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 Th42:
  cos.1>0 & sin.1>0 & cos.1<sin.1
proof
A1: Partial_Sums(1 P_cos) is convergent by Th35;
A2: cos.1=Sum(1 P_cos) by Th36;
 lim ((Partial_Sums(1 P_cos))*bq )=lim(Partial_Sums(1 P_cos)) by A1,SEQ_4:17;
then A3: lim ((Partial_Sums(1 P_cos))*bq )= cos.1 by A2,SERIES_1:def 3;
 for n holds ((Partial_Sums(1 P_cos))*bq).n >=1/2
  proof
    let n;
    defpred X[Nat] means ((Partial_Sums(1 P_cos))*bq).$1 >= 1/2;
 ((Partial_Sums(1 P_cos))*bq).0=(Partial_Sums(1 P_cos)).(bq.0) by FUNCT_2:15
      .= (Partial_Sums(1 P_cos)).(2*0+1) by Lm6
      .=(Partial_Sums(1 P_cos)).0+(1 P_cos).(0+1) by SERIES_1:def 1
      .= (1 P_cos).0 +(1 P_cos).(0+1) by SERIES_1:def 1
      .=(-1)|^ 0 * (1)|^ (2*0)/((2*0)!)+(1 P_cos).(1) by Def21
      .= (-1)|^ 0 * (1)|^ (2*0)/((2*0)!)+
    (-1)|^ 1 * (1)|^ (2*1)/((2*1)!) by Def21
      .=1* (1)|^ (2*0)/((2*0)!)+
    (-1)|^ 1 * (1)|^ (2*1)/((2*1)!) by Lm7
      .=1/1 + (-1)|^ 1 * (1)|^ (2*1)/((2*1)!) by NEWTON:12
      .=1+ (-1)* (1)|^ (2*1)/((2*1)!)
      .=1+ (-1)*1/((2*1)!)
      .=1+(-1)/(1! * (1+1)) by NEWTON:15
      .=1+(-1)/(0! * (0+1) *2) by NEWTON:15
      .=1/2 by NEWTON:12;
then A4: X[0];
A5: for k st X[k] holds X[k+1]
    proof
      let k;
A6:  k in NAT by ORDINAL1:def 12;
      assume
A7:   ((Partial_Sums(1 P_cos))*bq).k >= 1/2;
  ((Partial_Sums(1 P_cos))*bq).(k+1)
      =(Partial_Sums(1 P_cos)).(bq.(k+1)) by FUNCT_2:15
        .=(Partial_Sums(1 P_cos)).(2*(k+1)+1) by Lm6
        .=(Partial_Sums(1 P_cos)).(2*k+1+1)+(1 P_cos).(2*(k+1)+1) by
SERIES_1:def 1
        .=(Partial_Sums(1 P_cos)).(2*k+1) + (1 P_cos).(2*k+1+1)+
      (1 P_cos).(2*(k+1)+1) by SERIES_1:def 1
        .= (Partial_Sums(1 P_cos)).(bq.k)+ (1 P_cos).(2*k+1+1)+
      (1 P_cos).(2*(k+1)+1) by Lm6
        .=((Partial_Sums(1 P_cos))*bq).k+ (1 P_cos).(2*k+1+1)+
      (1 P_cos).(2*(k+1)+1) by FUNCT_2:15,A6;
      then
A8:  ( (Partial_Sums(1 P_cos))*bq).(k+1)-((Partial_Sums(1 P_cos))*bq).k
      =(1 P_cos).(2*k+1+1)+ (1 P_cos).(2*(k+1)+1);
A9:  (1 P_cos).(2*k+1+1)=(-1)|^ (2*(k+1)) * (1)|^ (2*(2*(k+1)))/((2*(2*
      (k+1)))!) by Def21
        .= 1 * 1|^ (2*(2*(k+1)))/((2*(2*(k+1)))!) by Lm7
        .=1/((2*(2*(k+1)))!);
A10:  (1 P_cos).(2*(k+1)+1)
      = (-1)|^ (2*(k+1)+1) * (1)|^ (2*(2*(k+1)+1))/((2*(2*(k+1)+1))!)
      by Def21
        .=(-1) * (1)|^ (2*(2*(k+1)+1))/((2*(2*(k+1)+1))!) by Lm7
        .=(-1) * 1/((2*(2*(k+1)+1))!)
        .=(-1)/((2*(2*(k+1)+1))!);
  2*(2*(k+1))<2*(2*(k+1)+1) by XREAL_1:29,68;
then   (2*(2*(k+1)))! <= (2*(2*(k+1)+1))! by Th38;
then   1/((2*(2*(k+1)))!) >= 1/((2*(2*(k+1)+1))!) by XREAL_1:85;
then   1/((2*(2*(k+1)))!)-1/(2*(2*(k+1)+1)!)>=0 by XREAL_1:48;
then   ((Partial_Sums(1 P_cos))*bq).(k+1)>=((Partial_Sums(1 P_cos))*bq).
      k by A8,A9,A10,XREAL_1:49;
      hence thesis by A7,XXREAL_0:2;
    end;
 for n holds X[n] from NAT_1:sch 2(A4,A5);
    hence thesis;
  end;
then A11: cos.1>=1/2 by A1,A3,PREPOWER:1,SEQ_4:16;
A12: Partial_Sums(1 P_sin) is convergent by Th35;
A13: sin.1=Sum(1 P_sin) by Th36;
 lim ((Partial_Sums(1 P_sin))*bq )=lim(Partial_Sums(1 P_sin)) by A12,SEQ_4:17;
then A14: lim ((Partial_Sums(1 P_sin))*bq )= sin.1 by A13,SERIES_1:def 3;
 for n holds ((Partial_Sums(1 P_sin))*bq).n >=5/6
  proof
    let n;
    defpred X[Nat] means ((Partial_Sums(1 P_sin))*bq).$1 >= 5/6;
 ((Partial_Sums(1 P_sin))*bq).0=(Partial_Sums(1 P_sin)).(bq.0)
    by FUNCT_2:15
      .= (Partial_Sums(1 P_sin)).(2*0+1) by Lm6
      .=(Partial_Sums(1 P_sin)).0+(1 P_sin).(0+1) by SERIES_1:def 1
      .= (1 P_sin).0 +(1 P_sin).(0+1) by SERIES_1:def 1
      .=(-1)|^ 0 * (1)|^ (2*0+1)/((2*0+1)!)+(1 P_sin).(1) by Def20
      .= (-1)|^ 0 * (1)|^ (2*0+1)/((2*0+1)!)+
    (-1)|^ 1 * (1)|^ (2*1+1)/((2*1+1)!) by Def20
      .=1* (1)|^ (2*0+1)/((2*0+1)!)+
    (-1)|^ 1 * (1)|^ (2*1+1)/((2*1+1)!) by Lm7
      .=1/((0+1)!)+(-1)|^ 1 * (1)|^ (2*1+1)/((2*1+1)!)
      .=1/(0!*1)+(-1)|^ 1 * (1)|^ (2*1+1)/((2*1+1)!) by NEWTON:15
      .=1+ (-1)* (1)|^ (2*1+1)/((2*1+1)!) by NEWTON:12
      .=1+ (-1)*1/((2*1+1)!)
      .=1+ (-1)/((2*1)!*(2*1+1)) by NEWTON:15
      .=1+(-1)/(1! * (1+1)*3) by NEWTON:15
      .=1+(-1)/((0+1)! *(2*3))
      .=1+(-1)/(1* 1 *6) by NEWTON:12,15
      .=5/6;
then A15: X[0];
A16: for k st X[k] holds X[k+1]
    proof
      let k;
A17: k in NAT by ORDINAL1:def 12;
      assume
A18:  ((Partial_Sums(1 P_sin))*bq).k >= 5/6;
  ((Partial_Sums(1 P_sin))*bq).(k+1)
      =(Partial_Sums(1 P_sin)).(bq.(k+1)) by FUNCT_2:15
        .=(Partial_Sums(1 P_sin)).(2*(k+1)+1) by Lm6
        .=(Partial_Sums(1 P_sin)).(2*k+1+1)+(1 P_sin).(2*(k+1)+1) by
SERIES_1:def 1
        .=(Partial_Sums(1 P_sin)).(2*k+1) + (1 P_sin).(2*k+1+1)+
      (1 P_sin).(2*(k+1)+1) by SERIES_1:def 1
        .= (Partial_Sums(1 P_sin)).(bq.k)+ (1 P_sin).(2*k+1+1)+
      (1 P_sin).(2*(k+1)+1) by Lm6
        .=((Partial_Sums(1 P_sin))*bq).k+ (1 P_sin).(2*k+1+1)+
      (1 P_sin).(2*(k+1)+1) by FUNCT_2:15,A17;
      then
A19:  ( (Partial_Sums(1 P_sin))*bq).(k+1)-((Partial_Sums(1 P_sin))*bq).k
      =(1 P_sin).(2*k+1+1)+ (1 P_sin).(2*(k+1)+1);
A20:  (1 P_sin).(2*k+1+1)=(-1)|^ (2*(k+1)) * (1)|^ (2*(2*(k+1))+1)/((2*
      (2*(k+1))+1)!) by Def20
        .=(1)* (1)|^ (2*(2*(k+1))+1)/((2*(2*(k+1))+1)!) by Lm7
        .=1/((2*(2*(k+1))+1)!);
A21:  (1 P_sin).(2*(k+1)+1)
      = (-1)|^ (2*(k+1)+1) * (1)|^ (2*(2*(k+1)+1)+1)/((2*(2*(k+1)+1)+1)!)
      by Def20
        .=(-1) * (1)|^ (2*(2*(k+1)+1)+1)/((2*(2*(k+1)+1)+1)!) by Lm7
        .=(-1) * 1/((2*(2*(k+1)+1)+1)!)
        .=(-1)/((2*(2*(k+1)+1)+1)!);
  2*(2*(k+1))<2*(2*(k+1)+1) by XREAL_1:29,68;
then   2*(2*(k+1))+1<2*(2*(k+1)+1) +1 by XREAL_1:6;
then   (2*(2*(k+1))+1)! <= (2*(2*(k+1)+1)+1)! by Th38;
then   1/((2*(2*(k+1))+1)!) >= 1/((2*(2*(k+1)+1)+1)!) by XREAL_1:85;
then   1/((2*(2*(k+1))+1)!)-1/((2*(2*(k+1)+1)+1)!)>=0 by XREAL_1:48;
then   ((Partial_Sums(1 P_sin))*bq).(k+1)>=((Partial_Sums(1 P_sin))*bq)
      .k by A19,A20,A21,XREAL_1:49;
      hence thesis by A18,XXREAL_0:2;
    end;
 for n holds X[n] from NAT_1:sch 2(A15,A16);
    hence thesis;
  end;
then A22: sin.1>=5/6 by A12,A14,PREPOWER:1,SEQ_4:16;
A23: (cos.(1))^2+(sin.(1))^2=1 by Th28;
A24: (sin.(1))^2>=(5/6)^2 by A22,SQUARE_1:15;
then  1-(1-(cos.(1))^2)<=1-25/36 by A23,XREAL_1:10;
then  (cos.(1))^2<25/36 by XXREAL_0:2;
then  (sin.(1))^2> (cos.(1))^2 by A24,XXREAL_0:2;
then A25: sqrt(cos.(1))^2<sqrt(sin.(1))^2 by SQUARE_1:27,XREAL_1:63;
 sqrt(cos.(1))^2 = cos.1 by A11,SQUARE_1:22;
  hence thesis by A11,A22,A25,SQUARE_1:22;
end;
