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 Th1:
   1 < s & s+1/s < sqrt 5 implies
   s < (sqrt 5+1)/2 & 1/s > (sqrt 5 -1)/2
   proof
     assume that
A1:  1 < s and
A2:  s + 1/s < sqrt 5;
A4:  (-1* sqrt 5)^2 = (-1)^2*(sqrt 5)^2 .= 5 by SQUARE_1:def 2;
     s*(s+1/s) < s* sqrt 5 by A1,A2,XREAL_1:68; then
     s*s + s*1/s < s* sqrt 5; then
     s*s + 1 < s* sqrt 5 by A1,XCMPLX_0:def 7; then
A5:  s*s + 1 - s* sqrt 5 < s* sqrt 5 - s * sqrt 5 by XREAL_1:14;
     set a = 1, b=-sqrt 5, c = 1;
A7:  a * s^2 + b * s + c < 0 by A5;
A6:  delta(1,-sqrt 5,1) = (- sqrt 5)^2 - 4*1*1 by QUIN_1:def 1
      .=1 by A4; then
     (-b+sqrt delta(a,b,c))/(2 * a) = (sqrt 5+1)/2; then
A8:  s < (sqrt 5+1)/2 by A7,A6,QUIN_1:26;
     sqrt 5-1 <> 0 by SQRT2; then
A9:  (sqrt 5-1)/(sqrt 5-1) = 1 by XCMPLX_1:60;
     1/((sqrt 5+1)/2)
   = (2/(sqrt 5+1)) * ((sqrt 5-1)/(sqrt 5-1)) by A9,XCMPLX_1:57
  .= (2*(sqrt 5-1))/((sqrt 5+1) * (sqrt 5-1)) by XCMPLX_1:76
  .= (2*(sqrt 5-1))/((sqrt 5)^2-1)
  .= (2*(sqrt 5-1))/(5-1) by SQUARE_1:def 2
  .= (sqrt 5 -1)/2;
     hence thesis by A1,A8,XREAL_1:88;
   end;
