reserve A, B, X, Y for set;
reserve R, S, T for non empty TopSpace;
reserve S1, S2, T1, T2 for non empty TopSpace,
  R for Refinement of [:S1,T1:], [:S2,T2:],
  R1 for Refinement of S1, S2,
  R2 for Refinement of T1, T2;

theorem Th48:
  the carrier of S1 = the carrier of S2 & the carrier of T1 = the
carrier of T2 implies { [:U1,V1:] /\ [:U2,V2:] where U1 is Subset of S1, U2 is
Subset of S2, V1 is Subset of T1, V2 is Subset of T2 : U1 is open & U2 is open
  & V1 is open & V2 is open } is Basis of R
proof
  assume
A1: the carrier of S1 = the carrier of S2 & the carrier of T1 = the
  carrier of T2;
  set Y = { [:U1,V1:] /\ [:U2,V2:] where U1 is Subset of S1, U2 is Subset of
  S2, V1 is Subset of T1, V2 is Subset of T2 : U1 is open & U2 is open & V1 is
  open & V2 is open };
A2: the carrier of [:S2,T2:] = [:the carrier of S2, the carrier of T2:] by
BORSUK_1:def 2;
A3: the carrier of [:S1,T1:] = [:the carrier of S1, the carrier of T1:] by
BORSUK_1:def 2;
  then reconsider
  BST = INTERSECTION(the topology of [:S1,T1:], the topology of [:
  S2,T2:]) as Basis of R by A1,A2,YELLOW_9:60;
A4: the carrier of R = (the carrier of [:S1,T1:]) \/ the carrier of [:S2,T2
  :] by YELLOW_9:def 6
    .= [:the carrier of S1,the carrier of T1:] \/ [:the carrier of S2,the
  carrier of T2:] by A3,BORSUK_1:def 2
    .= [:the carrier of S1,the carrier of T1:] by A1;
  Y c= bool the carrier of R
  proof
    let c be object;
    assume c in Y;
    then
    ex U1 being Subset of S1, U2 being Subset of S2, V1 being Subset of T1
, V2 being Subset of T2 st c = [:U1,V1:] /\ [:U2,V2:] & U1 is open & U2 is open
    & V1 is open & V2 is open;
    hence thesis by A1,A2,A4;
  end;
  then reconsider C1 = Y as Subset-Family of R;
  reconsider C1 as Subset-Family of R;
A5: the topology of [:S2,T2:] = { union A where A is Subset-Family of [:S2,
T2:]: A c= { [:X1,Y1:] where X1 is Subset of S2, Y1 is Subset of T2 : X1 in the
  topology of S2 & Y1 in the topology of T2}} by BORSUK_1:def 2;
A6: the topology of [:S1,T1:] = { union A where A is Subset-Family of [:S1,
T1:]: A c= { [:X1,Y1:] where X1 is Subset of S1, Y1 is Subset of T1 : X1 in the
  topology of S1 & Y1 in the topology of T1}} by BORSUK_1:def 2;
A7: for A being Subset of R st A is open for p being Point of R st p in A
  ex a being Subset of R st a in C1 & p in a & a c= A
  proof
    let A be Subset of R such that
A8: A is open;
    let p be Point of R;
    assume p in A;
    then consider X being Subset of R such that
A9: X in BST and
A10: p in X and
A11: X c= A by A8,YELLOW_9:31;
    consider X1, X2 be set such that
A12: X1 in the topology of [:S1,T1:] and
A13: X2 in the topology of [:S2,T2:] and
A14: X = X1 /\ X2 by A9,SETFAM_1:def 5;
    consider F1 being Subset-Family of [:S1,T1:] such that
A15: X1 = union F1 and
A16: F1 c= { [:K1,L1:] where K1 is Subset of S1, L1 is Subset of T1 :
    K1 in the topology of S1 & L1 in the topology of T1 } by A6,A12;
    p in X1 by A10,A14,XBOOLE_0:def 4;
    then consider G1 being set such that
A17: p in G1 and
A18: G1 in F1 by A15,TARSKI:def 4;
A19: G1 in { [:K1,L1:] where K1 is Subset of S1, L1 is Subset of T1 : K1
    in the topology of S1 & L1 in the topology of T1 } by A16,A18;
    consider F2 being Subset-Family of [:S2,T2:] such that
A20: X2 = union F2 and
A21: F2 c= { [:K2,L2:] where K2 is Subset of S2, L2 is Subset of T2 :
    K2 in the topology of S2 & L2 in the topology of T2 } by A5,A13;
    p in X2 by A10,A14,XBOOLE_0:def 4;
    then consider G2 being set such that
A22: p in G2 and
A23: G2 in F2 by A20,TARSKI:def 4;
    G2 in { [:K2,L2:] where K2 is Subset of S2, L2 is Subset of T2 : K2
    in the topology of S2 & L2 in the topology of T2 } by A21,A23;
    then consider K2 being Subset of S2, L2 being Subset of T2 such that
A24: G2 = [:K2,L2:] and
A25: K2 in the topology of S2 & L2 in the topology of T2;
A26: [:K2,L2:] c= X2 by A20,A23,A24,ZFMISC_1:74;
A27: K2 is open & L2 is open by A25;
    consider K1 being Subset of S1, L1 being Subset of T1 such that
A28: G1 = [:K1,L1:] and
A29: K1 in the topology of S1 & L1 in the topology of T1 by A19;
    reconsider a = [:K1,L1:] /\ [:K2,L2:] as Subset of R by A1,A4,
BORSUK_1:def 2;
    take a;
    K1 is open & L1 is open by A29;
    hence a in C1 by A27;
    thus p in a by A17,A22,A28,A24,XBOOLE_0:def 4;
    [:K1,L1:] c= X1 by A15,A18,A28,ZFMISC_1:74;
    then a c= X by A14,A26,XBOOLE_1:27;
    hence thesis by A11;
  end;
  Y c= the topology of R
  proof
    let c be object;
A30: BST c= the topology of R by TOPS_2:64;
    assume c in Y;
    then consider
    U1 being Subset of S1, U2 being Subset of S2, V1 being Subset of
    T1, V2 being Subset of T2 such that
A31: c = [:U1,V1:] /\ [:U2,V2:] and
A32: U1 is open and
A33: U2 is open and
A34: V1 is open and
A35: V2 is open;
    [:U2,V2:] is open by A33,A35,BORSUK_1:6;
    then
A36: [:U2,V2:] in the topology of [:S2,T2:];
    [:U1,V1:] is open by A32,A34,BORSUK_1:6;
    then [:U1,V1:] in the topology of [:S1,T1:];
    then c in BST by A31,A36,SETFAM_1:def 5;
    hence thesis by A30;
  end;
  hence thesis by A7,YELLOW_9:32;
end;
