reserve T1,T2,T3 for TopSpace,
  A1 for Subset of T1, A2 for Subset of T2, A3 for Subset of T3;
reserve n,k for Nat;
reserve M,N for non empty TopSpace;
reserve p,q,p1,p2 for Point of TOP-REAL n;
reserve r for Real;

theorem Th22:
  for V being Subset of TOP-REAL n st V = RN_Base n holds
  ex l being Linear_Combination of V st p = Sum l
proof
  let V be Subset of TOP-REAL n;
  assume
A1: V = RN_Base n;
  reconsider p0 = p as Element of RealVectSpace(Seg n) by Lm1;
  reconsider V0 = V as Subset of RealVectSpace(Seg n) by Lm1;
  consider l0 be Linear_Combination of V0 such that
A2: p0 = Sum l0 by A1,EUCLID_7:43;
  reconsider l=l0 as Linear_Combination of TOP-REAL n by Th17;
  Carrier l0 c= V0 by RLVECT_2:def 6;
  then reconsider l as Linear_Combination of V by RLVECT_2:def 6;
  take l;
  thus p = Sum l by A2,Th20;
end;
