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 Th70:
  tan|[.0,1 .] is continuous
proof
 for th be Real st th in dom(tan|[.0,1 .]) holds
  tan|[.0,1 .] is_continuous_in th
  proof
    let th be Real;
    assume
A1: th in dom(tan|[.0,1 .]);
then A2: th in [.0,1 .] by RELAT_1:57;
 dom sin = REAL by FUNCT_2:def 1;
then A3: th in dom sin by XREAL_0:def 1;
A4: th in [#]REAL by XREAL_0:def 1;
then  sin is_differentiable_in th by Th67,FDIFF_1:9;
then A5: sin is_continuous_in th by FDIFF_1:24;
 cos is_differentiable_in th by A4,Th66,FDIFF_1:9;
then A6: cos is_continuous_in th by FDIFF_1:24;
 cos.(th) <> 0 by A2,Th68;
then A7: tan is_continuous_in th by A3,A5,A6,FCONT_1:11;
 now
      let rseq;
      assume that
A8:  rng rseq c= dom (tan|[.0,1 .]) and
A9:  rseq is convergent & lim rseq = th;
A10:  rng rseq c= dom tan by A8,Lm14,Th69,XBOOLE_1:1;
then A11:  tan.th = lim (tan/*rseq) by A7,A9,FCONT_1:def 1;
  now
        let k1 be Element of NAT;
    dom (rseq) = NAT by SEQ_1:1;
then     rseq.k1 in rng rseq by FUNCT_1:def 3;
then A12:    (tan|[.0,1 .]).(rseq.k1)=tan.(rseq.k1) by A8,Lm14;
    (tan|[.0,1 .]).(rseq.k1)=((tan|[.0,1 .])/*rseq).k1 by A8,FUNCT_2:108;
        hence ((tan|[.0,1 .])/*rseq).k1=(tan/*rseq).k1
        by A8,A12,Lm14,Th69,FUNCT_2:108,XBOOLE_1:1;
      end;
then   (tan|[.0,1 .])/*rseq =tan/*rseq;
      hence (tan|[.0,1 .])/*rseq is convergent
      & (tan|[.0,1 .]).th = lim ((tan|[.0,1 .])/*rseq)
      by A1,A7,A9,A10,A11,Lm14,FCONT_1:def 1;
    end;
    hence thesis by FCONT_1:def 1;
  end;
  hence thesis by FCONT_1:def 2;
end;
