theorem Th47:
  vector(LC_Z_Module(V),L1) + vector(LC_Z_Module(V),L2) = L1 + L2
  proof
    set v2 = vector(LC_Z_Module(V),L2);
    A1: L1 = @@L1 & L2 = @@L2;
    L2 in the carrier of LC_Z_Module(V) by Def29;
    then
    A2: L2 in LC_Z_Module(V);
    L1 in the carrier of LC_Z_Module(V) by Def29;
    then L1 in LC_Z_Module(V);
    hence vector(LC_Z_Module(V),L1)
    + vector(LC_Z_Module(V),L2) = LCAdd(V). [L1,v2] by RLVECT_2:def 1
    .= LCAdd(V).(@L1,@L2) by A2,RLVECT_2:def 1
    .= L1 + L2 by A1,Def32;
  end;
