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

  rr for set,

  rseq for Real_Sequence;

theorem Th1:
  [.0,PI/2.[ c= dom sec
proof
  [.0,PI/2.[ /\ cos"{0} = {}
  proof
    assume [.0,PI/2.[ /\ cos"{0} <> {};
    then consider rr being object such that
A1: rr in [.0,PI/2.[ /\ 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 [.0,PI/2.[ by A1,XBOOLE_0:def 4;
    then cos.rr <> 0 by Lm1,COMPTRIG:11;
    hence contradiction by A2,TARSKI:def 1;
  end;
  then
  [.0,PI/2.[ \ cos"{0} c= dom cos \ cos"{0} & [.0,PI/2.[ misses cos"{0} by
SIN_COS:24,XBOOLE_0:def 7,XBOOLE_1:33;
  then [.0,PI/2.[ c= dom cos \ cos"{0} by XBOOLE_1:83;
  hence thesis by RFUNCT_1:def 2;
end;
