
theorem Th40:
  for T1,T2 being non empty TopSpace
  for B1 being Basis of T1, B2 being Basis of T2 holds
  {[:a,b:] where a is Subset of T1, b is Subset of T2: a in B1 & b in B2}
  is Basis of [:T1,T2:]
proof
  let T1,T2 be non empty TopSpace;
  let B1 be Basis of T1, B2 be Basis of T2;
  set BB = {[:a,b:] where a is Subset of T1, b is Subset of T2:
  a in B1 & b in B2};
  set T = [:T1,T2:];
A1: the carrier of T = [:the carrier of T1, the carrier of T2:]
  by BORSUK_1:def 2;
  BB c= bool the carrier of T
  proof
    let x be object;
    assume x in BB;
    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 BB as Subset-Family of T;
A2: B1 c= the topology of T1 by TOPS_2:64;
A3: B2 c= the topology of T2 by TOPS_2:64;
  BB is Basis of T
  proof
A4:  BB is open
      proof
      let x be Subset of T;
      assume x in BB;
      then consider a being Subset of T1, b being Subset of T2 such that
A5:   x = [:a,b:] and
A6:   a in B1 and
A7:   b in B2;
A8:   a is open by A2,A6;
      b is open by A3,A7;
      hence x is open by A5,A8,BORSUK_1:6;
    end;
    BB is quasi_basis
    proof
    let x be object;
    assume
A9: x in the topology of T;
    then reconsider X = x as Subset of T;
    X is open by A9;
    then
A10: X = union Base-Appr X by BORSUK_1:13;
    set Y = BB /\ Base-Appr X;
A11: Y c= BB by XBOOLE_1:17;
    reconsider Y as Subset-Family of T;
    union Y = X
    proof
      thus union Y c= X by A10,XBOOLE_1:17,ZFMISC_1:77;
      let z be object;
      assume
A12:  z in X;
      then reconsider p = z as Point of T;
      consider z1,z2 being object such that
A13:  z1 in the carrier of T1 and
A14:  z2 in the carrier of T2 and
A15:  p = [z1,z2] by A1,ZFMISC_1:def 2;
      reconsider z1 as Point of T1 by A13;
      reconsider z2 as Point of T2 by A14;
      consider Z being set such that
A16:  z in Z and
A17:  Z in Base-Appr X by A10,A12,TARSKI:def 4;
A18:  Base-Appr X = {[:a,b:] where a is Subset of T1, b is Subset of T2:
      [:a,b:] c= X & a is open & b is open} by BORSUK_1:def 3;
      then consider a being Subset of T1, b being Subset of T2 such that
A19:  Z = [:a,b:] and
A20:  [:a,b:] c= X and
A21:  a is open and
A22:  b is open by A17;
A23:  z1 in a by A15,A16,A19,ZFMISC_1:87;
A24:  z2 in b by A15,A16,A19,ZFMISC_1:87;
      consider a9 being Subset of T1 such that
A25:  a9 in B1 and
A26:  z1 in a9 and
A27:  a9 c= a by A21,A23,Th31;
      consider b9 being Subset of T2 such that
A28:  b9 in B2 and
A29:  z2 in b9 and
A30:  b9 c= b by A22,A24,Th31;
      [:a9,b9:] c= [:a,b:] by A27,A30,ZFMISC_1:96;
      then
A31:  [:a9,b9:] c= X by A20;
A32:  a9 is open by A2,A25;
      b9 is open by A3,A28;
      then
A33:  [:a9,b9:] in Base-Appr X by A18,A31,A32;
A34:  [:a9,b9:] in BB by A25,A28;
A35:  p in [:a9,b9:] by A15,A26,A29,ZFMISC_1:87;
      [:a9,b9:] in Y by A33,A34,XBOOLE_0:def 4;
      hence thesis by A35,TARSKI:def 4;
    end;
    hence thesis by A11,CANTOR_1:def 1;
  end;
  hence thesis by A4;
  end;
  hence thesis;
end;
