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 Th19:
  Carrier(L1 + L2) c= Carrier L1 \/ Carrier L2
proof
  let x be object;
  assume x in Carrier(L1 + L2);
  then consider u such that
A1: x = u and
A2: (L1 + L2).u <> 0c;
  (L1 + L2).u = L1.u + L2.u by Def8;
  then L1.u <> 0c or L2.u <> 0c by A2;
  then x in Carrier L1 or x in Carrier L2 by A1;
  hence thesis by XBOOLE_0:def 3;
end;
