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 Th51:
  -1 <= r & r <= 1 implies tan arctan r = r
proof
A1: [.-PI/4,PI/4.] c= ].-PI/2,PI/2.[ by Lm7,Lm8,XXREAL_2:def 12;
  assume that
A2: -1 <= r and
A3: r <= 1;
A4: r in [.-1,1.] by A2,A3,XXREAL_1:1;
  then
A5: r in dom (arctan | [.-1,1.]) by Th23,RELAT_1:62;
  arctan.r in [.-PI/4,PI/4.] by A4,Th49;
  hence tan (arctan r) = tan.(arctan.r) by A1,Th13
    .= ((tan|[.-PI/4,PI/4.]) qua Function).(arctan.r) by A4,Th49,FUNCT_1:49
    .= ((tan|[.-PI/4,PI/4.]) qua Function).((arctan | [.-1,1.]).r) by A4,
FUNCT_1:49
    .= ((tan | [.-PI/4,PI/4.]) qua Function * (arctan | [.-1,1.])).r by A5,
FUNCT_1:13
    .= (id [.-1,1.]).r by Th21,Th25,FUNCT_1:39
    .= r by A4,FUNCT_1:18;
end;
