reserve a,b,c,x,y,z for object,X,Y,Z for set,
  n for Nat,
  i,j for Integer,
  r,r1,r2,r3,s for Real,
  c1,c2 for Complex,
  p for Point of TOP-REAL n;

theorem Th2:
  frac(r) >= frac(s) implies frac(r-s) = frac(r) - frac(s)
  proof
    assume
A1: frac(r) >= frac(s);
    set a = r-s - frac(r) + frac(s);
A2: a = r-frac(r) - (s-frac(s));
A3: a = r-s+(-frac(r)+frac(s));
    -frac(r) <= -frac(s) by A1,XREAL_1:24;
    then -frac(r)+frac(s) <= -frac(s)+frac(s) by XREAL_1:6;
    then
A4: a <= r-s+Q by A3,XREAL_1:6;
A5: a = r-s-(frac(r)-frac(s));
A6: 0 <= frac(s) by INT_1:43;
    frac(r) < 1 by INT_1:43;
    then 1+frac(s) > frac(r)+Q by A6,XREAL_1:8;
    then 1 > frac(r)-frac(s) by XREAL_1:19;
    then r-s-1 < a by A5,XREAL_1:10;
    then a = [\r-s/] by A4,A2,INT_1:def 6;
    hence thesis;
  end;
