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
  for U be Subspace of n-VectSp_over F_Real,
      W be Subspace of TOP-REAL n st [#]U = [#]W
  holds
    X is Linear_Combination of U
  iff
    X is Linear_Combination of W
proof
  set V=n-VectSp_over F_Real;
  set T=TOP-REAL n;
  let U be Subspace of V,W be Subspace of T such that
   A1: [#]U=[#]W;
  hereby assume X is Linear_Combination of U;
   then reconsider L=X as Linear_Combination of U;
   ex S be finite Subset of U st
     for v be Element of U st not v in S holds L.v=0.F_Real
     by VECTSP_6:def 1;
   hence X is Linear_Combination of W by A1,RLVECT_2:def 3;
  end;
  assume X is Linear_Combination of W;
  then reconsider L=X as Linear_Combination of W;
  consider S be finite Subset of W such that
   A2: for v be Element of W st not v in S holds L.v=0 by RLVECT_2:def 3;
  for v be Element of U st not v in S holds 0.F_Real=L.v by A1,A2;
  hence thesis by A1,VECTSP_6:def 1;
end;
