reserve r1,r2,r3 for non negative Real;
reserve n,m1 for Nat;
reserve s for Real;
reserve cn,cd,i1,j1 for Integer;
reserve r for irrational Real;
reserve q for Rational;

theorem Th3:
  for r,n st n > 1 & |.r-c_n(r).n/c_d(r).n.|>= 1/(sqrt 5*c_d(r).n|^2) &
    |.r-c_n(r).(n+1)/c_d(r).(n+1).| >= 1/(sqrt 5 * c_d(r).(n+1)|^2)
   holds sqrt 5 > c_d(r).(n+1)/c_d(r).n + 1/(c_d(r).(n+1)/c_d(r).n)
   proof
     let r, n;
     set s5 = sqrt 5;
     assume that
A2:  n > 1 and
A3:  |.r-c_n(r).n/c_d(r).n .| >= 1/(sqrt 5 * c_d(r).n|^2) and
A4:  |.r-c_n(r).(n+1)/c_d(r).(n+1).| >= 1/(sqrt 5 * c_d(r).(n+1)|^2);
     |.r-c_n(r).n/c_d(r).n .| +  |.r - c_n(r).(n+1)/c_d(r).(n+1).| =
     |.c_n(r).n/c_d(r).n - c_n(r).(n+1)/c_d(r).(n+1).| by A2,DIOPHAN1:17;
       then
A6:  |.c_n(r).n/c_d(r).n - c_n(r).(n+1)/c_d(r).(n+1).|
     >= 1/(s5 * c_d(r).n|^2) + 1/(s5 * c_d(r).(n+1)|^2)
       by A3,A4,XREAL_1:7;
A7:  for n holds c_d(r).n<>0;
A8:  |.c_n(r).n/c_d(r).n-c_n(r).(n+1)/c_d(r).(n+1).| =
     |.c_n(r).(n+1)/c_d(r).(n+1)-c_n(r).n/c_d(r).n.| by COMPLEX1:60;
     1/|.c_d(r).(n+1)*c_d(r).n.| = 1* (c_d(r).(n+1)*c_d(r).n)"; then
     (c_d(r).(n+1)*c_d(r).n)" >=
     1/(s5*c_d(r).n|^2)+1/(s5*c_d(r).(n+1)|^2) by A6,A7,A8,REAL_3:69; then
A10: (c_d(r).(n+1)*c_d(r).n)*(c_d(r).(n+1)*c_d(r).n)" >=
     (c_d(r).(n+1)*c_d(r).n)
     *(1/(s5*c_d(r).n|^2)+1/(s5*c_d(r).(n+1)|^2)) by XREAL_1:64;
     c_d(r).n|^2 = c_d(r).n*c_d(r).n by WSIERP_1:1; then
A15: ((c_d(r).(n+1)*c_d(r).n))/(s5*c_d(r).n|^2)
      =(c_d(r).(n+1)*c_d(r).n)*((s5)"*(c_d(r).n*c_d(r).n)") by XCMPLX_1:204
     .=(c_d(r).(n+1)*c_d(r).n)*((s5)"*((c_d(r).n)"*(c_d(r).n)"))
      by XCMPLX_1:204
     .=(s5)"*c_d(r).(n+1)*(c_d(r).n*(c_d(r).n)")*(c_d(r).n)"
     .=(s5)"*c_d(r).(n+1)*1*(c_d(r).n)" by XCMPLX_0:def 7
     .=(s5)"*c_d(r).(n+1)*(c_d(r).n)";
     c_d(r).(n+1)|^2 = c_d(r).(n+1)*c_d(r).(n+1) by WSIERP_1:1; then
     ((c_d(r).(n+1)*c_d(r).n)* 1)/(s5*c_d(r).(n+1)|^2)
      =(c_d(r).(n+1)*c_d(r).n)*((s5)"*(c_d(r).(n+1)*c_d(r).(n+1))")
      by XCMPLX_1:204
     .=(c_d(r).(n+1)*c_d(r).n)*((s5)"*((c_d(r).(n+1))"*(c_d(r).(n+1))"))
      by XCMPLX_1:204
     .=(sqrt 5)"*c_d(r).n*(c_d(r).(n+1)*(c_d(r).(n+1))")*(c_d(r).(n+1))"
     .=(sqrt 5)"*c_d(r).n*1*(c_d(r).(n+1))" by XCMPLX_0:def 7
     .=(sqrt 5)"*c_d(r).n*(c_d(r).(n+1))"; then
A17: 1>=(sqrt 5)"*c_d(r).(n+1)*(c_d(r).n)"+(s5)"*c_d(r).n*(c_d(r).(n+1))"
     by A15,A10,XCMPLX_0:def 7;
     0 < sqrt 5 by SQUARE_1:20,27; then
A18: s5 * 1>= s5 *
     ((s5)"*c_d(r).(n+1)*(c_d(r).n)"+(s5)"*c_d(r).n*(c_d(r).(n+1))")
     by A17,XREAL_1:64;
A20: s5 *
     ((s5)"*c_d(r).(n+1)*(c_d(r).n)"+(s5)"*c_d(r).n*(c_d(r).(n+1))")
      =s5 *(s5)"*c_d(r).(n+1)*(c_d(r).n)"
       +s5 *(s5)"*c_d(r).n*(c_d(r).(n+1))"
     .=1*c_d(r).(n+1)*(c_d(r).n)"+ (s5 *(s5)")*c_d(r).n*(c_d(r).(n+1))"
     by SQRT2,XCMPLX_0:def 7
     .=c_d(r).(n+1)*(c_d(r).n)"+1*c_d(r).n*(c_d(r).(n+1))"
     by SQRT2,XCMPLX_0:def 7
     .=c_d(r).(n+1)/c_d(r).n +c_d(r).n/c_d(r).(n+1);
     s5 <> c_d(r).(n+1)/c_d(r).n +c_d(r).n/c_d(r).(n+1)
       by PEPIN:59,IRRAT_1:1; then
     s5 > c_d(r).(n+1)/c_d(r).n +c_d(r).n/c_d(r).(n+1)
     by A18,A20,XXREAL_0:1;
     hence thesis by XCMPLX_1:213;
   end;
