
theorem Th13:
  for S, T being TopSpace holds the TopStruct of [:S,T:] = [:the
  TopStruct of S,the TopStruct of T:]
proof
  let S, T be TopSpace;
A1: the carrier of [:S,T:] = [:the carrier of S,the carrier of T:] by
BORSUK_1:def 2
    .= the carrier of [:the TopStruct of S,the TopStruct of T:] by
BORSUK_1:def 2;
  for C1 being Subset of [:S,T:], C2 being Subset of [:the TopStruct of S,
  the TopStruct of T:] st C1 = C2 holds C1 is open iff C2 is open
  proof
    let C1 be Subset of [:S,T:];
    let C2 be Subset of [:the TopStruct of S,the TopStruct of T:];
    assume
A2: C1 = C2;
    hereby
      assume C1 is open;
      then consider A being Subset-Family of [:S,T:] such that
A3:   C1 = union A and
A4:   for e being set st e in A ex X1 being Subset of S, Y1 being
      Subset of T st e = [:X1,Y1:] & X1 is open & Y1 is open by BORSUK_1:5;
      reconsider AA = A as Subset-Family of [:the TopStruct of S,the TopStruct
      of T:] by A1;
      for e being set st e in AA ex X1 being Subset of the TopStruct of S,
Y1 being Subset of the TopStruct of T st e = [:X1,Y1:] & X1 is open & Y1 is
      open
      proof
        let e be set;
        assume e in AA;
        then consider X1 being Subset of S, Y1 being Subset of T such that
A5:     e = [:X1,Y1:] & X1 is open & Y1 is open by A4;
        reconsider Y2 = Y1 as Subset of the TopStruct of T;
        reconsider X2 = X1 as Subset of the TopStruct of S;
        take X2, Y2;
        thus thesis by A5,TOPS_3:76;
      end;
      hence C2 is open by A2,A3,BORSUK_1:5;
    end;
    assume C2 is open;
    then consider
    A being Subset-Family of [:the TopStruct of S,the TopStruct of T
    :] such that
A6: C2 = union A and
A7: for e being set st e in A ex X1 being Subset of the TopStruct of
S, Y1 being Subset of the TopStruct of T st e = [:X1,Y1:] & X1 is open & Y1 is
    open by BORSUK_1:5;
    reconsider AA = A as Subset-Family of [:S,T:] by A1;
    for e being set st e in AA ex X1 being Subset of S, Y1 being Subset
    of T st e = [:X1,Y1:] & X1 is open & Y1 is open
    proof
      let e be set;
      assume e in AA;
      then consider
      X1 being Subset of the TopStruct of S, Y1 being Subset of the
      TopStruct of T such that
A8:   e = [:X1,Y1:] & X1 is open & Y1 is open by A7;
      reconsider Y2 = Y1 as Subset of T;
      reconsider X2 = X1 as Subset of S;
      take X2, Y2;
      thus thesis by A8,TOPS_3:76;
    end;
    hence thesis by A2,A6,BORSUK_1:5;
  end;
  hence thesis by A1,TOPS_3:72;
end;
