reserve a,b,p,k,l,m,n,s,h,i,j,t,i1,i2 for natural Number;

theorem
  k mod n = 0 & l = k - m * n implies l mod n = 0
proof
  assume that
A1: k mod n = 0 and
A2: l = k - m * n;
  per cases;
  suppose n = 0;
    hence thesis by A1,A2;
  end;
  suppose n <> 0;
    then consider t being Nat such that
A3: k = n * t + 0 and
A4: 0 < n by A1,Def2;
A5: l = n * (t - m) + 0 by A2,A3;
    now
      assume t < m + 0;
      then t - m < 0 by XREAL_1:19;
      then l < n * 0 by A4,A5,XREAL_1:68;
      hence contradiction;
    end;
    then t - m is Element of NAT by NAT_1:21;
    hence thesis by A4,A5,Def2;
  end;
end;
