b in B2;
        hence thesis by A24;
      end;
      [:a,cT2:] = union Y
      proof
        hereby
          let z be object;
          assume z in [:a,cT2:];
          then consider p1, p2 being object such that
A26:      p1 in a and
A27:      p2 in cT2 and
A28:      [p1,p2] = z by ZFMISC_1:def 2;
          consider b being set such that
A29:      p2 in b and
A30:      b in B2 by A2,A27,TARSKI:def 4;
          reconsider b as Subset of T2 by A30;
A31:      [:a,b:] in Y by A30;
          z in [:a,b:] by A26,A28,A29,ZFMISC_1:def 2;
          hence z in union Y by A31,TARSKI:def 4;
        end;
        let z be object;
        assume z in union Y;
        then consider y being set such that
A32:    z in y and
A33:    y in Y by TARSKI:def 4;
        ex b being Subset of T2 st y = [:a,b:] & b in B2 by A33;
        then y c= [:a,cT2:] by ZFMISC_1:95;
        hence thesis by A32;
      end;
      hence thesis by A25,CANTOR_1:def 1,A23;
    end;
  end;
  then FinMeetCl BB c= FinMeetCl UniCl CC by CANTOR_1:14;
  then UniCl CC is prebasis of T by A3,A10,CANTOR_1:def 4,TOPS_2:64;
  hence thesis by Th24;
end;
