
theorem MFR:
  for a be non zero Nat, b be Nat holds
    b mod a < [/a/2\] implies frac (b/a) < 1/2
  proof
    let a be non zero Nat, b be Nat;
    assume
    A1: b mod a < [/a/2\];
    A2: a/2 <= [/a/2\] <= a/2 + 1 by INT_1:def 7;
    per cases;
    suppose a is even; then
      reconsider c = a/2 as non zero Nat;
      B1: (a/2)/a = (1/2)*(a/a) by XCMPLX_1:104
      .= 1/2*1 by XCMPLX_1:60;
      a*frac (b/a) < c by A1,R3; then
      a*frac (b/a) < a*(c/a) by XCMPLX_1:87;
      hence thesis by B1,XREAL_1:64;
    end;
    suppose a is odd; then
      reconsider c = (a + 1)/2 as non zero Nat;
      a - 1 < a - 0 by XREAL_1:10; then
      (a - 1)/2 < a/2 by XREAL_1:74; then
      ((a - 1)/2)/a < (a/2)/a by XREAL_1:74; then
      ((a + 1)/2 - 1)/a < 1/2*(a/a) by XCMPLX_1:104; then
      B0: (c - 1)/a < 1/2*1 by XCMPLX_1:60;
      B1: a/2 - 1/2 < a/2 - 0 & a/2 + 0 < a/2 + 1/2 &
        (a/2 + 1) + 0 < (a/2 + 1) + 1/2 by XREAL_1:10,XREAL_1:6; then
      c - 1 < [/a/2\] by A2,XXREAL_0:2; then
      B2: (c - 1) + 1 <= [/a/2\] by INT_1:7;
      [/a/2\] < c + 1 by B1,A2,XXREAL_0:2; then
      [/a/2\] <= c by INT_1:7; then
      b mod a < (c - 1) + 1 by A1,B2,XXREAL_0:1; then
      b mod a <= c - 1 by INT_1:7; then
      a*frac (b/a) <= c - 1 by R3; then
      a*frac (b/a) <= a*((c-1)/a) by XCMPLX_1:87; then
      frac (b/a) <= (c - 1)/a by XREAL_1:68;
      hence thesis by B0,XXREAL_0:2;
    end;
  end;
