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
  for x be Real st sin.x <> 0 holds cot.x = cot x
proof
  let x be Real;
  assume
A1: sin.x <> 0;
A2: x in REAL by XREAL_0:def 1;
  not x in sin"{0}
  proof
    assume x in sin"{0};
    then sin.x in {0} by FUNCT_1:def 7;
    hence contradiction by A1,TARSKI:def 1;
  end;
  then x in dom sin \ sin"{0} by SIN_COS:24,XBOOLE_0:def 5,A2;
  then x in dom cos /\ (dom sin \ sin"{0}) by SIN_COS:24,XBOOLE_0:def 4,A2;
  then x in dom (cos/sin) by RFUNCT_1:def 1;
  then cot.x = (cos x)/(sin x) by RFUNCT_1:def 1
    .= cot x by SIN_COS4:def 2;
  hence thesis;
end;
