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 Th46:
  a * (L1 + L2) = a * L1 + a * L2
proof
  let v;
  thus (a * (L1 + L2)).v = a * (L1 + L2).v by Def11
    .= a * (L1.v + L2.v) by Def10
    .= a * L1.v + a * L2.v
    .= (a * L1).v + a * L2.v by Def11
    .= (a * L1).v + (a * L2). v by Def11
    .= ((a * L1) + (a * L2)).v by Def10;
end;
