theorem
  -1 <= r & r <= 1 implies tan arccot r = 1/r
proof
  set x = arccot r;
  assume that
A1: -1 <= r and
A2: r <= 1;
A3: cos x /sin x = cot x by SIN_COS4:def 2
    .= r by A1,A2,Th52;
  tan x = sin x/cos x by SIN_COS4:def 1
    .= 1/r by A3,XCMPLX_1:57;
  hence thesis;
end;
