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 Th68:
  th in [.0,1 .] implies 0<cos.th & cos.th>=1/2
proof
  assume th in [.0,1 .];
then A1: 0<=th & th<=1 by XXREAL_1:1;
A2: Partial_Sums(th P_cos) is convergent by Th35;
A3: cos.th=Sum (th P_cos) by Th36;
 lim ((Partial_Sums(th P_cos))*bq )=lim(Partial_Sums(th P_cos)) by A2,SEQ_4:17;
then A4: lim ((Partial_Sums(th P_cos))*bq )= cos.th by A3,SERIES_1:def 3;
 for n holds ((Partial_Sums(th P_cos))*bq).n >=1/2
  proof
    let n;
A5: ((Partial_Sums(th P_cos))*bq).0=(Partial_Sums(th P_cos)).(bq.0)
    by FUNCT_2:15
      .= (Partial_Sums(th P_cos)).(2*0+1) by Lm6
      .= (Partial_Sums(th P_cos)).0+(th P_cos).(0+1) by SERIES_1:def 1
      .= (th P_cos).0 +(th P_cos).(0+1) by SERIES_1:def 1
      .= (-1)|^ 0 * (th)|^ (2*0)/((2*0)!)+(th P_cos).1 by Def21
      .= (-1)|^ 0 * (th)|^ (2*0)/((2*0)!)+
    (-1)|^ 1 * (th)|^ (2*1)/((2*1)!) by Def21
      .= 1* (th)|^ (2*0)/((2*0)!)+
    (-1)|^ 1 * (th)|^ (2*1)/((2*1)!) by Lm7
      .= 1/1 + (-1)|^ 1 * (th)|^ (2*1)/((2*1)!) by NEWTON:4,12
      .= 1+ (-1)* (th)|^ (2*1)/((2*1)!)
      .= 1+ (-1)*(th*th)/((2*1)!) by NEWTON:81
      .= 1-(th^2)/2 by NEWTON:14;
    defpred X[Nat] means
((Partial_Sums(th P_cos))*bq).$1 >= 1/2;
 th^2 <= 1^2 by A1,SQUARE_1:15;
then  1 - 1/2 = 1/2 & (th^2)/2<=1/2 by XREAL_1:72;
then A6: X[0] by A5,XREAL_1:10;
A7: for k st X[k] holds X[k+1]
    proof
      let k;
A8: k in NAT by ORDINAL1:def 12;
      assume
A9:  ((Partial_Sums(th P_cos))*bq).k >= 1/2;
  ((Partial_Sums(th P_cos))*bq).(k+1)
      =(Partial_Sums(th P_cos)).(bq.(k+1)) by FUNCT_2:15
        .=(Partial_Sums(th P_cos)).(2*(k+1)+1) by Lm6
        .=(Partial_Sums(th P_cos)).(2*k+1+1)+(th P_cos).(2*(k+1)+1) by
SERIES_1:def 1
        .=(Partial_Sums(th P_cos)).(2*k+1) + (th P_cos).(2*k+1+1)+
      (th P_cos).(2*(k+1)+1) by SERIES_1:def 1
        .= (Partial_Sums(th P_cos)).(bq.k)+ (th P_cos).(2*k+1+1)+
      (th P_cos).(2*(k+1)+1) by Lm6
        .=((Partial_Sums(th P_cos))*bq).k+ (th P_cos).(2*k+1+1)+
      (th P_cos).(2*(k+1)+1) by FUNCT_2:15,A8;
      then A10:  (
(Partial_Sums(th P_cos))*bq).(k+1)-((Partial_Sums(th P_cos))* bq).k
      =(th P_cos).(2*k+1+1)+ (th P_cos).(2*(k+1)+1);
A11:  (th P_cos).(2*k+1+1)=(-1)|^ (2*(k+1)) * (th)|^ (2*(2*(k+1)))/((2*(
      2*(k+1)))!) by Def21
        .=(1)* (th)|^ (2*(2*(k+1)))/((2*(2*(k+1)))!) by Lm7
        .=(th)|^ (2*(2*(k+1)))/((2*(2*(k+1)))!);
A12:  (th P_cos).(2*(k+1)+1)
      = (-1)|^ (2*(k+1)+1) * (th)|^ (2*(2*(k+1)+1))/((2*(2*(k+1)+1))!)
      by Def21
        .=((-1) * (th)|^ (2*(2*(k+1)+1)))/((2*(2*(k+1)+1))!) by Lm7;
A13:  2*(2*(k+1))<2*(2*(k+1)+1) by XREAL_1:29,68;
then A14:  (2*(2*(k+1)))! <= (2*(2*(k+1)+1))! by Th38;
A15:  (th)|^ (2*(2*(k+1)+1)) <= (th)|^ (2*(2*(k+1))) by A1,A13,Th39;
      A16:  0 <= (th)|^ (2*(2*(k+1)+1)) & (2*(2*(k+1)+1))! >0 by POWER:3;
  1/((2*(2*(k+1)+1))!)<=1/((2*(2*(k+1)))!) by A14,XREAL_1:85;
then   (th)|^ (2*(2*(k+1)+1))*(1/((2*(2*(k+1)+1))!))<=
      (th)|^ (2*(2*(k+1)))*(1/((2*(2*(k+1)))!)) by A15,A16,XREAL_1:66;
then   ((th)|^ (2*(2*(k+1)))*1)/((2*(2*(k+1)))!)-
      ((th)|^ (2*(2*(k+1)+1))*1)/((2*(2*(k+1)+1))!)>=0 by XREAL_1:48;
then   ((Partial_Sums(th P_cos))*bq).(k+1)>=((Partial_Sums(th P_cos))*bq
      ).k by A10,A11,A12,XREAL_1:49;
      hence thesis by A9,XXREAL_0:2;
    end;
 for n holds X[n] from NAT_1:sch 2(A6,A7);
    hence thesis;
  end;
  hence thesis by A2,A4,PREPOWER:1,SEQ_4:16;
end;
