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 Th3:
  frac(r) < frac(s) implies frac(r-s) = frac(r) - frac(s) + 1
  proof
    assume
A1: frac(r) < frac(s);
    set a = r-s - frac(r) + frac(s) - 1;
A2: a = r-frac(r) - (s-frac(s)) - 1;
A3: a = r-s+(-frac(r)+frac(s)-1);
    frac(s) < 1 by INT_1:43;
    then
A4: frac(s)-1 < 1-1 by XREAL_1:9;
    0 <= frac(r) by INT_1:43;
    then frac(s)-1-frac(r) <= frac(r)-frac(r) by A4;
    then
A5: r-s-frac(r)+frac(s)-1 <= r-s+Q by A3,XREAL_1:6;
A6: a = r-s-1-(frac(r)-frac(s));
    frac(r)-frac(r) > frac(r)-frac(s) by A1,XREAL_1:10;
    then r-s-1-Q < a by A6,XREAL_1:10;
    then a = [\r-s/] by A5,A2,INT_1:def 6;
    hence thesis;
  end;
