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 Th52:
  -1 <= r & r <= 1 implies cot arccot r = r
proof
A1: [.PI/4,3/4*PI.] c= ].0,PI.[ by Lm9,Lm10,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 (arccot | [.-1,1.]) by Th24,RELAT_1:62;
  arccot.r in [.PI/4,3/4*PI.] by A4,Th50;
  hence cot (arccot r) = cot.(arccot.r) by A1,Th14
    .= ((cot|[.PI/4,3/4*PI.]) qua Function).(arccot.r) by A4,Th50,FUNCT_1:49
    .= ((cot|[.PI/4,3/4*PI.]) qua Function).((arccot | [.-1,1.]).r) by A4,
FUNCT_1:49
    .= ((cot|[.PI/4,3/4*PI.]) qua Function * (arccot | [.-1,1.])).r by A5,
FUNCT_1:13
    .= (id [.-1,1.]).r by Th22,Th26,FUNCT_1:39
    .= r by A4,FUNCT_1:18;
end;
