 reserve n,m for Nat,
         p,q for Point of TOP-REAL n, r for Real;

theorem Th3: ::MFOLD_1:9
  for r,s being positive Real holds
     Tball(p,r), Tball(q,s) are_homeomorphic
proof
  set TR=TOP-REAL n;
  let r,s be positive Real;
  Ball(q,s) c= Int cl_Ball(q,s) by TOPREAL9:16,TOPS_1:24;
  then A2: cl_Ball(q,s) is non boundary compact;
  Ball(p,r) c= Int cl_Ball(p,r) by TOPREAL9: 16,TOPS_1:24;
  then cl_Ball(p,r) is non boundary compact;
  then consider h be Function of TR |cl_Ball(p,r),TR |cl_Ball(q,s) such that
  A4: h is being_homeomorphism & h.:Fr cl_Ball(p,r) = Fr cl_Ball(q,s)
  by A2,BROUWER2:7;
  A5:[#](TR|cl_Ball(q,s))=cl_Ball(q,s) by PRE_TOPC:def 5;
  A6:h.:(dom h) = rng h by RELAT_1:113
               .= [#](TR|cl_Ball(q,s)) by FUNCT_2:def 3,A4;
  A7: Fr cl_Ball(p,r) = Sphere(p,r) by BROUWER2:5;
  A8:[#](TR|cl_Ball(p,r)) = cl_Ball(p,r) by PRE_TOPC:def 5;
  A9:Ball(q,s) \/ Sphere(q,s) = cl_Ball(q,s) & Ball(q,s) misses Sphere(q,s)
      by TOPREAL9:18,19;
  A10:Ball(p,r) \/ Sphere(p,r) = cl_Ball(p,r)& Ball(p,r) misses Sphere(p,r)
      by TOPREAL9:18,19;
  reconsider Br=Ball(p,r) as Subset of TR |cl_Ball(p,r) by A10,A8,XBOOLE_1:7;
  reconsider Bs=Ball(q,s) as Subset of TR |cl_Ball(q,s) by A9,A5,XBOOLE_1:7;
  A12:dom h = [#] (TR|cl_Ball(p,r)) by FUNCT_2:def 1;
  A13:Ball(q,s) \/ Sphere(q,s) = cl_Ball(q,s) &
  Ball(q,s) misses Sphere(q,s) by TOPREAL9:18,19;
  A14:Ball(p,r) \/ Sphere(p,r) = cl_Ball(p,r) &
  Ball(p,r) misses Sphere(p,r) by TOPREAL9:18,19;
  dom h \ Sphere(p,r) =  Ball(p,r) by A12,A8,XBOOLE_1:88,A14;
  then h.:Br = h.:(dom h) \ (Fr cl_Ball(q,s)) by A4,FUNCT_1:64,A7
  .= cl_Ball(q,s) \ Sphere(q,s) by BROUWER2:5,A6,A5
  .=Bs by A13,XBOOLE_1:88;
  then TR |cl_Ball(p,r)|Br,TR |cl_Ball(q,s)|Bs are_homeomorphic
    by A4,METRIZTS:3,METRIZTS:def 1;
  then TR|Ball(p,r),TR|cl_Ball(q,s)|Bs are_homeomorphic by PRE_TOPC:7,A8;
  hence thesis by PRE_TOPC:7,A5;
end;
