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
  for k be positive Nat holds k+1 is prime iff
    ex a,b,c,d,e,f,g,h,i,j,l,m,n,o,p,q,r,s,t,u,w,v,x,y,z be Nat st
      0 = (w*z+h+j -q)^2 +
          ((g*k+g+k)*(h+j)+h-z)^2 +
          ((2*k) |^3 *(2*k+2)*(n+1) |^2+1 - f^2)^2 +
          (p+q+z+2*n-e)^2 +
          (e |^3 *(e+2)*(a+1) |^2+1 -o^2)^2 +
          (x^2-(a^2-'1)*y^2-1)^2 +
          (16*(a^2-1)*r^2*y^2*y^2+1 -u^2)^2+
          (((a+u^2*(u^2-a))^2-1)*(n+4*d*y)^2+1 -(x+c*u)^2)^2 +
          (m^2-(a^2-'1)*l^2-1)^2 +
          (k+i*(a-1)-l)^2 +
          (n+l+v-y)^2 +
          (p+l*(a-n-1)+b*(2*a*(n+1)-(n+1)^2-1)-m)^2+
          (q+y*(a-p-1)+s*(2*a*(p+1)-(p+1)^2-1)-x)^2+
          (z+p*l*(a-p)+t*(2*a*p-p^2-1)-p*m)^2
proof
  let k be positive Nat;
  thus k+1 is prime implies
    ex a,b,c,d,e,f,g,h,i,j,l,m,n,o,p,q,r,s,t,u,w,v,x,y,z be Nat st
      0=(w*z+h+j -q)^2 + ((g*k+g+k)*(h+j)+h-z)^2 +
        ((2*k) |^3 *(2*k+2)*(n+1) |^2+1 - f^2)^2 + (p+q+z+2*n-e)^2 +
        (e |^3 *(e+2)*(a+1) |^2+1 -o^2)^2 + (x^2-(a^2-'1)*y^2-1)^2 +
        (16*(a^2-1)*r^2*y^2*y^2+1 -u^2)^2+
        (((a+u^2*(u^2-a))^2-1)*(n+4*d*y)^2+1 -(x+c*u)^2)^2+
        (m^2-(a^2-'1)*l^2-1)^2 + (k+i*(a-1)-l)^2 +
        (n+l+v-y)^2 +(p+l*(a-n-1)+b*(2*a*(n+1)-(n+1)^2-1)-m)^2+
        (q+y*(a-p-1)+s*(2*a*(p+1)-(p+1)^2-1)-x)^2+
        (z+p*l*(a-p)+t*(2*a*p-p^2-1)-p*m)^2
  proof
    assume k+1 is prime;
    then consider a,b,c,d,e,f,g,h,i,j,l,m,n,o,p,q,r,s,t,u,w,v,x,y,z be Nat
      such that
A1:   q = w*z+h+j and
A2:   z = (g*k+g+k)*(h+j)+h and
A3:   (2*k) |^3 *(2*k+2)*(n+1) |^2+1 =f^2 and
A4:   e = p+q+z+2*n and
A5:   e |^3 *(e+2)*(a+1) |^2+1 =o^2 and
A6:   [x,y] is Pell's_solution of a^2-'1 and
A7:   u^2 = 16*(a^2-1)*r^2*y^2*y^2+1 and
A8:   (x+c*u)^2 = ((a+u^2*(u^2-a))^2-1)*(n+4*d*y)^2+1 and
A9:   [m,l] is Pell's_solution of a^2-'1 and
A10:  l = k+i*(a-1) and
A11:  n+l+v = y and
A12:  m = p+l*(a-n-1)+b*(2*a*(n+1)-(n+1)^2-1) and
A13:  x = q+y*(a-p-1)+s*(2*a*(p+1)-(p+1)^2-1) and
A14:  p*m = z+p*l*(a-p)+t*(2*a*p-p^2-1) by Th30;
    reconsider x1=w*z+h+j -q,
               x2=(g*k+g+k)*(h+j)+h-z,
               x3=(2*k) |^3 *(2*k+2)*(n+1) |^2+1 - f^2,
               x4=p+q+z+2*n-e,
               x5=e |^3 *(e+2)*(a+1) |^2+1 -o^2 as Integer;
    reconsider x6=x^2-(a^2-'1)*y^2-1,
               x7=16*(a^2-1)*r^2*y^2*y^2+1 -u^2,
               x8=((a+u^2*(u^2-a))^2-1)*(n+4*d*y)^2+1 -(x+c*u)^2,
               x9=m^2-(a^2-'1)*l^2-1,
               x10=k+i*(a-1)-l as Integer;
    reconsider x11= n+l+v-y,
               x12=p+l*(a-n-1)+b*(2*a*(n+1)-(n+1)^2-1)-m,
               x13=q+y*(a-p-1)+s*(2*a*(p+1)-(p+1)^2-1)-x,
               x14=z+p*l*(a-p)+t*(2*a*p-p^2-1)-p*m  as Integer;
    take a,b,c,d,e,f,g,h,i,j,l,m,n,o,p,q,r,s,t,u,w,v,x,y,z;
A15:  x1^2=x1*x1 & x2^2=x2*x2 & x3^2=x3*x3 & x4^2=x4*x4 & x5^2=x5*x5 &
      x6^2=x6*x6 & x7^2=x7*x7 & x8^2=x8*x8 & x9^2=x9*x9 & x10^2=x10*x10 &
      x11^2=x11*x11 & x12^2=x12*x12 & x13^2=x13*x13 & x14^2=x14*x14
      by SQUARE_1:def 1;
    m^2-(a^2-'1)*l^2=1 by A9,Lm1;
    hence thesis by A1,A2,A3,A4,A15,A5,A6,Lm1,A7,A8,A10,A11,A12,A13,A14;
  end;
  given a,b,c,d,e,f,g,h,i,j,l,m,n,o,p,q,r,s,t,u,w,v,x,y,z be Nat such that
A16: 0= (w*z+h+j -q)^2 + ((g*k+g+k)*(h+j)+h-z)^2 +
        ((2*k) |^3 *(2*k+2)*(n+1) |^2+1 - f^2)^2 +
        (p+q+z+2*n-e)^2+ (e |^3 *(e+2)*(a+1) |^2+1 -o^2)^2 +
        (x^2-(a^2-'1)*y^2-1)^2 +
        (16*(a^2-1)*r^2*y^2*y^2+1 -u^2)^2+
        (((a+u^2*(u^2-a))^2-1)*(n+4*d*y)^2+1 -(x+c*u)^2)^2+
        (m^2-(a^2-'1)*l^2-1)^2 + (k+i*(a-1)-l)^2 +
        (n+l+v-y)^2 + (p+l*(a-n-1)+b*(2*a*(n+1)-(n+1)^2-1)-m)^2+
        (q+y*(a-p-1)+s*(2*a*(p+1)-(p+1)^2-1)-x)^2+
        (z+p*l*(a-p)+t*(2*a*p-p^2-1)-p*m)^2;
  reconsider x1=w*z+h+j -q,
             x2=(g*k+g+k)*(h+j)+h-z,
             x3=(2*k) |^3 *(2*k+2)*(n+1) |^2+1 - f^2,
             x4 = p+q+z+2*n-e,
             x5=e |^3 *(e+2)*(a+1) |^2+1 -o^2 as Integer;
  reconsider x6=x^2-(a^2-'1)*y^2-1,
             x7=16*(a^2-1)*r^2*y^2*y^2+1 -u^2,
             x8=((a+u^2*(u^2-a))^2-1)*(n+4*d*y)^2+1 -(x+c*u)^2,
             x9=m^2-(a^2-'1)*l^2-1,
             x10=k+i*(a-1)-l as Integer;
  reconsider x11= n+l+v-y,
             x12=p+l*(a-n-1)+b*(2*a*(n+1)-(n+1)^2-1)-m,
             x13=q+y*(a-p-1)+s*(2*a*(p+1)-(p+1)^2-1)-x,
             x14=z+p*l*(a-p)+t*(2*a*p-p^2-1)-p*m as Integer;
A17: x1^2=x1*x1 & x2^2=x2*x2 & x3^2=x3*x3 & x4^2=x4*x4 & x5^2=x5*x5 &
     x6^2=x6*x6 & x7^2=x7*x7 & x8^2=x8*x8 & x9^2=x9*x9 & x10^2=x10*x10 &
     x11^2=x11*x11 & x12^2=x12*x12 & x13^2=x13*x13 & x14^2=x14*x14
     by SQUARE_1:def 1;
  ex a,b,c,d,e,f,g,h,i,j,l,m,n,o,p,q,r,s,t,u,w,v,x,y,z be Nat st
      q = w*z+h+j &
      z = (g*k+g+k)*(h+j)+h &
      (2*k) |^3 *(2*k+2)*(n+1) |^2+1 =f^2 &
      e=p+q+z+2*n &
      e |^3 *(e+2)*(a+1) |^2+1 =o^2 &
      [x,y] is Pell's_solution of a^2-'1 &
      u^2 = 16*(a^2-1)*r^2*y^2*y^2+1 &
      (x+c*u)^2 = ((a+u^2*(u^2-a))^2-1)*(n+4*d*y)^2+1 &
      [m,l] is Pell's_solution of a^2-'1 &
      l=k+i*(a-1) &
      n+l+v=y &
      m=p+l*(a-n-1)+b*(2*a*(n+1)-(n+1)^2-1) &
      x= q+y*(a-p-1)+s*(2*a*(p+1)-(p+1)^2-1) &
      p*m = z+p*l*(a-p)+t*(2*a*p-p^2-1)
    proof
      take a,b,c,d,e,f,g,h,i,j,l,m,n,o,p,q,r,s,t,u,w,v,x,y,z;
      x1=0 by A16,A17;
      hence q = w*z+h+j;
      x2=0 by A16,A17;
      hence z = (g*k+g+k)*(h+j)+h;
      x3=0 by A16,A17;
      hence (2*k) |^3 *(2*k+2)*(n+1) |^2+1 =f^2;
      x4=0 by A16,A17;
      hence e=p+q+z+2*n;
      x5=0 by A16,A17;
      hence e |^3 *(e+2)*(a+1) |^2+1 =o^2;
      x6=0 by A16,A17;
      hence [x,y] is Pell's_solution of a^2-'1 by Lm1;
      x7=0 by A16,A17;
      hence u^2 = 16*(a^2-1)*r^2*y^2*y^2+1;
      x8=0 by A16,A17;
      hence (x+c*u)^2 = ((a+u^2*(u^2-a))^2-1)*(n+4*d*y)^2+1;
      x9=0 by A16,A17;
      hence [m,l] is Pell's_solution of a^2-'1 by Lm1;
      x10=0 by A16,A17;
      hence l=k+i*(a-1);
      x11=0 by A16,A17;
      hence n+l+v=y;
      x12=0 by A16,A17;
      hence m=p+l*(a-n-1)+b*(2*a*(n+1)-(n+1)^2-1);
      x13=0 by A16,A17;
      hence x= q+y*(a-p-1)+s*(2*a*(p+1)-(p+1)^2-1);
      x14=0 by A16,A17;
      hence p*m = z+p*l*(a-p)+t*(2*a*p-p^2-1);
    end;
    hence thesis by Th30;
end;
