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;
