reserve X for set,
        n,m,k for Nat,
        K for Field,
        f for n-element real-valued FinSequence,
        M for Matrix of n,m,F_Real;

theorem Th1:
  X is Linear_Combination of n-VectSp_over F_Real
iff
  X is Linear_Combination of TOP-REAL n
proof
  set V=n-VectSp_over F_Real;
  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.F_Real
    by VECTSP_6:def 1;
   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.F_Real=L.v by A4,A6;
  end;
  L is Element of Funcs(the carrier of V,the carrier of F_Real) &
   S is finite Subset of V by Lm1;
  hence thesis by A5,VECTSP_6:def 1;
end;
