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 Th7:
  tan|].-PI/2,PI/2.[ is increasing
proof
A1: for x st x in ].-PI/2,PI/2.[ holds diff(tan,x) > 0
  proof
    let x;
    assume
A2: x in ].-PI/2,PI/2.[;
    then 0 < cos.x by COMPTRIG:11;
    then (cos.x)^2 > 0;
    then 1/(cos.x)^2 > 0 /(cos.x)^2;
    hence thesis by A2,Lm3;
  end;
  ].-PI/2,PI/2.[ c= dom tan by Lm1,FDIFF_1:def 6;
  hence thesis by A1,Lm1,ROLLE:9;
end;
