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 Th45:
  arctan|(tan.:].-PI/2,PI/2.[) is increasing
proof
  set f = tan | ].-PI/2,PI/2.[;
A1: f.:].-PI/2,PI/2.[ = rng(f|].-PI/2,PI/2.[) by RELAT_1:115
    .= rng(tan|].-PI/2,PI/2.[) by RELAT_1:73
    .= tan.:].-PI/2,PI/2.[ by RELAT_1:115;
  f|].-PI/2,PI/2.[ = f by RELAT_1:73;
  hence thesis by A1,Th7,FCONT_3:9;
end;
