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

theorem Th3:
  j divides i iff i = j * (i div j)
proof
  thus j divides i implies i = j * (i div j)
  proof
    assume j divides i;
    then consider k being Nat such that
A1: i = j * k;
A2: i = j * k + 0 by A1;
    now
      assume
A3:   j <> 0;
      then
A4:   i = j * (i div j) + (i mod j) by INT_1:59;
      i mod j < j by A3,Th1;
      hence thesis by A2,A4,NAT_1:18;
    end;
    hence thesis by A1;
  end;
  thus thesis;
end;
