
theorem Th41:
  for T1,T2 being non empty TopSpace
  for B1 being prebasis of T1, B2 being prebasis of T2 holds
  {[:the carrier of T1, b:] where b is Subset of T2: b in B2} \/
  {[:a, the carrier of T2:] where a is Subset of T1: a in B1}
  is prebasis of [:T1,T2:]
proof
  let T1,T2 be non empty TopSpace;
  set T = [:T1,T2:];
  let B1 be prebasis of T1, B2 be prebasis of T2;
  set C2 = {[:the carrier of T1, b:] where b is Subset of T2: b in B2};
  set C1 = {[:a, the carrier of T2:] where a is Subset of T1: a in B1};
  reconsider D1 = FinMeetCl B1 as Basis of T1 by Th23;
  reconsider D2 = FinMeetCl B2 as Basis of T2 by Th23;
  reconsider D = {[:a,b:] where a is Subset of T1, b is Subset of T2:
  a in D1 & b in D2} as Basis of T by Th40;
  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;
A1: cT1 in the topology of T1 by PRE_TOPC:def 1;
A2: cT2 in the topology of T2 by PRE_TOPC:def 1;
A3: B1 c= the topology of T1 by TOPS_2:64;
A4: B2 c= the topology of T2 by TOPS_2:64;
  C1 c= bool the carrier of T
  proof
    let x be object;
    assume x in C1;
    then ex a being Subset of T1 st x = [:a, cT2:] & a in B1;
    hence thesis;
  end;
  then reconsider C1 as Subset-Family of T;
  reconsider C1 as Subset-Family of T;
  C2 c= bool the carrier of T
  proof
    let x be object;
    assume x in C2;
    then ex a being Subset of T2 st x = [:cT1, a:] & a in B2;
    hence thesis;
  end;
  then reconsider C2 as Subset-Family of T;
  reconsider C2 as Subset-Family of T;
  reconsider C = C2 \/ C1 as Subset-Family of T;
  C is prebasis of T
  proof
A5: C is open
     proof
      let x be Subset of T;
      assume x in C;
      then x in C1 or x in C2 by XBOOLE_0:def 3;
      then (ex a being Subset of T1 st x = [:a, cT2:] & a in B1) or
      ex a being Subset of T2 st x = [:cT1, a:] & a in B2;
      then consider a being Subset of T1, b being Subset of T2 such that
A6:   x = [:a,b:] and
A7:   a in the topology of T1 and
A8:   b in the topology of T2 by A1,A2,A3,A4;
A9:   a is open by A7;
      b is open by A8;
      hence x is open by A6,A9,BORSUK_1:6;
    end;
    C is quasi_prebasis
    proof
    take D;
    let d be object;
    assume d in D;
    then consider a being Subset of T1, b being Subset of T2 such that
A10: d = [:a,b:] and
A11: a in D1 and
A12: b in D2;
    consider aY being Subset-Family of T1 such that
A13: aY c= B1 and
A14: aY is finite and
A15: a = Intersect aY by A11,CANTOR_1:def 3;
    consider bY being Subset-Family of T2 such that
A16: bY c= B2 and
A17: bY is finite and
A18: b = Intersect bY by A12,CANTOR_1:def 3;
    deffunc F(Subset of T1) = [:$1, cT2:];
A19: {F(s) where s is Subset of T1: s in aY} is finite
    from FRAENKEL:sch 21(A14);
    deffunc G(Subset of T2) = [:cT1, $1:];
A20: {G(s) where s is Subset of T2: s in bY} is finite
    from FRAENKEL:sch 21(A17);
    set Y1 = {[:s, cT2:] where s is Subset of T1: s in aY};
    set Y2 = {[:cT1, s:] where s is Subset of T2: s in bY};
A21: Y1 c= C
    proof
      let x be object;
      assume x in Y1;
      then ex s being Subset of T1 st ( x = [:s, cT2:])&( s in aY);
      then x in C1 by A13;
      hence thesis by XBOOLE_0:def 3;
    end;
A22: Y2 c= C
    proof
      let x be object;
      assume x in Y2;
      then ex s being Subset of T2 st ( x = [:cT1, s:])&( s in bY);
      then x in C2 by A16;
      hence thesis by XBOOLE_0:def 3;
    end;
    set Y = Y1 \/ Y2;
A23: Y c= C by A21,A22,XBOOLE_1:8;
    then reconsider Y as Subset-Family of T by XBOOLE_1:1;
    Intersect Y = [:a,b:]
    proof
      hereby
        let p be object;
        assume
A24:    p in Intersect Y;
        then consider p1 being Point of T1, p2 being Point of T2 such that
A25:    p = [p1,p2] by BORSUK_1:10;
        now
          let z be set;
          assume
A26:      z in aY;
          then reconsider z9 = z as Subset of T1;
          [:z9, cT2:] in Y1 by A26;
          then [:z9, cT2:] in Y by XBOOLE_0:def 3;
          then p in [:z9, cT2:] by A24,SETFAM_1:43;
          hence p1 in z by A25,ZFMISC_1:87;
        end;
        then
A27:    p1 in a by A15,SETFAM_1:43;
        now
          let z be set;
          assume
A28:      z in bY;
          then reconsider z9 = z as Subset of T2;
          [:cT1, z9:] in Y2 by A28;
          then [:cT1, z9:] in Y by XBOOLE_0:def 3;
          then p in [:cT1, z9:] by A24,SETFAM_1:43;
          hence p2 in z by A25,ZFMISC_1:87;
        end;
        then p2 in b by A18,SETFAM_1:43;
        hence p in [:a,b:] by A25,A27,ZFMISC_1:87;
      end;
      let p be object;
      assume p in [:a,b:];
      then consider p1,p2 being object such that
A29:  p1 in a and
A30:  p2 in b and
A31:  [p1,p2] = p by ZFMISC_1:def 2;
      reconsider p1 as Point of T1 by A29;
      reconsider p2 as Point of T2 by A30;
      now
        let z be set;
        assume
A32:    z in Y;
        per cases by A32,XBOOLE_0:def 3;
        suppose z in Y1;
          then consider s being Subset of T1 such that
A33:      z = [:s, cT2:] and
A34:      s in aY;
          p1 in s by A15,A29,A34,SETFAM_1:43;
          hence [p1,p2] in z by A33,ZFMISC_1:87;
        end;
        suppose z in Y2;
          then consider s being Subset of T2 such that
A35:      z = [:cT1, s:] and
A36:      s in bY;
          p2 in s by A18,A30,A36,SETFAM_1:43;
          hence [p1,p2] in z by A35,ZFMISC_1:87;
        end;
      end;
      hence thesis by A31,SETFAM_1:43;
    end;
    hence thesis by A19,A20,A23,CANTOR_1:def 3,A10;
  end;
  hence thesis by A5;
end;
hence thesis;
end;
