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 Th62:
  vector(LC_RLSpace(V),L1) + vector(LC_RLSpace(V),L2) = L1 + L2
proof
  set v2 = vector(LC_RLSpace(V),L2);
A1: L1 = @@L1 & L2 = @@L2;
  L2 in the carrier of LC_RLSpace(V) by Def14;
  then
A2: L2 in LC_RLSpace(V);
  L1 in the carrier of LC_RLSpace(V) by Def14;
  then L1 in LC_RLSpace(V);
  hence vector(LC_RLSpace(V),L1) + vector(LC_RLSpace(V),L2) = LCAdd(V).[L1,v2]
  by Def1
    .= LCAdd(V).(@L1,@L2) by A2,Def1
    .= L1 + L2 by A1,Def17;
end;
