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 Th22:
  (L1+L2).v = L1.v + L2.v
  proof
    dom L1 = the carrier of V & dom L2 = the carrier of V by FUNCT_2:def 1;
    then
A1: L1/.v = L1.v & L2/.v = L2.v by PARTFUN1:def 6;
A2: dom (L1+L2) = the carrier of V by FUNCT_2:def 1;
    hence (L1+L2).v = (L1+L2)/.v by PARTFUN1:def 6
    .= L1.v + L2.v by A1,A2,VFUNCT_1:def 1;
  end;
