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;

theorem
  for V be non empty addLoopStr, L be Linear_Combination of V, v be
  Element of V holds L.v = 0 iff not v in Carrier(L)
proof
  let V be non empty addLoopStr, L be Linear_Combination of V, v be Element of
  V;
  thus L.v = 0 implies not v in Carrier(L)
  proof
    assume
A1: L.v = 0;
    assume not thesis;
    then ex u be Element of V st u = v & L.u <> 0;
    hence thesis by A1;
  end;
  thus thesis;
end;
