reserve x,x0, r,r1,r2 for Real,
      th for Real,

  rr for set,

  rseq for Real_Sequence;

theorem Th3:
  [.-PI/2,0.[ c= dom cosec
proof
  [.-PI/2,0.[ /\ sin"{0} = {}
  proof
    assume [.-PI/2,0.[ /\ sin"{0} <> {};
    then consider rr being object such that
A1: rr in [.-PI/2,0.[ /\ sin"{0} by XBOOLE_0:def 1;
A2: rr in sin"{0} by A1,XBOOLE_0:def 4;
A3: rr in [.-PI/2,0.[ by A1,XBOOLE_0:def 4;
    reconsider rr as Real by A1;
    rr < 0 by A3,Lm3,XXREAL_1:4;
    then
A4: rr+2*PI < 0+2*PI by XREAL_1:8;
    -PI < rr by A3,Lm3,XXREAL_1:4;
    then -PI+2*PI < rr+2*PI by XREAL_1:8;
    then rr+2*PI in ].PI,2*PI.[ by A4;
    then sin.(rr+2*PI) < 0 by COMPTRIG:9;
    then
A5: sin.rr <> 0 by SIN_COS:78;
    sin.rr in {0} by A2,FUNCT_1:def 7;
    hence contradiction by A5,TARSKI:def 1;
  end;
  then
  [.-PI/2,0.[ \ sin"{0} c= dom sin \ sin"{0} & [.-PI/2,0.[ misses sin"{0}
  by SIN_COS:24,XBOOLE_0:def 7,XBOOLE_1:33;
  then [.-PI/2,0.[ c= dom sin \ sin"{0} by XBOOLE_1:83;
  hence thesis by RFUNCT_1:def 2;
end;
