reserve x for set,
  K for Ring,
  r for Scalar of K,
  V for LeftMod of K,
  a,b,a1,a2 for Vector of V,
  A,A1,A2 for Subset of V,
  l for Linear_Combination of A,
  W for Subspace of V,
  Li for FinSequence of Submodules(V);

theorem
  K is trivial implies (for l holds Carrier(l) = {}) & Lin A is trivial
proof
  assume
A1: K is trivial;
  thus
A2: for l holds Carrier l = {}
  proof
    let l;
    assume
A3: Carrier l <> {};
    set x = the Element of Carrier l;
    ex a st ( x = a)&( l.a <> 0.K) by A3,VECTSP_6:1;
    hence contradiction by A1;
  end;
  now
    let a be Vector of Lin A;
    a in Lin A;
    then consider l such that
A4: a = Sum(l) by MOD_3:4;
    Carrier l = {} by A2;
    then a = 0.V by A4,VECTSP_6:19;
    hence a=0.(Lin A) by VECTSP_4:11;
  end;
  hence thesis;
end;
