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 Th35:
  -PI/2 < r & r < PI/2 implies arctan tan.r = r & arctan tan r = r
proof
  assume that
A1: -PI/2 < r and
A2: r < PI/2;
A3: dom (tan|].-PI/2,PI/2.[) = ].-PI/2,PI/2.[ by Th1,RELAT_1:62;
A4: r in ].-PI/2,PI/2.[ by A1,A2,XXREAL_1:4;
  then arctan tan.r = arctan.((tan|].-PI/2,PI/2.[).r) by FUNCT_1:49
    .= (id ].-PI/2,PI/2.[).r by A4,A3,Th31,FUNCT_1:13
    .= r by A4,FUNCT_1:18;
  hence thesis by A4,Th13;
end;
