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 Th19:
  for x be set st x in [.-PI/4,PI/4.] holds tan.x in [.-1,1.]
proof
  let x be set;
  assume x in [.-PI/4,PI/4.];
  then x in ].-PI/4,PI/4.[ \/ {-PI/4,PI/4} by XXREAL_1:128;
  then
A1: x in ].-PI/4,PI/4.[ or x in {-PI/4,PI/4} by XBOOLE_0:def 3;
  per cases by A1,TARSKI:def 2;
  suppose
A2: x in ].-PI/4,PI/4.[;
    then x in { s where s is Real: -PI/4 < s & s < PI/4 }
         by RCOMP_1:def 2;
    then
A3: ex s be Real st s=x & -PI/4 < s & s < PI/4;
A4: ].-PI/4,PI/4.[ c= [.-PI/4,PI/4.] by XXREAL_1:25;
    -PI/4 in {s where s is Real: -PI/4 <= s & s <= PI/4};
    then
A5: -PI/4 in [.-PI/4,PI/4.] by RCOMP_1:def 1;
A6: [.-PI/4,PI/4.] c= ].-PI/2,PI/2.[ by Lm7,Lm8,XXREAL_2:def 12;
    then
A7: tan|[.-PI/4,PI/4.] is increasing by Th7,RFUNCT_2:28;
A8: [.-PI/4,PI/4.] /\ dom tan = [.-PI/4,PI/4.] by A6,Th1,XBOOLE_1:1,28;
    PI/4 in {s where s is Real: -PI/4 <= s & s <= PI/4};
    then PI/4 in [.-PI/4,PI/4.] /\ dom tan by A8,RCOMP_1:def 1;
    then tan.x < tan.(PI/4) by A2,A7,A8,A4,A3,RFUNCT_2:20;
    then
A9: tan.x < 1 by SIN_COS:def 28;
    x in { s where s is Real: -PI/4 < s & s < PI/4 }
     by A2,RCOMP_1:def 2;
    then ex s be Real st s=x & -PI/4 < s & s < PI/4;
    then -1 < tan.x by A2,A7,A5,A8,A4,Th17,RFUNCT_2:20;
    then tan.x in { s where s is Real: -1 < s & s < 1 } by A9;
    then
A10: tan.x in ].-1,1.[ by RCOMP_1:def 2;
    ].-1,1.[ c= [.-1,1.] by XXREAL_1:25;
    hence thesis by A10;
  end;
  suppose
    x = -PI/4;
    then tan.x in { s where s is Real: -1 <= s & s <= 1 } by Th17;
    hence thesis by RCOMP_1:def 1;
  end;
  suppose
    x = PI/4;
    then tan.x = 1 by SIN_COS:def 28;
    then tan.x in { s where s is Real: -1 <= s & s <= 1 };
    hence thesis by RCOMP_1:def 1;
  end;
end;
