 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 Th155:
  for R being right_zeroed non empty addLoopStr, a being Element of R,
  i be Integer st i = 1
  holds (Int-mult-left(R)).(i,a) = a
  proof
    let R be right_zeroed non empty addLoopStr,
        a be Element of R,
        i be Integer;
    assume i=1;
    hence (Int-mult-left(R)).(i,a) = 1 * a by Def23
    .= a by BINOM:13;
  end;
