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;

theorem Th20:
  L1 is C_Linear_Combination of A & L2 is C_Linear_Combination of
  A implies L1 + L2 is C_Linear_Combination of A
proof
  assume L1 is C_Linear_Combination of A & L2 is C_Linear_Combination of A;
  then Carrier L1 c= A & Carrier L2 c= A by Def4;
  then
A1: Carrier L1 \/ Carrier L2 c= A by XBOOLE_1:8;
  Carrier(L1 + L2) c= Carrier L1 \/ Carrier L2 by Th19;
  hence Carrier(L1 + L2) c= A by A1;
end;
