theorem
  -1 <= r & r <= 1 implies cot arctan r = 1/r
proof
  set x = arctan r;
  assume that
A1: -1 <= r and
A2: r <= 1;
A3: sin x /cos x = tan x by SIN_COS4:def 1
    .= r by A1,A2,Th51;
  cot x = cos x/sin x by SIN_COS4:def 2
    .= 1/r by A3,XCMPLX_1:57;
  hence thesis;
end;
