reserve p,q,r for FinSequence,
  x,y,y1,y2 for set,
  i,k for Element of NAT,
  GF for add-associative right_zeroed right_complementable Abelian associative
  well-unital distributive non empty doubleLoopStr,
  V for Abelian
  add-associative right_zeroed right_complementable vector-distributive
  scalar-distributive scalar-associative scalar-unital
   non empty ModuleStr over GF,
  u,v,v1,v2,v3,w for Element of V,
  a,b for Element of GF,
  F,G ,H for FinSequence of V,
  A,B for Subset of V,
  f for Function of V, GF;
reserve L,L1,L2,L3 for Linear_Combination of V;
reserve l for Linear_Combination of A;

theorem
  (a + b) * L = a * L + b * L
proof
  let v;
  thus ((a + b) * L).v = (a + b) * L.v by Def9
    .= a * L.v + b * L.v by VECTSP_1:def 7
    .= (a * L).v + b * L.v by Def9
    .= (a * L).v + (b * L). v by Def9
    .= ((a * L) + (b * L)).v by Th22;
end;
