reserve q,th,r for Real,
  a,b,p for Real,
  w,z for Complex,
  k,l,m,n,n1,n2 for Nat,
  seq,seq1,seq2,cq1 for Complex_Sequence,
  rseq,rseq1,rseq2 for Real_Sequence,
  rr for set,
  hy1 for 0-convergent non-zero Real_Sequence;
reserve d for Real;
reserve th,th1,th2 for Real;

theorem Th69:
  [.0,1 .] c= dom tan & ].0,1 .[ c= dom tan
proof
A1: [.0,1 .] \ cos"{0} c=dom cos \ cos"{0} by Th24,XBOOLE_1:33;
 [.0,1 .] /\ cos"{0}={}
  proof
    assume [.0,1 .] /\ cos"{0}<>{};
    then consider rr being object such that
A2: rr in [.0,1 .] /\ cos"{0} by XBOOLE_0:def 1;
A3: rr in [.0,1 .] by A2,XBOOLE_0:def 4;
A4: rr in cos"{0} by A2,XBOOLE_0:def 4;
A5: cos.(rr) <>0 by A3,Th68;
 cos.(rr) in {0} by A4,FUNCT_1:def 7;
    hence contradiction by A5,TARSKI:def 1;
  end;
then  [.0,1 .] misses cos"{0} by XBOOLE_0:def 7;
then  [.0,1 .] c= dom cos \ cos"{0} by A1,XBOOLE_1:83;
then  [.0,1 .] c= dom sin /\ (dom cos \ cos"{0}) by Th24,XBOOLE_1:19;
then A6: [.0,1 .] c= dom tan by RFUNCT_1:def 1;
 ].0,1 .[c=[.0,1 .] by XXREAL_1:25;
  hence thesis by A6,XBOOLE_1:1;
end;
