 reserve x for set,
         n,m,k for Nat,
         r for Real,
         V for RealLinearSpace,
         v,u,w,t for VECTOR of V,
         Av for finite Subset of V,
         Affv for finite affinely-independent Subset of V;
reserve pn for Point of TOP-REAL n,
        An for Subset of TOP-REAL n,
        Affn for affinely-independent Subset of TOP-REAL n,
        Ak for Subset of TOP-REAL k;

theorem Th11:
  0*n in An implies Affin An = [#]Lin An
proof
  set TR=TOP-REAL n;
  set A=An;
  A1: 0*n=0.TR & A c=Affin A by EUCLID:66,RLAFFIN1:49;
  assume 0*n in A;
  hence Affin A=0.TR+Up Lin(-0.TR+A) by A1,RLAFFIN1:57
   .=Up Lin(-0.TR+A) by RLAFFIN1:6
   .=Up Lin(0.TR+A) by RLVECT_1:12
   .=Up Lin A by RLAFFIN1:6
   .=[#]Lin A by RUSUB_4:def 5;
end;
