reserve x,y,y1,y2 for set,
  p for FinSequence,
  i,k,l,n for Nat,
  V for RealLinearSpace,
  u,v,v1,v2,v3,w for VECTOR of V,
  a,b for Real,
  F,G,H1,H2 for FinSequence of V,
  A,B for Subset of V,
  f for Function of the carrier of V, REAL;
reserve K,L,L1,L2,L3 for Linear_Combination of V;
reserve l,l1,l2 for Linear_Combination of A;

theorem Th37:
  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 <> 0;
  (L1 + L2).u = L1.u + L2.u by Def10;
  then L1.u <> 0 or L2.u <> 0 by A2;
  then x in {v1 : L1.v1 <> 0} or x in {v2 : L2.v2 <> 0} by A1;
  hence thesis by XBOOLE_0:def 3;
end;
