reserve x,x0, r, s, h for Real,

  n for Element of NAT,
  rr, y for set,
  Z for open Subset of REAL,

  f, f1, f2 for PartFunc of REAL,REAL;

theorem
  -1 < r & r < 1 implies -PI/4 < arctan r & arctan r < PI/4
proof
A1: tan (PI/4) = tan.(PI/4) by Lm8,Th13
    .= 1 by SIN_COS:def 28;
  assume that
A2: -1 < r and
A3: r < 1;
A4: arctan r <= PI/4 by A2,A3,Th63;
  -PI/4 <= arctan r by A2,A3,Th63;
  then
  -PI/4 < arctan r & arctan r < PI/4 or -PI/4 = arctan r or arctan r = PI/
  4 by A4,XXREAL_0:1;
  hence thesis by A2,A3,A1,Th17,Th51;
end;
