 reserve x,X for set,
         n, m, i for Nat,
         p, q for Point of TOP-REAL n,
         A, B for Subset of TOP-REAL n,
         r, s for Real;
reserve N for non zero Nat,
        u,t for Point of TOP-REAL(N+1);

theorem Th6:
  A is non boundary implies ind A = n
proof
  set TR=TOP-REAL n;
  set E=the empty Subset of TR;
  consider Ia be affinely-independent Subset of TR such that
    E c= Ia & Ia c= [#]TR
    and
A1:   Affin Ia = Affin [#]TR by RLAFFIN1:60;
A2: the TopStruct of TR=TopSpaceMetr Euclid n by EUCLID:def 8;
  then reconsider IA=Ia as finite Subset of Euclid n by TOPMETR:12;
  IA <> {} by A1;
  then consider X be object such that
A3: X in IA by XBOOLE_0:def 1;
  reconsider X as Point of Euclid n by A3;
  reconsider x=X as Point of TR by A2, TOPMETR:12;
A4: dim TR = n by RLAFFIN3:4;
  [#]TR c= Affin [#]TR by RLAFFIN1:49;
  then card Ia = (dim TR)+1 by A1,XBOOLE_0:def 10,RLAFFIN3:6;
  then
A5: ind conv Ia = n by A4,SIMPLEX2:25;
  set d=diameter IA;
A6: ind TR = n by SIMPLEX2:26;
  Ia c= conv Ia by RLAFFIN1:2;
  then
A7: conv Ia c= cl_Ball(X,d) by A3,SIMPLEX2:13;
  assume A is non boundary;
  then Int A <> {} by TOPS_1:48;
  then consider y be object such that
A8: y in Int A by XBOOLE_0:def 1;
  reconsider y as Point of TR by A8;
  reconsider Y=y as Point of Euclid n by A2, TOPMETR:12;
  consider r be Real such that
A9:   r>0
    and
A10: Ball(Y,r) c= A by A8,GOBOARD6:5;
  set r2=r/2;
A11: n in NAT by ORDINAL1:def 12;
A12: Ball(Y,r)=Ball(y,r) by TOPREAL9:13;
  d+0 < d+1 by XREAL_1:6;
  then
A13: cl_Ball(x,d) c= Ball(x,d+1) by A11,JORDAN:21;
  cl_Ball(X,d)=cl_Ball(x,d) by TOPREAL9:14;
  then conv Ia c= Ball(x,d+1) by A13,A7;
  then
A14: n <= ind Ball(x,d+1) by A5,TOPDIM_1:19;
  d>=0 by TBSP_1:21;
  then
A15: cl_Ball(x,d+1) is compact non boundary by Lm2;
  cl_Ball(y,r2) c= Ball(y,r) by A9,XREAL_1:216,A11,JORDAN:21;
  then cl_Ball(y,r2) c= A by A10,A12;
  then
A16:ind cl_Ball(y,r2) <= ind A by TOPDIM_1:19;
  cl_Ball(y,r2) is compact non boundary by A9, Lm2;
  then ex h be Function of TR |cl_Ball(x,d+1),TR |cl_Ball(y,r2) st
    h is being_homeomorphism & h.:Fr cl_Ball(x,d+1) = Fr cl_Ball(y,r2)
    by A15,BROUWER2:7;
  then cl_Ball(x,d+1),cl_Ball(y,r2) are_homeomorphic
    by T_0TOPSP:def 1,METRIZTS:def 1;
  then
A17:ind cl_Ball(x,d+1)=ind cl_Ball(y,r2) by TOPDIM_1:27;
  Ball(x,d+1) c= cl_Ball(x,d+1) by TOPREAL9:16;
  then ind Ball(x,d+1) <= ind cl_Ball(x,d+1) by TOPDIM_1:19;
  then n <= ind cl_Ball(y,r2) by A17,A14, XXREAL_0:2;
  then
A18: n<= ind A by A16,XXREAL_0:2;
  ind A <= ind TR by TOPDIM_1:20;
  hence thesis by A6,XXREAL_0:1,A18;
end;
