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 Th71:
  th1 in ].0,1 .[ & th2 in ].0,1 .[ & tan.th1=tan.th2 implies th1=th2
proof
  assume that
A1: th1 in ].0,1 .[ and
A2: th2 in ].0,1 .[ and
A3: tan.(th1)=tan.(th2);
A4: 0<th1 by A1,XXREAL_1:4;
A5: th1<1 by A1,XXREAL_1:4;
A6: 0<th2 by A2,XXREAL_1:4;
A7: th2<1 by A2,XXREAL_1:4;
  assume
A8: th1<>th2;
 now per cases by A8,XXREAL_0:1;
    suppose
A9:  th1<th2;
A10:  for th be Element of REAL st th in ].th1,th2.[ holds th in ].0,1 .[
      proof
        let th be Element of REAL;
        assume
A11:    th in ].th1,th2.[;
then A12:    th1<th by XXREAL_1:4;
    th< th2 by A11,XXREAL_1:4;
then     th <1 by A7,XXREAL_0:2;
        hence thesis by A4,A12,XXREAL_1:4;
      end;
A13:  for th be Element of REAL st th in [.th1,th2.] holds th in [.0,1 .]
      proof
        let th be Element of REAL;
        assume
A14:    th in [.th1,th2.];
then A15:    th1<=th by XXREAL_1:1;
    th<= th2 by A14,XXREAL_1:1;
then     th <=1 by A7,XXREAL_0:2;
        hence thesis by A4,A15,XXREAL_1:1;
      end;
  ].th1,th2.[c=].0,1 .[ by A10,SUBSET_1:2;
then A16:  tan is_differentiable_on ].th1,th2.[ by Lm15,FDIFF_1:26;
        [.th1,th2.]c=[. 0,1 .] & tan|[.th1,th2.] is continuous by A13,Th70,
FCONT_1:16,SUBSET_1:2;
      then consider r such that
A17:  r in ].th1,th2.[ & diff(tan,r)=0
      by A3,A9,A16,Th69,ROLLE:1,XBOOLE_1:1;
      take th=r;
      thus th in ].0,1 .[ & diff(tan,(th))=0 by A10,A17;
    end;
    suppose
A18:  th2<th1;
A19:  for th be Element of REAL st th in ].th2,th1 .[ holds th in ].0,1 .[
      proof
        let th be Element of REAL;
        assume
A20:    th in ].th2,th1 .[;
then A21:    th2<th by XXREAL_1:4;
    th< th1 by A20,XXREAL_1:4;
then     th <1 by A5,XXREAL_0:2;
        hence thesis by A6,A21,XXREAL_1:4;
      end;
A22:  for th be Element of REAL st th in [.th2,th1 .] holds th in [.0,1 .]
      proof
        let th be Element of REAL;
        assume
A23:    th in [.th2,th1 .];
then A24:    th2<=th by XXREAL_1:1;
    th<= th1 by A23,XXREAL_1:1;
then     th <=1 by A5,XXREAL_0:2;
        hence thesis by A6,A24,XXREAL_1:1;
      end;
  ].th2,th1 .[c=].0,1 .[ by A19,SUBSET_1:2;
then A25:  tan is_differentiable_on ].th2,th1 .[ by Lm15,FDIFF_1:26;
        [.
th2,th1 .] c=[.0,1 .] & tan|[.th2,th1.] is continuous by A22,Th70,FCONT_1:16
,SUBSET_1:2;
      then consider r such that
A26:  r in ].th2,th1 .[ & diff(tan,r)=0
      by A3,A18,A25,Th69,ROLLE:1,XBOOLE_1:1;
      take th=r;
      thus th in ].0,1 .[ & diff(tan,(th))=0 by A19,A26;
    end;
  end;
  hence thesis by Lm15;
end;
