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 Th2:
  ].0,PI.[ c= dom cot
proof
  ].0,PI.[ /\ sin"{0}={}
  proof
    assume ].0,PI.[ /\ sin"{0}<>{};
    then consider rr being object such that
A1: rr in ].0,PI.[ /\ sin"{0} by XBOOLE_0:7;
    rr in sin"{0} by A1,XBOOLE_0:def 4;
    then
A2: sin.rr in {0} by FUNCT_1:def 7;
    rr in ].0,PI.[ by A1,XBOOLE_0:def 4;
    then sin.rr <>0 by COMPTRIG:7;
    hence contradiction by A2,TARSKI:def 1;
  end;
  then
A3: ].0,PI.[ misses sin"{0} by XBOOLE_0:def 7;
  ].0,PI.[ \ sin"{0} c= dom sin \ sin"{0} by SIN_COS:24,XBOOLE_1:33;
  then ].0,PI.[ c= dom sin \ sin"{0} by A3,XBOOLE_1:83;
  then ].0,PI.[ c= dom cos /\ (dom sin \ sin"{0}) by SIN_COS:24,XBOOLE_1:19;
  hence thesis by RFUNCT_1:def 1;
end;
