reserve a, b, n for Nat,
  r for Real,
  f for FinSequence of REAL;
reserve p for Prime;

theorem Th31:
  for a, b being non zero Nat st b divides a holds
  p |-count (a div b) = (p |-count a) -' (p |-count b)
proof
  let a,b be non zero Nat;
  set x = p |-count a;
  set y = p |-count b;
  set z = p |-count (a div b);
  assume
A1: b divides a;
  then a = b * (a div b) by NAT_D:3;
  then a div b <> 0;
  then p |-count (b*(a div b)) = y + z by Th28;
  then
A2: z + y = x + 0 by A1,NAT_D:3;
  y <= x by A1,Th30;
  then y-y <= x-y by XREAL_1:13;
  hence thesis by A2,XREAL_0:def 2;
end;
