  reserve n,m,i for Nat,
          p,q for Point of TOP-REAL n,
          r,s for Real,
          R for real-valued FinSequence;
reserve T1,T2,S1,S2 for non empty TopSpace,
        t1 for Point of T1, t2 for Point of T2,
        pn,qn for Point of TOP-REAL n,
        pm,qm for Point of TOP-REAL m;

theorem
  r >0 & s>0 &
    T1,(TOP-REAL n)|Ball(pn,r) are_homeomorphic &
    T2,(TOP-REAL m)|Ball(pm,s) are_homeomorphic
implies
  [:T1,T2:],(TOP-REAL (n+m))|Ball(0.TOP-REAL (n+m),1) are_homeomorphic
proof
  reconsider N=n,M=m as Element of NAT by ORDINAL1:def 12;
  set TRn=TOP-REAL N,TRm=TOP-REAL M, NM=N+M,TRnm=TOP-REAL NM;
  reconsider Pn=pn as Point of TRn;
  reconsider Pm=pm as Point of TRm;
  assume that
A1: r >0
  and
A2: s>0;
  reconsider Bm=Ball(Pm,s) as non empty Subset of TRm by A2;
A3: [#]Tdisk(Pm,s) = cl_Ball(Pm,s) by BROUWER:3;
  then reconsider BBm = Ball(Pm,s) as Subset of Tdisk(Pm,s) by TOPREAL9:16;
A4:Tdisk(Pm,s) = TRm|cl_Ball(Pm,s) by BROUWER:def 2;
  then
A5:Tdisk(Pm,s) |BBm = TRm|Ball(Pm,s) by A3,PRE_TOPC:7;
  reconsider Bn=Ball(Pn,r) as non empty Subset of TRn by A1;
  reconsider Bnm=Ball(0.TRnm,1) as non empty Subset of TRnm;
A6: [#]Tdisk(Pn,r) = cl_Ball(pn,r) by BROUWER:3;
  then reconsider BBn = Ball(Pn,r) as Subset of Tdisk(Pn,r) by TOPREAL9:16;
A7: [#]Tdisk(0.TRnm,1) = cl_Ball(0.TRnm,1) by BROUWER:3;
  then reconsider BBnm = Ball(0.TRnm,1) as Subset of Tdisk(0.TRnm,1)
    by TOPREAL9:16;
A8:Tdisk(0.TRnm,1) = TRnm|cl_Ball(0.TRnm,1) by BROUWER:def 2;
  then
A9:Tdisk(0.TRnm,1) |BBnm = TRnm|Ball(0.TRnm,1) by A7, PRE_TOPC:7;
  assume T1,(TOP-REAL n)|Ball(pn,r) are_homeomorphic;
  then consider f1 be Function of T1,TRn|Bn such that
A10: f1 is being_homeomorphism by T_0TOPSP:def 1;
  assume T2,(TOP-REAL m)|Ball(pm,s) are_homeomorphic;
  then consider f2 be Function of T2,TRm|Bm such that
A11: f2 is being_homeomorphism by T_0TOPSP:def 1;
A12: [:f1,f2:] is being_homeomorphism by A10,A11,Th14;
  consider h be Function of [: Tdisk(Pn,r),Tdisk(Pm,s):], Tdisk(0.TRnm,1) such
    that
A13: h is being_homeomorphism
  and
A14: h.:[: Ball(Pn,r), Ball(Pm,s):] = Ball(0.TRnm,1) by A1,A2, Th18;
A15:Tdisk(Pn,r) = TRn|cl_Ball(Pn,r) by BROUWER:def 2;
  then Tdisk(Pn,r) |BBn = TRn|Ball(Pn,r) by A6,PRE_TOPC:7;
  then
A16:[: Tdisk(Pn,r),Tdisk(Pm,s):] | [:BBn,BBm:] = [: TRn|Bn,TRm|Bm:] by A5,
  BORSUK_3:22;
  then reconsider h1=h| [:BBn,BBm:] as Function of [: TRn|Bn,TRm|Bm:],TRnm|Bnm
    by JORDAN24:12,A9,A14, A1,A15, A2,A4, A8;
  reconsider hf=h1*[:f1,f2:] as Function of [:T1,T2:],TRnm|Bnm;
  h1 is being_homeomorphism by A9,A16,A13,A14, METRIZTS:2;
  then hf is being_homeomorphism by A12, TOPS_2:57;
  hence thesis by T_0TOPSP:def 1;
end;
