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 Th17:
  for X being set holds
  X is Linear_Combination of RealVectSpace(Seg n)
iff
  X is Linear_Combination of TOP-REAL n
proof
  let X be set;
  set V=RealVectSpace(Seg n);
  set T=TOP-REAL n;
  hereby assume X is Linear_Combination of V;
    then reconsider L=X as Linear_Combination of V;
    consider S be finite Subset of V such that
A1: for v be Element of V st not v in S holds L.v = 0
    by RLVECT_2:def 3;
A2: now let v be Element of T;
      assume A3: not v in S;
      v is Element of V by Lm1;
      hence 0=L.v by A1,A3;
    end;
    (L is Element of Funcs(the carrier of T,REAL)) & S is finite Subset of T
    by Lm1;
    hence X is Linear_Combination of T by A2,RLVECT_2:def 3;
  end;
  assume X is Linear_Combination of T;
  then reconsider L=X as Linear_Combination of T;
  consider S be finite Subset of T such that
A4: for v be Element of T st not v in S holds L.v=0 by RLVECT_2:def 3;
A5: now let v be Element of V;
   assume
A6: not v in S;
    v is Element of T by Lm1;
    hence 0=L.v by A4,A6;
  end;
  L is Element of Funcs(the carrier of V,REAL) &
  S is finite Subset of V by Lm1;
  hence thesis by A5,RLVECT_2:def 3;
end;
