reserve R for Ring,
  V for RightMod of R,
  a,b for Scalar of R,
  x,y for set,
  p,q ,r for FinSequence,
  i,k for Nat,
  u,v,v1,v2,v3,w for Vector of V,
  F,G,H for FinSequence of V,
  A,B for Subset of V,
  f for Function of V, R,
  S,T for finite Subset of V;
reserve L,L1,L2,L3 for Linear_Combination of V;
reserve l for Linear_Combination of A;
reserve RR for domRing;
reserve VV for RightMod of RR;
reserve LL for Linear_Combination of VV;
reserve aa for Scalar of RR;
reserve uu, vv for Vector of VV;

theorem Th52:
  Carrier(- L) = Carrier(L)
proof
  set a = -1_R;
  Carrier(L * a) = Carrier(L)
  proof
    set S = {v : L.v <> 0.R};
    set T = {u : (L * a).u <> 0.R};
    T = S
    proof
      thus T c= S
      proof
        let x be object;
        assume x in T;
        then consider u such that
A1:     x = u and
A2:     (L * a).u <> 0.R;
        (L * a).u = L.u * a by Def10;
        then L.u <> 0.R by A2;
        hence thesis by A1;
      end;
      let x be object;
      assume x in S;
      then consider v such that
A3:   x = v and
A4:   L.v <> 0.R;
      (L * a).v = L.v * a by Def10
        .= -(L.v) by VECTSP_2:28;
      then (L * a).v <> 0.R by A4,VECTSP_2:3;
      hence thesis by A3;
    end;
    hence thesis;
  end;
  hence thesis;
end;
