 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 Th5:
  for V be finite-dimensional RealLinearSpace
  for A be affinely-independent Subset of V holds card A <= 1+dim V
proof
  let V be finite-dimensional RealLinearSpace;
  let A be affinely-independent Subset of V;
  per cases;
  suppose A is empty;
   hence thesis;
  end;
  suppose A is non empty;
   then consider v be VECTOR of V such that
    v in A and
    A1: -v+A\{0.V} is linearly-independent by RLAFFIN1:def 4;
   set vA=-v+A;
   vA\{0.V} misses {0.V} by XBOOLE_1:79;
   then
   A2: card{0.V}=1 & card(vA\{0.V}\/{0.V})=card(vA\{0.V})+card{0.V}
     by CARD_2:40,42;
   A3: card vA=card A by RLAFFIN1:7;
   reconsider vA as finite set;
   card(vA\{0.V})=dim Lin(-v+A\{0.V}) by A1,RLVECT_5:29;
   then card(vA\{0.V})<=dim V by RLVECT_5:28;
   then A4: card(vA\{0.V}\/{0.V})<=1+dim V by A2,XREAL_1:7;
   vA\{0.V}\/{0.V}=vA\/{0.V} by XBOOLE_1:39;
   then card A<=card(vA\{0.V}\/{0.V}) by A3,NAT_1:43,XBOOLE_1:7;
   hence thesis by A4,XXREAL_0:2;
  end;
end;
