 reserve R for Ring;
 reserve x, y, y1 for set;
 reserve a, b for Element of R;
 reserve V for LeftMod of R;
 reserve v, w for Vector of V;
 reserve u,v,w for Vector of V;
 reserve F,G,H,I for FinSequence of V;
 reserve j,k,n for Nat;
 reserve f,f9,g for sequence of V;
 reserve R for Ring;
 reserve V, X, Y for LeftMod of R;
 reserve u, u1, u2, v, v1, v2 for Vector of V;
 reserve a for Element of R;
 reserve V1, V2, V3 for Subset of V;
 reserve x for set;
 reserve W, W1, W2 for Submodule of V;
 reserve w, w1, w2 for Vector of W;
 reserve D for non empty set;
 reserve d1 for Element of D;
 reserve A for BinOp of D;
 reserve M for Function of [:the carrier of R,D:],D;
reserve B,C for Coset of W;
 reserve V for LeftMod of R;
 reserve W, W1, W2, W3 for Submodule of V;
 reserve u, u1, u2, v, v1, v2 for Vector of V;
 reserve a, a1, a2 for Element of R;
 reserve X, Y, y, y1, y2 for set;
 reserve C for Coset of W;
 reserve C1 for Coset of W1;
 reserve C2 for Coset of W2;
reserve A1,A2,B for Element of Submodules(V);

theorem Th157:
  for R being Abelian right_zeroed add-associative right_complementable
  non empty addLoopStr,
  a being Element of R, i being Element of NAT
  holds -(Nat-mult-left(R)).(i,a) = (Nat-mult-left(R)).(i,-a)
  proof
    let R be Abelian right_zeroed add-associative right_complementable
    non empty addLoopStr,
    a being Element of R, i being Element of NAT;
    defpred P[Nat] means
    (Nat-mult-left(R)).($1,a) + (Nat-mult-left(R)).($1,-a) =0.R;
    A1: P[0]
    proof
      (Nat-mult-left(R)).(0,a) + (Nat-mult-left(R)).(0,-a)
      = 0.R + (Nat-mult-left(R)).(0,-a) by BINOM:def 3
      .= 0.R + 0.R by BINOM:def 3
      .= 0.R by RLVECT_1:4;
      hence thesis;
    end;
    A2: for n be Nat st P[n] holds P[n+1]
    proof
      let n be Nat;
      assume A3:P[n];
      (Nat-mult-left(R)).(n+1,a) + (Nat-mult-left(R)).(n+1,-a)
      =a + (Nat-mult-left(R)).(n,a) + (Nat-mult-left(R)).(n+1,-a)
      by BINOM:def 3
      .= a + (Nat-mult-left(R)).(n,a) + (-a + (Nat-mult-left(R)).(n,-a))
      by BINOM:def 3
      .= a + (Nat-mult-left(R)).(n,a) + (Nat-mult-left(R)).(n,-a) + -a
      by RLVECT_1:def 3
      .= a + ((Nat-mult-left(R)).(n,a) + (Nat-mult-left(R)).(n,-a)) + -a
      by RLVECT_1:def 3
      .= a + -a by A3,RLVECT_1:4
      .= 0.R by RLVECT_1:5;
      hence P[n+1];
    end;
    for n be Nat holds P[n] from NAT_1:sch 2(A1,A2); then
    (Nat-mult-left(R)).(i,a) + (Nat-mult-left(R)).(i,-a) =0.R;
    hence thesis by RLVECT_1:6;
  end;
