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 Th36:
  0 < r & r < PI implies arccot cot.r = r & arccot cot r = r
proof
  assume that
A1: 0 < r and
A2: r < PI;
A3: dom (cot|].0,PI.[) = ].0,PI.[ by Th2,RELAT_1:62;
A4: r in ].0,PI.[ by A1,A2,XXREAL_1:4;
  then arccot cot.r = arccot.((cot|].0,PI.[).r) by FUNCT_1:49
    .= (id ].0,PI.[).r by A4,A3,Th32,FUNCT_1:13
    .= r by A4,FUNCT_1:18;
  hence thesis by A4,Th14;
end;
