reserve x,y for set,
  i for Nat;
reserve V for non empty CLSStruct,
  u,v,v1,v2,v3 for VECTOR of V,
  A for Subset of V,
  l, l1, l2 for C_Linear_Combination of A,
  x,y,y1,y2 for set,
  a,b for Complex,
  F for FinSequence of the carrier of V,
  f for Function of the carrier of V, COMPLEX;
reserve K,L,L1,L2,L3 for C_Linear_Combination of V;
reserve e,e1,e2 for Element of C_LinComb V;

theorem Th38:
  vector(LC_CLSpace V,L1) + vector(LC_CLSpace V,L2) = L1 + L2
proof
  set v2 = vector(LC_CLSpace V,L2);
A1: L1 = @@L1 & L2 = @@L2;
  L2 in the carrier of LC_CLSpace V by Def12;
  then
A2: L2 in LC_CLSpace V;
  L1 in the carrier of LC_CLSpace V by Def12;
  then L1 in LC_CLSpace V;
  hence
  vector(LC_CLSpace V,L1) + vector(LC_CLSpace V,L2) = (C_LCAdd V).[L1,v2]
  by RLVECT_2:def 1
    .= (C_LCAdd V).(@L1,@L2) by A2,RLVECT_2:def 1
    .= L1 + L2 by A1,Def15;
end;
