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;
reserve e,e1,e2 for Element of LinComb(V);

theorem Th63:
  a * vector(LC_RLSpace(V),L) = a * L
proof
A1: @@L = L;
  L in the carrier of LC_RLSpace(V) by Def14;
  then L in LC_RLSpace(V);
  hence a * vector(LC_RLSpace(V),L) = LCMult(V).[a,@L] by Def1
    .= a * L by A1,Def18;
end;
