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 Th27:
  (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
    .= (a * L).v + b * L.v by Def9
    .= (a * L).v + (b * L). v by Def9
    .= ((a * L) + (b * L)).v by Def8;
end;
