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

theorem Th10:
  i divides j & i divides j + h implies i divides h
proof
  assume that
A1: i divides j and
A2: i divides j + h;
  consider t being Nat such that
A3: j + h = i * t by A2;
  consider u being Nat such that
A4: j = i * u by A1;
  now
    assume
A5: i <> 0;
    then
A6: h = i * (h div i) + (h mod i) by INT_1:59;
A7: j + (i * (h div i) + (h mod i)) = i * (u + (h div i)) + (h mod i) by A4;
A8: h mod i < i by A5,Th1;
    j + h = i * t + 0 by A3;
    then h mod i = 0 by A6,A7,A8,NAT_1:18;
    hence thesis by A6;
  end;
  hence thesis by A3;
end;
