  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 Th18:
  for pn be Point of TOP-REAL n,pm be Point of TOP-REAL m
  for r,s st r>0 & s>0
   ex h be Function of [: Tdisk(pn,r),Tdisk(pm,s):],Tdisk(0.TOP-REAL (n+m),1)
    st h is being_homeomorphism &
       h.:[:Ball(pn,r), Ball(pm,s):] = Ball(0.TOP-REAL (n+m),1)
proof
A1:n in NAT & m in NAT & n+m in NAT by ORDINAL1:def 12;
set TRn=TOP-REAL n,TRm=TOP-REAL m, nm=n+m,TRnm=TOP-REAL nm;
  let pn be Point of TRn,pm be Point of TRm;
  let r,s such that
A2: r>0
  and
A3: s>0;
  reconsider CLn=cl_Ball(pn,r) as non empty Subset of TRn by A2;
A4:Int CLn = CLn\Fr CLn by TOPS_1:40
          .=CLn \Sphere(pn,r) by A2, BROUWER2:5
          .= CLn \(CLn\Ball(pn,r)) by A1,JORDAN:19
          .= CLn /\Ball(pn,r) by XBOOLE_1:48
          .=Ball(pn,r) by TOPREAL9: 16,XBOOLE_1:28;
  reconsider CLm=cl_Ball(pm,s) as non empty Subset of TRm by A3;
A5:Int CLm = CLm\Fr CLm by TOPS_1:40
          .= CLm \Sphere(pm,s) by A3, BROUWER2:5
          .= CLm \(CLm\Ball(pm,s)) by A1,JORDAN:19
          .= CLm /\Ball(pm,s) by XBOOLE_1:48
          .= Ball(pm,s) by TOPREAL9: 16,XBOOLE_1:28;
  reconsider CLnm=cl_Ball(0.TRnm,1) as non empty Subset of TRnm;
A6: TRn |CLn = Tdisk(pn,r) by BROUWER:def 2;
A7: Ball(0.TRnm,1) c= CLnm by TOPREAL9: 16;
A8: [#]Tdisk(0.TRnm,1) = CLnm by A1,BROUWER:3;
A9: TRnm|CLnm = Tdisk(0.TRnm, 1) by BROUWER:def 2;
A10:Int CLnm = CLnm\Fr CLnm by TOPS_1:40
           .= CLnm \Sphere(0.TRnm,1) by BROUWER2:5
           .= CLnm \(CLnm\Ball(0.TRnm,1)) by A1,JORDAN:19
           .= CLnm /\Ball(0.TRnm,1) by XBOOLE_1:48
           .=Ball(0.TRnm,1) by TOPREAL9: 16,XBOOLE_1:28;
  set Rn=ClosedHypercube(0.TRn,n|->1),Rm=ClosedHypercube(0.TRm,m|->1),
      Rnm=ClosedHypercube(0.TRnm,nm|->1);
A11: [#](TRm |Rm) = Rm by PRE_TOPC:def 5;
A12: Ball(pn,r) c= CLn by TOPREAL9: 16;
  CLn is non boundary convex compact Subset of TRn by A2,Lm1;
  then consider fn be Function of TRn |CLn,TRn |Rn such that
A13: fn is being_homeomorphism
  and
A14: fn.:Fr CLn = Fr Rn by BROUWER2:7;
A15:fn.:Int CLn misses fn.:Fr CLn by TOPS_1:39, A13,FUNCT_1:66;
  [#](TRn |Rn) = Rn by PRE_TOPC:def 5;
  then
A16: rng fn = Rn by A13, TOPS_2:def 5;
  CLm is non boundary convex compact by A3,Lm1;
  then consider gm be Function of TRm |CLm,TRm |Rm such that
A17: gm is being_homeomorphism
  and
A18: gm.:Fr CLm = Fr Rm by BROUWER2:7;
A19: TRm |CLm = Tdisk(pm,s) by BROUWER:def 2;
  [#](TRm |Rm) = Rm by PRE_TOPC:def 5;
  then
A20: rng gm = Rm by A17, TOPS_2:def 5;
A21: Ball(pm,s) c= CLm by TOPREAL9: 16;
A22: [#]Tdisk(pn,r) = CLn by A1,BROUWER:3;
  then
A23: dom fn = CLn by A6,A13, TOPS_2:def 5;
  CLnm is non boundary convex compact by Lm1;
  then consider P be Function of [:Tdisk(pn,r),Tdisk(pm,s):], Tdisk(0.TRnm,1)
    such that
A24: P is being_homeomorphism
  and
A25: for t1 be Point of TRn |CLn, t2 be Point of TRm |CLm holds fn.t1 in
   Int Rn & gm.t2 in Int Rm iff P. (t1,t2) in Int CLnm
  by A6,A19,A9,Th17,A13,A17;
  take P;
  thus P is being_homeomorphism by A24;
A26:gm.:Int CLm misses gm.:Fr CLm by TOPS_1:39, A17,FUNCT_1:66;
A27: [#]Tdisk(pm,s) = CLm by A1,BROUWER:3;
  then
A28: dom gm = CLm by A19,A17, TOPS_2:def 5;
A29: dom gm = CLm by A19,A17,A27, TOPS_2:def 5;
  thus P.:[: Ball(pn,r), Ball(pm,s):] c= Ball(0.TRnm,1)
  proof
    let y be object;
    assume y in P.:[: Ball(pn,r), Ball(pm,s):];
    then consider x be object such that
      x in dom P
    and
A30:  x in [: Ball(pn,r), Ball(pm,s):]
    and
A31:  P.x=y by FUNCT_1:def 6;
    consider y1,y2 be object such that
A32:  y1 in Ball(pn,r)
    and
A33:  y2 in Ball(pm,s)
    and
A34:  x = [y1,y2] by A30,ZFMISC_1:def 2;
    reconsider y1 as Point of TRn |CLn by A32,A12,A1,BROUWER:3,A6;
    fn.y1 in fn.:Int CLn by A32,A4,A12,A23,FUNCT_1:def 6;
    then
A35:  not fn.y1 in Fr Rn by A15,XBOOLE_0:3,A14;
    reconsider y2 as Point of TRm |CLm by A33,A21,A1,BROUWER:3,A19;
    gm.y2 in gm.:Int CLm by A33,A5,A21,A29,FUNCT_1:def 6;
    then not gm.y2 in Fr Rm by XBOOLE_0:3,A26,A18;
    then gm.y2 in Rm\Fr Rm by A11,XBOOLE_0:def 5;
    then
A36:  gm.y2 in Int Rm by TOPS_1:40;
    fn.y1 in Rn by A32,A12,A23,FUNCT_1:def 3,A16;
    then fn.y1 in Rn\Fr Rn by A35,XBOOLE_0:def 5;
    then fn.y1 in Int Rn by TOPS_1:40;
    then P. (y1,y2) in Int CLnm by A25,A36;
    hence thesis by A31,A34,A10;
  end;
  let y be object;
  assume
A37:y in Ball(0.TRnm,1);
  rng P = [#]Tdisk(0.TRnm,1) by TOPS_2:def 5,A24;
  then consider x be object such that
A38: x in dom P
  and
A39: P.x=y by A37,A7,A8,FUNCT_1:def 3;
  [#][:Tdisk(pn,r),Tdisk(pm,s):] = [: [#]Tdisk(pn,r),[#]Tdisk(pm,s):] by
  BORSUK_1:def 2;
  then consider y1,y2 be object such that
A40: y1 in CLn
  and
A41: y2 in CLm
  and
A42: x=[y1,y2] by A38,A22,A27,ZFMISC_1:def 2;
  reconsider y2 as Point of TRm |CLm by A19, A1,BROUWER:3,A41;
  reconsider y1 as Point of TRn |CLn by A6,A40,A1,BROUWER:3;
A43: P.x = P.(y1,y2) by A42;
  then fn.y1 in Int Rn by A25,A39, A37,A10;
  then fn.y1 in Rn\fn.:Fr CLn by TOPS_1:40,A14;
  then fn.y1 in (fn.:CLn)\fn.:Fr CLn by RELAT_1: 113,A23,A16;
  then fn.y1 in fn.:(CLn\Fr CLn) by A13,FUNCT_1: 64;
  then fn.y1 in fn.:(Int CLn) by TOPS_1:40;
  then ex z1 be object st z1 in dom fn & z1 in Int CLn & fn.z1 = fn.y1 by
  FUNCT_1:def 6;
  then
A44:y1 in Int CLn by A23,A40, A13, FUNCT_1:def 4;
  gm.y2 in Int Rm by A43,A25,A39, A37,A10;
  then gm.y2 in Rm\gm.:Fr CLm by TOPS_1:40,A18;
  then gm.y2 in (gm.:CLm)\gm.:Fr CLm by RELAT_1: 113,A28,A20;
  then gm.y2 in gm.:(CLm\Fr CLm) by A17,FUNCT_1: 64;
  then gm.y2 in gm.:(Int CLm) by TOPS_1:40;
  then ex z2 be object st z2 in dom gm & z2 in Int CLm & gm.z2 = gm.y2
    by FUNCT_1:def 6;
  then y2 in Int CLm by A28,A41, A17, FUNCT_1:def 4;
  then [y1,y2] in [:Ball(pn,r), Ball(pm,s):] by A44,A5,A4,ZFMISC_1:87;
  hence thesis by A38,A39,A42,FUNCT_1:def 6;
end;
