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 Th49:
  the carrier of S1 = the carrier of S2 & the carrier of T1 = the
  carrier of T2 implies the carrier of [:R1,R2:] = the carrier of R & the
  topology of [:R1,R2:] = the topology of R
proof
  assume that
A1: the carrier of S1 = the carrier of S2 and
A2: the carrier of T1 = the carrier of T2;
A3: the carrier of R1 = (the carrier of S1) \/ the carrier of S2 by
YELLOW_9:def 6
    .= the carrier of S1 by A1;
  set C = { [: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 };
  reconsider BT = INTERSECTION(the topology of T1, the topology of T2) as
  Basis of R2 by A2,YELLOW_9:60;
  reconsider BS = INTERSECTION(the topology of S1, the topology of S2) as
  Basis of R1 by A1,YELLOW_9:60;
  reconsider Bpr = {[:a,b:] where a is Subset of R1, b is Subset of R2: a in
  BS & b in BT} as Basis of [:R1,R2:] by YELLOW_9:40;
A4: C = Bpr
  proof
    hereby
      let c be object;
      assume c in C;
      then consider
      U1 being Subset of S1, U2 being Subset of S2, V1 being Subset
      of T1, V2 being Subset of T2 such that
A5:   c = [:U1,V1:] /\ [:U2,V2:] and
A6:   U1 is open & U2 is open and
A7:   V1 is open & V2 is open;
      U1 in the topology of S1 & U2 in the topology of S2 by A6;
      then
A8:   U1 /\ U2 in BS by SETFAM_1:def 5;
      V1 in the topology of T1 & V2 in the topology of T2 by A7;
      then
A9:   V1 /\ V2 in BT by SETFAM_1:def 5;
      c = [:U1 /\ U2, V1 /\ V2:] by A5,ZFMISC_1:100;
      hence c in Bpr by A8,A9;
    end;
    let c be object;
    assume c in Bpr;
    then consider a being Subset of R1, b being Subset of R2 such that
A10: c = [:a,b:] and
A11: a in BS and
A12: b in BT;
    consider a1, a2 being set such that
A13: a1 in the topology of S1 and
A14: a2 in the topology of S2 and
A15: a = a1 /\ a2 by A11,SETFAM_1:def 5;
    reconsider a2 as Subset of S2 by A14;
    reconsider a1 as Subset of S1 by A13;
A16: a1 is open & a2 is open by A13,A14;
    consider b1, b2 being set such that
A17: b1 in the topology of T1 and
A18: b2 in the topology of T2 and
A19: b = b1 /\ b2 by A12,SETFAM_1:def 5;
    reconsider b2 as Subset of T2 by A18;
    reconsider b1 as Subset of T1 by A17;
A20: b1 is open & b2 is open by A17,A18;
    c = [:a1,b1:] /\ [:a2,b2:] by A10,A15,A19,ZFMISC_1:100;
    hence thesis by A16,A20;
  end;
A21: the carrier of R2 = (the carrier of T1) \/ the carrier of T2 by
YELLOW_9:def 6
    .= the carrier of T1 by A2;
A22: the carrier of [:S1,T1:] = [:the carrier of S1, the carrier of T1:] by
BORSUK_1:def 2;
  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 A22,BORSUK_1:def 2
    .= [:the carrier of S1,the carrier of T1:] by A1,A2;
  hence
A23: the carrier of [:R1,R2:] = the carrier of R by A3,A21,BORSUK_1:def 2;
  C is Basis of R by A1,A2,Th48;
  hence thesis by A23,A4,YELLOW_9:25;
end;
