reserve m,n for Nat;
reserve i,j for Integer;
reserve S for non empty addMagma;
reserve r,r1,r2,s,s1,s2,t,t1,t2 for Element of S;
reserve G for addGroup-like non empty addMagma;
reserve e,h for Element of G;
reserve G for addGroup;
reserve f,g,h for Element of G;
reserve u for UnOp of G;
reserve A for Abelian addGroup;
reserve a,b for Element of A;

theorem
  for L be add-associative Abelian add-unital non empty addMagma for x,y be
Element of L for n be Nat holds (mult L).(n,x+y) = (mult L).(n,x)
  + (mult L).(n,y)
proof
  let L be add-associative Abelian add-unital non empty addMagma;
  let x,y be Element of L;
  defpred P[Nat] means
   (mult L).($1,x+y) = (mult L).($1,x) + (mult L).($1,y);
A1: now
    let n be Nat;
    assume P[n];
    then (mult L).(n+1,x+y) = (mult L).(n,x) + (mult L).(n,y) + (x+y) by
Def7
      .= (mult L).(n,x) + ((mult L).(n,y) + (x+y)) by RLVECT_1:def 3
      .= (mult L).(n,x) + (x+((mult L).(n,y)+y)) by RLVECT_1:def 3
      .= (mult L).(n,x) + (x+(mult L).(n+1,y)) by Def7
      .= (mult L).(n,x)+x + (mult L).(n+1,y) by RLVECT_1:def 3
      .= (mult L).(n+1,x) + (mult L).(n+1,y) by Def7;
    hence P[n+1];
  end;
  (mult L).(0,x+y) = 0_L by Def7
    .= 0_L + 0_L by Def4
    .= (mult L).(0,x) + 0_L by Def7
    .= (mult L).(0,x) + (mult L).(0,y) by Def7;
  then
A2: P[0];
  thus for n be Nat holds P[n] from NAT_1:sch 2(A2,A1);
end;
