reserve M for non empty MetrSpace,
        F,G for open Subset-Family of TopSpaceMetr M;
reserve L for Lebesgue_number of F;
reserve n,k for Nat,
        r for Real,
        X for set,
        M for Reflexive non empty MetrStruct,
        A for Subset of M,
        K for SimplicialComplexStr;
reserve V for RealLinearSpace,
        Kv for non void SimplicialComplex of V;
reserve A for Subset of TOP-REAL n;
reserve A for affinely-independent Subset of TOP-REAL n;

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;
