
theorem Th1:
  for a, b being Real st b > 0 ex r being Real st r =
  b*-[\ a/b /]+a & 0 <= r & r < b
proof
  let a, b be Real such that
A1: b > 0;
  set ab = [\ a/b /];
  set i = -ab;
  take r = b*i+a;
  thus r = b*i+a;
  ab <= a/b by INT_1:def 6;
  then ab*b <= a/b*b by A1,XREAL_1:64;
  then ab*b <= a by A1,XCMPLX_1:87;
  then -(ab*b) >= -a by XREAL_1:24;
  then b*i+a >= a+-a by XREAL_1:6;
  hence 0 <= r;
  a/b-1 < ab by INT_1:def 6;
  then -(a/b-1) > i by XREAL_1:24;
  then (-(a/b)+1)*b > i*b by A1,XREAL_1:68;
  then -(a/b*b)+b > i*b;
  then -a+b > i*b by A1,XCMPLX_1:87;
  then -a+b+a > r by XREAL_1:8;
  hence thesis;
end;
