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

  rr for set,

  rseq for Real_Sequence;

theorem Th4:
  ].0,PI/2.] c= dom cosec
proof
  ].0,PI/2.] /\ sin"{0} = {}
  proof
    assume ].0,PI/2.] /\ sin"{0} <> {};
    then consider rr being object such that
A1: rr in ].0,PI/2.] /\ sin"{0} by XBOOLE_0:def 1;
    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/2.] by A1,XBOOLE_0:def 4;
    then sin.rr <> 0 by Lm4,COMPTRIG:7;
    hence contradiction by A2,TARSKI:def 1;
  end;
  then
  ].0,PI/2.] \ sin"{0} c= dom sin \ sin"{0} & ].0,PI/2.] misses sin"{0} by
SIN_COS:24,XBOOLE_0:def 7,XBOOLE_1:33;
  then ].0,PI/2.] c= dom sin \ sin"{0} by XBOOLE_1:83;
  hence thesis by RFUNCT_1:def 2;
end;
