reserve x,y for set,
  i for Nat;
reserve V for non empty CLSStruct,
  u,v,v1,v2,v3 for VECTOR of V,
  A for Subset of V,
  l, l1, l2 for C_Linear_Combination of A,
  x,y,y1,y2 for set,
  a,b for Complex,
  F for FinSequence of the carrier of V,
  f for Function of the carrier of V, COMPLEX;
reserve K,L,L1,L2,L3 for C_Linear_Combination of V;

theorem Th28:
  a * (L1 + L2) = a * L1 + a * L2
proof
  let v;
  thus (a * (L1 + L2)).v = a * (L1 + L2).v by Def9
    .= a * (L1.v + L2.v) by Def8
    .= a * L1.v + a * L2.v
    .= (a * L1).v + a * L2.v by Def9
    .= (a * L1).v + (a * L2). v by Def9
    .= ((a * L1) + (a * L2)).v by Def8;
end;
