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 Th47:
  arctan|[.-1,1.] is increasing
proof
A1: [.-1,1.] = tan.:[.-PI/4,PI/4.] by Th21,RELAT_1:115;
  [.-PI/4,PI/4.] c= ].-PI/2,PI/2.[ by Lm7,Lm8,XXREAL_2:def 12;
  hence thesis by A1,Th45,RELAT_1:123,RFUNCT_2:28;
end;
