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

  rr for set,

  rseq for Real_Sequence;

theorem Th2:
  ].PI/2,PI.] c= dom sec
proof
  ].PI/2,PI.] /\ cos"{0} = {}
  proof
    assume ].PI/2,PI.] /\ cos"{0} <> {};
    then consider rr being object such that
A1: rr in ].PI/2,PI.] /\ cos"{0} by XBOOLE_0:def 1;
    rr in cos"{0} by A1,XBOOLE_0:def 4;
    then
A2: cos.(rr) in {0} by FUNCT_1:def 7;
    rr in ].PI/2,PI.] by A1,XBOOLE_0:def 4;
    then cos.rr <> 0 by Lm2,COMPTRIG:13;
    hence contradiction by A2,TARSKI:def 1;
  end;
  then
  ].PI/2,PI.] \ cos"{0} c= dom cos \ cos"{0} & ].PI/2,PI.] misses cos"{0}
  by SIN_COS:24,XBOOLE_0:def 7,XBOOLE_1:33;
  then ].PI/2,PI.] c= dom cos \ cos"{0} by XBOOLE_1:83;
  hence thesis by RFUNCT_1:def 2;
end;
