reserve x,y,y1,y2 for set,
  p for FinSequence,
  i,k,l,n for Nat,
  V for RealLinearSpace,
  u,v,v1,v2,v3,w for VECTOR of V,
  a,b for Real,
  F,G,H1,H2 for FinSequence of V,
  A,B for Subset of V,
  f for Function of the carrier of V, REAL;
reserve K,L,L1,L2,L3 for Linear_Combination of V;
reserve l,l1,l2 for Linear_Combination of A;

theorem Th45:
  (a + b) * L = a * L + b * L
proof
  let v;
  thus ((a + b) * L).v = (a + b) * L.v by Def11
    .= a * L.v + b * L.v
    .= (a * L).v + b * L.v by Def11
    .= (a * L).v + (b * L). v by Def11
    .= ((a * L) + (b * L)).v by Def10;
end;
