reserve n for Nat,
        i,j,i1,i2,i3,i4,i5,i6 for Element of n,
        p,q,r for n-element XFinSequence of NAT;
reserve i,j,n,n1,n2,m,k,l,u,e,p,t for Nat,
        a,b for non trivial Nat,
        x,y for Integer,
        r,q for Real;

theorem Th14:
  0 < p|^n < a implies p|^n + Py(a,n)*(a-p) <= Px(a,n)
proof
  assume
A1: 0 < p|^n < a;
A2: Px(a,0)= 1 & Py(a,0)= 0 & p|^0 =1 by HILB10_1:3,NEWTON:4;
A3: Py(a,0+1) = 1+0*a & Px(a,0+1) = 1*a +0 *(a^2-'1) by A2,HILB10_1:6;
A4: Py(a,1+1) = Px(a,1) + Py(a,1)*a by HILB10_1:6;
A5: a*a>=0+1 & a*a =a^2 by INT_1:7,SQUARE_1:def 1;
  Px(a,n)^2 - (a^2-'1) *Py(a,n)^2 = 1 by HILB10_1:7;
  then
A6: Px(a,n)^2 = (a^2-'1) *Py(a,n)^2 +1;
A7: ((a-1) *Py(a,n))^2=((a-1) *Py(a,n))* ((a-1) *Py(a,n)) &
    Py(a,n)^2 = Py(a,n)* Py(a,n) by SQUARE_1:def 1;
  - (2*a - 1) <= -1 by XREAL_1:24,NAT_1:14;
  then
A8: a^2+-(2*a -1) <= a^2+-1 & (a-1)^2=(a-1)*(a-1)
    by XREAL_1:6,SQUARE_1:def 1;
  then (a-1)^2 *Py(a,n)^2 <= (a^2-1)*Py(a,n)^2 by A5,XREAL_1:64;
  then ((a-1) *Py(a,n))^2 <= (a^2-'1)*Py(a,n)^2 by A8,A7,XREAL_1:233,A5;
  then ((a-1) *Py(a,n))^2 < Px(a,n)^2 by A6,NAT_1:13;
  then
A9: (a-1) *Py(a,n) < Px(a,n) by SQUARE_1:15;
  per cases by NAT_1:23;
  suppose n=0 or n=1;
    hence thesis by A2,A3;
  end;
  suppose p=1;
    hence thesis by A9,INT_1:7;
  end;
  suppose
A10:  n >=1+1 & p <>1;
    then n>=1 by NAT_1:13;
    then p<>0 by A1,NEWTON:11;
    then
A11:  Py(a,n)*p >= Py(a,n)*2 by XREAL_1:64,A10,NAT_1:23;
    n >2 or n=2 by A10,XXREAL_0:1;
    then Py(a,n) > Py(a,2) or Py(a,n) = Py(a,2) by HILB10_1:11;
    then Py(a,n)+Py(a,n) >= Py(a,n)+(a+a) by A3,A4,XREAL_1:6;
    then
A12:  Py(a,n)*p >= Py(a,n)+(a+a) by A11,XXREAL_0:2;
    Py(a,n)+a+a >= Py(a,n)+a+1 by NAT_1:14, XREAL_1:6;
    then Py(a,n)*p >= Py(a,n)+a+1 by A12,XXREAL_0:2;
    then Py(a,n)*p > Py(a,n)+a by NAT_1:13;
    then -(Py(a,n)*p -a) < - Py(a,n) by XREAL_1:20,24;
    then Py(a,n)*a + -(Py(a,n)*p -a) < Py(a,n)*a + - Py(a,n) by XREAL_1:8;
    then
A13:  Py(a,n)*(a - p) +a < Px(a,n) by A9,XXREAL_0:2;
    Py(a,n)*(a - p) +a > Py(a,n)*(a - p) + p|^n by A1, XREAL_1:8;
    hence thesis by A13,XXREAL_0:2;
  end;
end;
