reserve X for set, x,y,z for object,
  k,l,n for Nat,
  r for Real;
reserve i,i0,i1,i2,i3,i4,i5,i8,i9,j for Integer;
reserve r1,p,p1,g,g1,g2 for Real,
  Y for Subset of REAL;
reserve r, s for Real;
reserve i for Integer,
  a, b, r, s for Real;

theorem Th69:
  a >= [\r/]+1 & a < r+1 implies frac a < frac r
proof
  assume
A1: a >= [\r/]+1;
  assume
A2: a < r+1;
  then a-1 < r by XREAL_1:19;
  then
A3: frac a = a-[\a/] & a-1-[\r/] < r-[\r/] by XREAL_1:14;
  [\a/] = [\r/]+1 by A1,A2,Th68;
  hence thesis by A3;
end;
