
theorem Th4:
  for n,k being Element of NAT st k>0 & n mod (2*k)<k holds n mod (
  2*k)= n mod k
proof
  let n,k be Element of NAT;
  assume that
A1: k>0 and
A2: n mod (2*k)<k;
  ( ex t be Nat st n = 2*k * t + (n mod (2*k)) & n mod (2*k) < 2*k ) or n
  mod (2*k) = 0 & 2*k = 0 by NAT_D:def 2;
  then consider t be Nat such that
A3: n=2*k*t + (n mod (2*k)) by A1;
  reconsider t as Element of NAT by ORDINAL1:def 12;
  n=k*(2*t)+ (n mod (2*k)) by A3;
  hence thesis by A2,NAT_D:def 2;
end;
