theorem
  ind TOP-REAL n = n
  proof
   set T=TOP-REAL n;
   consider I be affinely-independent Subset of T such that
   A1:  {}T c= I & I c= [#]T & Affin I = Affin [#]T by RLAFFIN1:60;
   [#]T is Affine by RUSUB_4:22;
   then dim T = n & Affin [#]T = [#]T by RLAFFIN1:50,RLAFFIN3:4;
   then card I = n+1 by A1,RLAFFIN3:6;
   then ind conv I=n by Th25;
   then ind T >= n & ind T <= n by TOPDIM_1:20,TOPDIM_2:21;
   hence thesis by XXREAL_0:1;
 end;
