
theorem Th3:
  for n,k being Element of NAT st k>0 & n mod (2*k)>=k holds n div
  k= (n div (2*k))*2+1
proof
  let n,k be Element of NAT;
  assume that
A1: k>0 and
A2: n mod (2*k)>=k;
  2*k>2*0 by A1,XREAL_1:68;
  then
A3: n=2*k*( n div (2*k))+ (n mod (2*k)) by NAT_D:2
    .=2*k*( n div (2*k))+((n mod k) +k) by A1,A2,Th2
    .=k*(2*( n div (2*k))+1)+(n mod k);
  n mod k<k by A1,NAT_D:1;
  hence thesis by A3,NAT_D:def 1;
end;
