reserve i,j,k,n,m for Nat,
        X for set,
        b,s for bag of X,
        x for object;
reserve O for Ordinal,
        R for right_zeroed add-associative right_complementable
          right_unital distributive non trivial doubleLoopStr,
        p for Polynomial of O, R;
reserve O for Ordinal,
        R for right_zeroed add-associative right_complementable
             right_unital distributive non trivial doubleLoopStr,
        p for Polynomial of O, R;

theorem Th82:
  ex Z be INT -valued Polynomial of 17,F_Real st
    vars Z c= {0}\/(17\8) &
  for xk be Nat st xk > 0 holds
    xk+1 is prime iff
    ex x being INT -valued Function of 17, F_Real st
       x/.8 = xk & x/.9 is positive Nat & x/.10 is positive Nat &
       x/.11 is positive Nat & x/.12 is positive Nat & x/.13 is positive Nat &
       x/.14 is Nat & x/.15 is Nat & x/.16 is Nat & x/.0 is Nat &
       eval(Z,x) = 0.F_Real
proof
  set N= 17;
  set EB = EmptyBag N;
  set VARs = N\8;
A1: n in VARs iff 8 <= n < N
  proof
    thus n in VARs implies 8 <= n < N
    proof
      assume n in VARs;
      then n in Segm N & not n in Segm 8 by XBOOLE_0:def 5;
      hence thesis by NAT_1:44;
    end;
    assume 8 <= n < N;
    then n in Segm N & not n in Segm 8 by NAT_1:44;
    hence thesis by XBOOLE_0:def 5;
  end;
  set k=8, Pk = Monom(1.F_Real,EB +*(k,1));
A2:vars Pk c= VARs
  proof
    vars Pk c= {k} c= VARs by A1,Th48,ZFMISC_1:31;
    hence vars Pk c= VARs;
  end;
A3:for x be Function of N,F_Real holds eval(Pk,x) = x /. k
  proof
    let x be Function of N,F_Real;
    k in VARs by A1;
    then eval(Pk,x) = 1.F_Real * (x/.k) by Th27;
    hence thesis;
  end;
  set f=9, Pf = Monom(1.F_Real,EB +*(f,1));
A4:vars Pf c= VARs
  proof
    vars Pf c= {f} c= VARs by A1,Th48,ZFMISC_1:31;
    hence thesis;
  end;
A5:for x be Function of N,F_Real holds eval(Pf,x) = x /. f
  proof
    let x be Function of N,F_Real;
    f in VARs by A1;
    then eval(Pf,x) = 1.F_Real * (x/.f) by Th27;
    hence thesis;
  end;
  set i=10, Pi = Monom(1.F_Real,EB +*(i,1));
A6:vars Pi c= VARs
  proof
    vars Pi c= {i} c= VARs by A1,Th48,ZFMISC_1:31;
    hence thesis;
  end;
A7:for x be Function of N,F_Real holds eval(Pi,x) = x /. i
  proof
    let x be Function of N,F_Real;
    i in VARs by A1;
    then eval(Pi,x) = 1.F_Real * (x/.i) by Th27;
    hence thesis;
  end;
  set j=11, Pj = Monom(1.F_Real,EB +*(j,1));
A8:vars Pj c= VARs
  proof
    vars Pj c= {j} c= VARs by A1,Th48,ZFMISC_1:31;
    hence thesis;
  end;
A9:for x be Function of N,F_Real holds eval(Pj,x) = x /. j
  proof
    let x be Function of N,F_Real;
    j in VARs by A1;
    then eval(Pj,x) = 1.F_Real * (x/.j) by Th27;
    hence thesis;
  end;
  set m=12, Pm = Monom(1.F_Real,EB +*(m,1));
A10:vars Pm c= VARs
  proof
    vars Pm c= {m} c= VARs by A1,Th48,ZFMISC_1:31;
    hence thesis;
  end;
A11:for x be Function of N,F_Real holds eval(Pm,x) = x /. m
  proof
    let x be Function of N,F_Real;
    m in VARs by A1;
    then eval(Pm,x) = 1.F_Real * (x/.m) by Th27;
    hence thesis;
  end;
  set u=13, Pu = Monom(1.F_Real,EB +*(u,1));
A12:vars Pu c= VARs
  proof
    vars Pu c= {u} c= VARs by A1,Th48,ZFMISC_1:31;
    hence thesis;
  end;
A13:for x be Function of N,F_Real holds eval(Pu,x) = x /. u
  proof
    let x be Function of N,F_Real;
    u in VARs by A1;
    then eval(Pu,x) = 1.F_Real * (x/.u) by Th27;
    hence thesis;
  end;
  set r=14, Pr = Monom(1.F_Real,EB +*(r,1));
A14:vars Pr c= VARs
  proof
    vars Pr c= {r} c= VARs by A1,Th48,ZFMISC_1:31;
    hence thesis;
  end;
A15:for x be Function of N,F_Real holds eval(Pr,x) = x /. r
  proof
    let x be Function of N,F_Real;
    r in VARs by A1;
    then eval(Pr,x) = 1.F_Real * (x/.r) by Th27;
    hence thesis;
  end;
  set s=15, Ps = Monom(1.F_Real,EB +*(s,1));
A16:vars Ps c= VARs
  proof
    vars Ps c= {s} c= VARs by A1,Th48,ZFMISC_1:31;
    hence thesis;
  end;
A17:for x be Function of N,F_Real holds eval(Ps,x) = x /. s
  proof
    let x be Function of N,F_Real;
    s in VARs by A1;
    then eval(Ps,x) = 1.F_Real * (x/.s) by Th27;
    hence thesis;
  end;
  set t=16, Pt = Monom(1.F_Real,EB +*(t,1));
A18:vars Pt c= VARs
  proof
    vars Pt c= {t} c= VARs by A1,Th48,ZFMISC_1:31;
    hence thesis;
  end;
A19:for x be Function of N,F_Real holds eval(Pt,x) = x /. t
  proof
    let x be Function of N,F_Real;
    t in VARs by A1;
    then eval(Pt,x) = 1.F_Real * (x/.t) by Th27;
    hence thesis;
  end;
  reconsider Hund = 100 as integer Element of F_Real by XREAL_0:def 1;
  set O = 1_(N,F_Real);
A20: vars O c= VARs by Th38;
  reconsider W = Hund *(Pf*'Pk*'(Pk+O)) as INT -valued Polynomial of N,F_Real;
A21: vars W c= VARs
  proof
    vars (Pf*'Pk) c= VARs & vars (Pk+O) c= VARs by A20,A4,A2,Th79,Th78;
    then vars (Pf*'Pk*'(Pk+O)) c= VARs by Th79;
    hence thesis by Th80;
  end;
A22:for x be Function of N,F_Real holds
    eval(W,x) = Hund * x /. f * x/.k * (x/.k+1.F_Real)
  proof
    let x be Function of N,F_Real;
    thus eval(W,x)
    = Hund * eval(Pf*'Pk*'(Pk+O),x) by POLYNOM7:29
    .= Hund * (eval(Pf*'Pk,x) * eval(Pk+O,x)) by POLYNOM2:25
    .= Hund * (eval(Pf*'Pk,x) * (eval(Pk,x)+eval(O,x)))
    by POLYNOM2:23
    .= Hund * (eval(Pf*'Pk,x) * (eval(Pk,x)+1.F_Real))
    by POLYNOM2:21
    .= Hund * (eval(Pf*'Pk,x) * (x/.k+1.F_Real)) by A3
    .= Hund * (eval(Pf,x) * eval(Pk,x) * (x/.k+1.F_Real))
    by POLYNOM2:25
    .= Hund * (eval(Pf,x) * x/.k * (x/.k+1.F_Real)) by A3
    .= Hund * (x/.f * x/.k * (x/.k+1.F_Real)) by A5
    .= Hund * x /. f * x/.k * (x/.k+1.F_Real);
  end;
  reconsider U = (Hund *((Pu*'Pu*'Pu)*'(W*'W*'W)))+O
  as INT -valued Polynomial of N,F_Real;
A23: vars U c= VARs
  proof
    vars (Pu*'Pu) c= VARs & vars (W*'W) c= VARs by A21,A12,Th79;
    then vars (Pu*'Pu*'Pu) c= VARs & vars (W*'W*'W) c= VARs
    by A21,A12,Th79;
    then vars ((Pu*'Pu*'Pu)*'(W*'W*'W)) c= VARs by Th79;
    then vars (Hund *((Pu*'Pu*'Pu)*'(W*'W*'W))) c= VARs by Th80;
    hence thesis by A20,Th78;
  end;
A24:for x be Function of N,F_Real holds
    eval(U,x) = Hund * ((x /. u)|^3 ) * eval(W,x)|^3 + 1.F_Real
  proof
    let x be Function of N,F_Real;
A25: eval(Pu,x) = x /. u by A13;
A26: eval(Pu*'Pu*'Pu,x) = eval(Pu*'Pu,x) * eval(Pu,x) by POLYNOM2:25
    .= (x/.u) * (x/.u) * (x/.u) by A25,POLYNOM2:25
    .= (x/.u)|^(1+1) * (x/.u)|^1 by NEWTON:6
    .= (x/.u)|^(1+1+1) by NEWTON:6;
A27: eval(W*'W*'W,x) = eval(W*'W,x) * eval(W,x) by POLYNOM2:25
    .= eval(W,x) * eval(W,x) * eval(W,x) by POLYNOM2:25
    .= eval(W,x)|^(1+1) * eval(W,x)|^1 by NEWTON:6
    .= eval(W,x)|^(1+1+1) by NEWTON:6;
    thus eval(U,x)
    =eval(Hund *((Pu*'Pu*'Pu)*'(W*'W*'W)),x) + eval(O,x) by POLYNOM2:23
    .=eval(Hund *((Pu*'Pu*'Pu)*'(W*'W*'W)),x) + 1.F_Real by POLYNOM2:21
    .=Hund *eval((Pu*'Pu*'Pu)*'(W*'W*'W),x) + 1.F_Real by POLYNOM7:29
    .=Hund *(eval(Pu*'Pu*'Pu,x)*
    eval((W*'W*'W),x)) + 1.F_Real by POLYNOM2:25
    .=Hund * ((x /. u)|^3 ) * eval(W,x)|^3 + 1.F_Real by A26,A27;
  end;
  reconsider M = Hund *(Pm*'U*'W)+O as INT -valued Polynomial of N,F_Real;
A28: vars M c= VARs
  proof
    vars (Pm*'U) c= VARs by A23,A10,Th79;
    then vars (Pm*'U*'W) c= VARs by A21,Th79;
    then vars (Hund *(Pm*'U*'W)) c= VARs by Th80;
    hence thesis by Th78,A20;
  end;
A29:for x be Function of N,F_Real holds
  eval(M,x) = Hund * (x /. m) * eval(U,x) * eval(W,x) + 1.F_Real
  proof
    let x be Function of N,F_Real;
    thus eval(M,x) = eval(Hund *(Pm*'U*'W),x) + eval(O,x) by POLYNOM2:23
    .=eval(Hund *(Pm*'U*'W),x) + 1.F_Real by POLYNOM2:21
    .=Hund * eval(Pm*'U*'W,x) + 1.F_Real by POLYNOM7:29
    .=Hund * (eval(Pm*'U,x)*eval(W,x)) + 1.F_Real by POLYNOM2:25
    .=Hund * (eval(Pm,x)*eval(U,x)* eval(W,x))+ 1.F_Real by POLYNOM2:25
    .=Hund * ((x/.m) *eval(U,x)* eval(W,x))+ 1.F_Real by A11
    .=Hund * (x /. m) * eval(U,x) * eval(W,x) + 1.F_Real;
  end;
  reconsider S= (M-O)*'Ps+Pk+O as INT -valued Polynomial of N,F_Real;
A30: vars S c= VARs
  proof
    vars (M-O) c= VARs by Th81,A28,A20;
    then vars((M-O)*'Ps) c= VARs by A16,Th79;
    then vars((M-O)*'Ps+Pk) c= VARs by A2,Th78;
    hence thesis by A20,Th78;
  end;
A31:for x be Function of N,F_Real holds
      eval(S,x) = (eval(M,x) - 1.F_Real) * (x/.s) + (x/.k) + 1.F_Real
  proof
    let x be Function of N,F_Real;
    thus eval(S,x)
    = eval((M-O)*'Ps+Pk,x) + eval(O,x) by POLYNOM2:23
    .= eval((M-O)*'Ps+Pk,x) +1.F_Real by POLYNOM2:21
    .= eval((M-O)*'Ps,x)+eval(Pk,x) +1.F_Real by POLYNOM2:23
    .= eval((M-O)*'Ps,x)+(x/.k) +1.F_Real by A3
    .= (eval(M-O,x)*eval(Ps,x))+(x/.k) +1.F_Real by POLYNOM2:25
    .= ((eval(M,x) - eval(O,x))*eval(Ps,x))+(x/.k) +1.F_Real by POLYNOM2:24
    .= ((eval(M,x) - 1.F_Real)*eval(Ps,x))+(x/.k) +1.F_Real by POLYNOM2:21
    .=(eval(M,x) - 1.F_Real) * (x/.s) + (x/.k) + 1.F_Real by A17;
  end;
  reconsider T= (M*'U-O)*'Pt+W-Pk+O as INT -valued Polynomial of N,F_Real;
A32: vars T c= VARs
  proof
    vars (M*'U) c= VARs by A23,A28,Th79;
    then vars (M*'U-O) c= VARs by A20,Th81;
    then vars ((M*'U-O)*'Pt) c= VARs by A18,Th79;
    then vars ((M*'U-O)*'Pt+W) c= VARs by A21,Th78;
    then vars ((M*'U-O)*'Pt+W-Pk) c= VARs by Th81,A2;
    hence thesis by A20,Th78;
  end;
A33:for x be Function of N,F_Real holds
    eval(T,x) = (eval(M,x)*eval(U,x) - 1.F_Real)*(x/.t)+
    eval(W,x)-(x/.k)+1.F_Real
  proof
    let x be Function of N,F_Real;
    thus eval(T,x)
    = eval((M*'U-O)*'Pt+W-Pk,x) + eval(O,x) by POLYNOM2:23
    .=eval((M*'U-O)*'Pt+W-Pk,x) + 1.F_Real by POLYNOM2:21
    .=eval((M*'U-O)*'Pt+W,x)-eval(Pk,x) + 1.F_Real by POLYNOM2:24
    .=eval((M*'U-O)*'Pt+W,x)-(x/.k)+1.F_Real by A3
    .=eval((M*'U-O)*'Pt,x)+eval(W,x)-(x/.k)+1.F_Real by POLYNOM2:23
    .=(eval(M*'U-O,x)*eval(Pt,x))+eval(W,x)-(x/.k)+1.F_Real by POLYNOM2:25
    .=(eval(M*'U-O,x)*(x/.t))+eval(W,x)-(x/.k)+1.F_Real by A19
    .=((eval(M*'U,x)-eval(O,x))* (x/.t))+eval(W,x)-(x/.k)+1.F_Real
    by POLYNOM2:24
    .=((eval(M*'U,x)-1.F_Real)* (x/.t))+eval(W,x)-(x/.k)+1.F_Real
    by POLYNOM2:21
    .=((eval(M,x)*eval(U,x)-1.F_Real)* (x/.t))+eval(W,x)-(x/.k)+1.F_Real
    by POLYNOM2:25;
  end;
  reconsider Two=2 as integer Element of F_Real by XREAL_0:def 1;
  reconsider Q= Two*(M*'W)-(W*'W)-O as INT -valued Polynomial of N,F_Real;
A34: vars Q c= VARs
  proof
    vars (M*'W) c= VARs by A28,A21,Th79;
    then vars (Two*(M*'W)) c= VARs & vars (W*'W) c= VARs by A21,Th79,Th80;
    then vars (Two*(M*'W)-(W*'W)) c= VARs by Th81;
    hence thesis by A20,Th81;
  end;
A35:for x be Function of N,F_Real holds
  eval(Q,x) = Two*eval(M,x)*eval(W,x) - eval(W,x)^2 - 1.F_Real
  proof
    let x be Function of N,F_Real;
    thus eval(Q,x) = eval(Two*(M*'W)-(W*'W),x) - eval(O,x) by POLYNOM2:24
    .= eval(Two*(M*'W)-(W*'W),x) - 1.F_Real by POLYNOM2:21
    .= eval(Two*(M*'W),x)-eval(W*'W,x) - 1.F_Real by POLYNOM2:24
    .= Two*eval(M*'W,x)-eval(W*'W,x) - 1.F_Real by POLYNOM7:29
    .= Two*eval(M*'W,x)-eval(W,x)*eval(W,x) - 1.F_Real by POLYNOM2:25
    .= Two*(eval(M,x)*eval(W,x))-eval(W,x)*eval(W,x) - 1.F_Real
    by POLYNOM2:25
    .=Two*eval(M,x)*eval(W,x) - eval(W,x)^2 - 1.F_Real by SQUARE_1:def 1;
  end;
  reconsider L= (Pk+O)*'Q as INT -valued Polynomial of N,F_Real;
A36: vars L c= VARs
  proof
    vars (Pk+O) c= VARs by A2,A20,Th78;
    hence thesis by A34,Th79;
  end;
A37:for x be Function of N,F_Real holds
      eval(L,x) = (x/.k + 1.F_Real)*eval(Q,x)
  proof
    let x be Function of N,F_Real;
    thus eval(L,x) = eval(Pk+O,x) * eval(Q,x) by POLYNOM2:25
    .= (eval(Pk,x)+eval(O,x)) * eval(Q,x) by POLYNOM2:23
    .= (eval(Pk,x)+1.F_Real) * eval(Q,x) by POLYNOM2:21
    .= (x/.k + 1.F_Real)*eval(Q,x) by A3;
  end;
  reconsider A= M*'(U+O) as INT -valued Polynomial of N,F_Real;
A38: vars A c= VARs
  proof
    vars (U+O) c= VARs by A20,A23,Th78;
    hence thesis by A28,Th79;
  end;
A39:for x be Function of N,F_Real holds
  eval(A,x) = eval(M,x)*(eval(U,x) + 1.F_Real)
  proof
    let x be Function of N,F_Real;
    thus eval(A,x) = eval(M,x)*eval(U+O,x) by POLYNOM2:25
    .= eval(M,x)*(eval(U,x)+eval(O,x)) by POLYNOM2:23
    .= eval(M,x)*(eval(U,x) + 1.F_Real) by POLYNOM2:21;
  end;
  reconsider B= W+O as INT -valued Polynomial of N,F_Real;
A40: vars B c= VARs by A20,A21,Th78;
A41:for x be Function of N,F_Real holds eval(B,x) = eval(W,x) + 1.F_Real
  proof
    let x be Function of N,F_Real;
    thus eval(B,x) = eval(W,x)+eval(O,x) by POLYNOM2:23
    .= eval(W,x) + 1.F_Real by POLYNOM2:21;
  end;
  reconsider C= Pr+W+O as INT -valued Polynomial of N,F_Real;
A42: vars C c= VARs
  proof
    vars (Pr+W) c= VARs by A14,A21,Th78;
    hence thesis by A20,Th78;
  end;
A43:for x be Function of N,F_Real holds
    eval(C,x) = (x/.r) + eval(W,x) + 1.F_Real
  proof
    let x be Function of N,F_Real;
    thus eval(C,x) = eval(Pr+W,x)+eval(O,x) by POLYNOM2:23
    .= eval(Pr+W,x)+1.F_Real by POLYNOM2:21
    .= eval(Pr,x) + eval(W,x) + 1.F_Real by POLYNOM2:23
    .= (x/.r) + eval(W,x) + 1.F_Real by A15;
  end;
  reconsider D= (A*'A-O)*'(C*'C)+O as INT -valued Polynomial of N,F_Real;
A44: vars D c= VARs
  proof
    vars (A*'A) c= VARs by A38,Th79;
    then vars (A*'A-O) c= VARs & vars (C*'C) c= VARs by A42,A20,Th81,Th79;
    then vars ((A*'A-O)*'(C*'C)) c= VARs by Th79;
    hence thesis by A20,Th78;
  end;
A45:for x be Function of N,F_Real holds
  eval(D,x) = (eval(A,x)^2 - 1.F_Real)* eval(C,x)^2+ 1.F_Real
  proof
    let x be Function of N,F_Real;
    thus eval(D,x) = eval((A*'A-O)*'(C*'C),x)+eval(O,x) by POLYNOM2:23
    .= eval((A*'A-O)*'(C*'C),x)+1.F_Real by POLYNOM2:21
    .= (eval(A*'A-O,x)*eval(C*'C,x))+1.F_Real by POLYNOM2:25
    .= (eval(A*'A-O,x)*(eval(C,x)*eval(C,x)))+1.F_Real by POLYNOM2:25
    .= ((eval(A*'A,x)-eval(O,x))*(eval(C,x)*eval(C,x)))+1.F_Real
    by POLYNOM2:24
    .= ((eval(A*'A,x)-eval(O,x))*eval(C,x)^2)+1.F_Real by SQUARE_1:def 1
    .= ((eval(A*'A,x)-1.F_Real)*eval(C,x)^2)+1.F_Real by POLYNOM2:21
    .= ((eval(A,x)*eval(A,x)-1.F_Real)*eval(C,x)^2)+1.F_Real
    by POLYNOM2:25
    .=(eval(A,x)^2 - 1.F_Real)* eval(C,x)^2+ 1.F_Real by SQUARE_1:def 1;
  end;
  reconsider E= Two*(Pi*'C*'C*'L*'D) as INT -valued Polynomial of N,F_Real;
A46: vars E c= VARs
  proof
    vars (Pi*'C) c= VARs by A42,A6,Th79;
    then vars (Pi*'C*'C) c= VARs by A42,Th79;
    then vars (Pi*'C*'C*'L) c= VARs by A36,Th79;
    then vars (Pi*'C*'C*'L*'D) c= VARs by A44,Th79;
    hence thesis by Th80;
  end;
A47:for x be Function of N,F_Real holds
  eval(E,x) = Two *(x/.i)*(eval(C,x)^2)* eval(L,x)*eval(D,x)
  proof
    let x be Function of N,F_Real;
A48:eval(C,x)*eval(C,x) = eval(C,x)^2 by SQUARE_1:def 1;
    thus eval(E,x) = Two* eval(Pi*'C*'C*'L*'D,x)
    by POLYNOM7:29
    .= Two* (eval(Pi*'C*'C*'L,x)*eval(D,x)) by POLYNOM2:25
    .= Two* (eval(Pi*'C*'C,x)*eval(L,x)*eval(D,x)) by POLYNOM2:25
    .= Two* (eval(Pi*'C,x)*eval(C,x)*eval(L,x)*eval(D,x))
    by POLYNOM2:25
    .= Two* (eval(Pi,x)*eval(C,x)*eval(C,x)*eval(L,x)*eval(D,x))
    by POLYNOM2:25
    .= Two* (eval(Pi,x)*(eval(C,x)*eval(C,x))*eval(L,x)*eval(D,x))
    .= Two* ((x/.i)*(eval(C,x)*eval(C,x))*eval(L,x)*eval(D,x)) by A7
    .=Two *(x/.i)*(eval(C,x)^2)* eval(L,x)*eval(D,x) by A48;
  end;
  reconsider F= (A*'A-O)*'(E*'E)+O as INT -valued Polynomial of N,F_Real;
A49: vars F c= VARs
  proof
    vars (A*'A) c= VARs by A38,Th79;
    then vars (A*'A-O) c= VARs & vars (E*'E) c= VARs by A46,A20,Th81,Th79;
    then vars ((A*'A-O)*'(E*'E)) c= VARs by Th79;
    hence thesis by A20,Th78;
  end;
A50:for x be Function of N,F_Real holds
  eval(F,x) = (eval(A,x)^2 - 1.F_Real)* eval(E,x)^2+ 1.F_Real
  proof
    let x be Function of N,F_Real;
    thus eval(F,x) = eval((A*'A-O)*'(E*'E),x)+eval(O,x) by POLYNOM2:23
    .= eval((A*'A-O)*'(E*'E),x)+1.F_Real by POLYNOM2:21
    .= (eval(A*'A-O,x)*eval(E*'E,x))+1.F_Real by POLYNOM2:25
    .= (eval(A*'A-O,x)*(eval(E,x)*eval(E,x)))+1.F_Real by POLYNOM2:25
    .= (eval(A*'A-O,x)*eval(E,x)^2)+1.F_Real by SQUARE_1:def 1
    .= ((eval(A*'A,x)-eval(O,x))*eval(E,x)^2)+1.F_Real by POLYNOM2:24
    .= ((eval(A*'A,x)-1.F_Real)*eval(E,x)^2)+1.F_Real by POLYNOM2:21
    .= ((eval(A,x)*eval(A,x)-1.F_Real)*eval(E,x)^2)+1.F_Real by POLYNOM2:25
    .= (eval(A,x)^2 - 1.F_Real)* eval(E,x)^2+ 1.F_Real by SQUARE_1:def 1;
  end;
  reconsider G= A+F*'(F-A) as INT -valued Polynomial of N,F_Real;
A51: vars G c= VARs
  proof
    vars (F-A) c= VARs by A38,A49,Th81;
    then vars (F*'(F-A)) c= VARs by A49,Th79;
    hence thesis by A38,Th78;
  end;
A52:for x be Function of N,F_Real holds
  eval(G,x) = eval(A,x)+eval(F,x)*(eval(F,x) - eval(A,x))
  proof
    let x be Function of N,F_Real;
    thus eval(G,x) = eval(A,x)+eval(F*'(F-A),x) by POLYNOM2:23
    .=eval(A,x)+eval(F,x) *eval(F-A,x) by POLYNOM2:25
    .= eval(A,x)+eval(F,x)*(eval(F,x) - eval(A,x)) by POLYNOM2:24;
  end;
  reconsider H= B+Two*((Pj-O)*'C) as INT -valued Polynomial of N,F_Real;
A53: vars H c= VARs
  proof
    vars (Pj-O) c= VARs by A8,A20,Th81;
    then vars ((Pj-O)*'C) c= VARs by A42,Th79;
    then vars (Two*((Pj-O)*'C)) c= VARs by Th80;
    hence thesis by A40,Th78;
  end;
A54:for x be Function of N,F_Real holds
  eval(H,x) = eval(B,x)+Two*(x/.j - 1.F_Real)*eval(C,x)
  proof
    let x be Function of N,F_Real;
    thus eval(H,x) = eval(B,x)+eval(Two*((Pj-O)*'C),x) by POLYNOM2:23
    .=eval(B,x)+Two*eval((Pj-O)*'C,x) by POLYNOM7:29
    .=eval(B,x)+Two*(eval(Pj-O,x)*eval(C,x)) by POLYNOM2:25
    .=eval(B,x)+Two*((eval(Pj,x)-eval(O,x))*eval(C,x)) by POLYNOM2:24
    .=eval(B,x)+Two*((eval(Pj,x)-1.F_Real)*eval(C,x)) by POLYNOM2:21
    .=eval(B,x)+Two*((x/.j-1.F_Real)*eval(C,x)) by A9
    .=eval(B,x)+Two*(x/.j - 1.F_Real)*eval(C,x);
  end;
  reconsider I= (G*'G-O)*'(H*'H)+O as INT -valued Polynomial of N,F_Real;
A55: vars I c= VARs
  proof
    vars (G*'G) c= VARs by A51,Th79;
    then vars (G*'G-O) c= VARs & vars (H*'H) c= VARs by A53,A20,Th81,Th79;
    then vars ((G*'G-O)*'(H*'H)) c= VARs by Th79;
    hence thesis by A20,Th78;
  end;
A56:for x be Function of N,F_Real holds
      eval(I,x) = (eval(G,x)^2 - 1.F_Real)* eval(H,x)^2+ 1.F_Real
  proof
    let x be Function of N,F_Real;
    thus eval(I,x) = eval((G*'G-O)*'(H*'H),x)+eval(O,x)
    by POLYNOM2:23
    .= eval((G*'G-O)*'(H*'H),x)+1.F_Real by POLYNOM2:21
    .= (eval(G*'G-O,x)*eval(H*'H,x))+1.F_Real by POLYNOM2:25
    .= (eval(G*'G-O,x)*(eval(H,x)*eval(H,x)))+1.F_Real by POLYNOM2:25
    .= (eval(G*'G-O,x)*eval(H,x)^2)+1.F_Real by SQUARE_1:def 1
    .= ((eval(G*'G,x)-eval(O,x))*eval(H,x)^2)+1.F_Real by POLYNOM2:24
    .= ((eval(G*'G,x)-1.F_Real)*eval(H,x)^2)+1.F_Real by POLYNOM2:21
    .= ((eval(G,x)*eval(G,x)-1.F_Real)*eval(H,x)^2)+1.F_Real
    by POLYNOM2:25
    .= (eval(G,x)^2 - 1.F_Real)* eval(H,x)^2+ 1.F_Real by SQUARE_1:def 1;
  end;
  reconsider X1= (M*'M-O)*'(S*'S)+O as INT -valued Polynomial of N,F_Real;
A57: vars X1 c= VARs
  proof
    vars (M*'M) c= VARs by A28,Th79;
    then vars (M*'M-O) c= VARs & vars (S*'S) c= VARs by A20,Th81,A30,Th79;
    then vars ((M*'M-O)*'(S*'S)) c= VARs by Th79;
    hence thesis by A20,Th78;
  end;
A58:for x be Function of N,F_Real holds
      eval(X1,x) = (eval(M,x)^2 - 1.F_Real)* eval(S,x)^2+ 1.F_Real
  proof
    let x be Function of N,F_Real;
    thus eval(X1,x) = eval((M*'M-O)*'(S*'S),x)+eval(O,x) by POLYNOM2:23
    .= eval((M*'M-O)*'(S*'S),x)+1.F_Real by POLYNOM2:21
    .= (eval(M*'M-O,x)*eval(S*'S,x))+1.F_Real by POLYNOM2:25
    .= (eval(M*'M-O,x)*(eval(S,x)*eval(S,x)))+1.F_Real by POLYNOM2:25
    .= (eval(M*'M-O,x)*(eval(S,x)^2))+1.F_Real by SQUARE_1:def 1
    .= ((eval(M*'M,x)-eval(O,x))*eval(S,x)^2)+1.F_Real by POLYNOM2:24
    .= ((eval(M*'M,x)-1.F_Real)*eval(S,x)^2)+1.F_Real by POLYNOM2:21
    .= ((eval(M,x)*eval(M,x)-1.F_Real)*eval(S,x)^2)+1.F_Real by POLYNOM2:25
    .= (eval(M,x)^2 - 1.F_Real)* eval(S,x)^2+ 1.F_Real by SQUARE_1:def 1;
  end;
  reconsider X2= (M*'U*'(M*'U)-O)*'(T*'T)+O as
   INT -valued Polynomial of N,F_Real;
A59: vars X2 c= VARs
  proof
    vars (M*'U) c= VARs by A23,A28,Th79;
    then vars ((M*'U)*'(M*'U)) c= VARs by Th79;
    then vars ((M*'U)*'(M*'U)-O) c= VARs & vars (T*'T) c= VARs
    by A20,Th81,A32,Th79;
    then vars ((M*'U*'(M*'U)-O)*'(T*'T)) c= VARs by Th79;
    hence thesis by A20,Th78;
  end;
A60:for x be Function of N,F_Real holds
      eval(X2,x) = ((eval(M,x)*eval(U,x))^2 - 1.F_Real)* eval(T,x)^2+ 1.F_Real
  proof
    let x be Function of N,F_Real;
    eval(M*'U,x) = eval(M,x) * eval(U,x) by POLYNOM2:25;
    then
A61: eval(M*'U*'(M*'U),x) =
    (eval(M,x) * eval(U,x))*(eval(M,x) * eval(U,x)) & eval(O,x) = 1.F_Real &
    eval(T*'T,x) = eval(T,x) * eval(T,x) by POLYNOM2:21,25;
    thus eval(X2,x) = eval((M*'U*'(M*'U)-O)*'(T*'T),x)+1.F_Real
    by POLYNOM2:23,A61
    .= eval(M*'U*'(M*'U)-O,x)*eval(T*'T,x)+1.F_Real by POLYNOM2:25
    .= (eval(M*'U*'(M*'U),x) - 1.F_Real)*eval(T*'T,x)+1.F_Real
    by A61,POLYNOM2:24
    .= ( (eval(M,x) * eval(U,x))*(eval(M,x) * eval(U,x)) - 1.F_Real) *
       eval(T,x)^2 +1.F_Real by A61,SQUARE_1:def 1
    .= ( (eval(M,x) * eval(U,x))^2 - 1.F_Real) * eval(T,x)^2 +1.F_Real
    by SQUARE_1:def 1;
  end;
  reconsider X3= D*'F*'I as INT -valued Polynomial of N,F_Real;
A62: vars X3 c= VARs
  proof
    vars (D*'F) c= VARs by A44,A49,Th79;
    hence thesis by A55,Th79;
  end;
A63:for x be Function of N,F_Real holds
  eval(X3,x) = eval(D,x) *eval(F,x) * eval(I,x)
  proof
    let x be Function of N,F_Real;
    eval(D*'F,x) = eval(D,x) * eval(F,x) by POLYNOM2:25;
    hence thesis by POLYNOM2:25;
  end;
  reconsider P= F*'L as INT -valued Polynomial of N,F_Real;
A64: vars P c= VARs by A49,A36,Th79;
A65: for x be Function of N,F_Real holds eval(P,x) = eval(F,x) * eval(L,x)
     by POLYNOM2:25;
  reconsider R= (H-C)*'L + F*'(Pf+O)*'Q +
    F*'(Pk+O) *'((W*'W-O)*'S*'Pu-W*'W*'(Pu*'Pu) +O)
    as INT -valued Polynomial of N,F_Real;
A66: vars R c= VARs
  proof
    vars(H-C) c= VARs by A53,A42,Th81;
    then
A67: vars((H-C)*'L) c= VARs by A36,Th79;
    vars(Pf+O) c= VARs by A4,A20,Th78;
    then vars (F*'(Pf+O)) c= VARs by A49,Th79;
    then vars (F*'(Pf+O)*'Q) c= VARs by A34,Th79;
    then
A68:vars ((H-C)*'L + F*'(Pf+O)*'Q) c= VARs by A67,Th78;
    vars(Pk+O) c= VARs by A20,A2,Th78;
    then
A69:vars (F*'(Pk+O)) c= VARs by A49,Th79;
A70: vars (W*'W) c= VARs by A21,Th79;
    then vars (W*'W-O) c= VARs by A20,Th81;
    then vars ((W*'W-O)*'S) c= VARs by A30,Th79;
    then
A71: vars ((W*'W-O)*'S*'Pu) c= VARs by A12,Th79;
    vars (Pu*'Pu) c= VARs by A12,Th79;
    then vars (W*'W*'(Pu*'Pu)) c= VARs by A70,Th79;
    then vars ((W*'W-O)*'S*'Pu-W*'W*'(Pu*'Pu)) c= VARs by A71,Th81;
    then vars ((W*'W-O)*'S*'Pu-W*'W*'(Pu*'Pu)+O) c= VARs by A20,Th78;
    then vars (F*'(Pk+O) *'((W*'W-O)*'S*'Pu-W*'W*'(Pu*'Pu) +O)) c= VARs
    by A69,Th79;
    hence thesis by A68,Th78;
  end;
A72:for x be Function of N,F_Real holds eval(R,x) =
  (eval(H,x)-eval(C,x))*eval(L,x) + eval(F,x)*(x/.f+1.F_Real)*eval(Q,x) +
  eval(F,x)*(x/.k+1.F_Real) *
  ((eval(W,x)^2-1.F_Real)*eval(S,x)*x/.u-eval(W,x)^2*(x/.u)^2 +1.F_Real)
  proof
    let x be Function of N,F_Real;
A73: eval((H-C)*'L,x) = eval(H-C,x) *eval(L,x) by POLYNOM2:25
    .= (eval(H,x)-eval(C,x))*eval(L,x) by POLYNOM2:24;
A74:eval(F*'(Pf+O)*'Q,x) = eval(F*'(Pf+O),x)*eval(Q,x) by POLYNOM2:25
    .=eval(F,x)*eval(Pf+O,x)*eval(Q,x) by POLYNOM2:25
    .=eval(F,x)*(eval(Pf,x)+eval(O,x))*eval(Q,x) by POLYNOM2:23
    .=eval(F,x)*(eval(Pf,x)+1.F_Real)*eval(Q,x) by POLYNOM2:21
    .=eval(F,x)*(x/.f+1.F_Real)*eval(Q,x) by A5;
A75:eval (F*'(Pk+O),x) = eval (F,x)*eval(Pk+O,x) by POLYNOM2:25
    .=eval (F,x)*(eval(Pk,x)+eval(O,x)) by POLYNOM2:23
    .=eval (F,x)*(eval(Pk,x)+1.F_Real) by POLYNOM2:21
    .=eval (F,x)*(x/.k+1.F_Real) by A3;
A76: eval(W*'W,x) = eval(W,x)*eval(W,x) = eval(W,x)^2
    by POLYNOM2:25,SQUARE_1:def 1;
    then eval(W*'W-O,x) = eval(W,x)*eval(W,x)-eval(O,x) by POLYNOM2:24;
    then eval((W*'W-O)*'S,x) = (eval(W,x)*eval(W,x)-eval(O,x)) *eval(S,x)
    by POLYNOM2:25;
    then
A77: eval((W*'W-O)*'S*'Pu,x) =
    (eval(W,x)*eval(W,x)-eval(O,x)) *eval(S,x)*eval(Pu,x) &
    eval(O,x)=1.F_Real & eval(Pu,x) = x/.u by A13,POLYNOM2:21,25;
A78:eval(Pu,x) = x/.u by A13;
A79: eval(W*'W*'(Pu*'Pu),x) = eval(W*'W,x)*eval(Pu*'Pu,x) by POLYNOM2:25
    .=eval(W*'W,x)*(x/.u * x/.u) by A78,POLYNOM2:25
    .=(eval(W,x)*eval(W,x))*(x/.u * x/.u) by POLYNOM2:25;
    A80:eval((W*'W-O)*'S*'Pu-W*'W*'(Pu*'Pu) +O,x) =
    eval((W*'W-O)*'S*'Pu-W*'W*'(Pu*'Pu),x) +eval(O,x)
    by POLYNOM2:23
    .=eval((W*'W-O)*'S*'Pu-W*'W*'(Pu*'Pu),x) +1.F_Real by POLYNOM2:21
    .=(eval((W*'W-O)*'S*'Pu,x)-eval(W*'W*'(Pu*'Pu),x))+1.F_Real by POLYNOM2:24
    .=(eval(W,x)^2-1.F_Real) *eval(S,x)*(x/.u)
    - (eval(W,x)*eval(W,x)*(x/.u * x/.u )) +1.F_Real by A77,A79,SQUARE_1:def 1;
    A81: eval (F*'(Pk+O) *'((W*'W-O)*'S*'Pu-W*'W*'(Pu*'Pu) +O),x)=
    eval (F*'(Pk+O),x)*eval((W*'W-O)*'S*'Pu-W*'W*'(Pu*'Pu)+O,x) by POLYNOM2:25
    .= eval (F,x)*(x/.k+1.F_Real)*
    ((eval(W,x)^2-1.F_Real) *eval(S,x)*(x/.u)
    - eval(W,x)^2*(x/.u)^2 +1.F_Real) by A76,A80,A75,SQUARE_1:def 1;
    eval(R,x) = eval((H-C)*'L + F*'(Pf+O)*'Q,x)+
    eval (F*'(Pk+O) *'((W*'W-O)*'S*'Pu-W*'W*'(Pu*'Pu) +O),x)
    by POLYNOM2:23;
    hence thesis by A81,A73,A74,POLYNOM2:23;
  end;
  reconsider Eight = 8 as integer Element of F_Real by XREAL_0:def 1;
  reconsider V1= Eight*(Pf*'Pu*'S*'T*'(Pr-Pm*'S*'T*'U))
               as INT -valued Polynomial of N,F_Real;
A82: vars V1 c= VARs
  proof
    vars(Pf*'Pu) c= VARs by A4,A12,Th79;
    then vars(Pf*'Pu*'S) c= VARs by A30,Th79;
    then
A83:vars(Pf*'Pu*'S*'T) c= VARs by A32,Th79;
    vars(Pm*'S) c= VARs by A10,A30,Th79;
    then vars(Pm*'S*'T) c= VARs by A32,Th79;
    then vars(Pm*'S*'T*'U) c= VARs by A23,Th79;
    then vars(Pr-Pm*'S*'T*'U) c= VARs by A14,Th81;
    then vars(Pf*'Pu*'S*'T*'(Pr-Pm*'S*'T*'U)) c= VARs by A83,Th79;
    hence thesis by Th80;
  end;
A84:for x be Function of N,F_Real holds
  eval(V1,x) = Eight*((x/.f)*(x/.u)*eval(S,x)*eval(T,x)*
  ((x/.r)-(x/.m)*eval(S,x)*eval(T,x)*eval(U,x)))
  proof
    let x be Function of N,F_Real;
    eval(Pm*'S*'T*'U,x) = eval(Pm*'S*'T,x)*eval(U,x) by POLYNOM2:25
    .= eval(Pm*'S,x)*eval(T,x)*eval(U,x) by POLYNOM2:25
    .= eval(Pm,x)*eval(S,x)*eval(T,x)*eval(U,x) by POLYNOM2:25
    .= (x/.m)*eval(S,x)*eval(T,x)*eval(U,x) by A11;
    then
A85:eval(Pr-Pm*'S*'T*'U,x) =
    eval(Pr,x) - (x/.m)*eval(S,x)*eval(T,x)*eval(U,x) by POLYNOM2:24
    .= x/.r - (x/.m)*eval(S,x)*eval(T,x)*eval(U,x) by A15;
    eval(Pf*'Pu,x) = eval(Pf,x)* eval(Pu,x) by POLYNOM2:25
    .= (x/.f)* eval(Pu,x) by A5
    .= (x/.f)* (x/.u) by A13;
    then eval(Pf*'Pu*'S,x) =
    (x/.f)* (x/.u) * eval(S,x) by POLYNOM2:25;
    then eval(Pf*'Pu*'S*'T,x) =
    (x/.f)* (x/.u) * eval(S,x) * eval(T,x) by POLYNOM2:25;
    then eval(Pf*'Pu*'S*'T*'(Pr-Pm*'S*'T*'U),x) =
    (x/.f)* (x/.u) * eval(S,x) * eval(T,x) *
    (x/.r - (x/.m)*eval(S,x)*eval(T,x)*eval(U,x)) by A85,POLYNOM2:25;
    hence thesis by POLYNOM7:29;
  end;
  reconsider Four = 4 as integer Element of F_Real by XREAL_0:def 1;
  reconsider V2= Four*((Pu*'Pu)*'(S*'S)*'(T*'T))
               as INT -valued Polynomial of N,F_Real;
A86: vars V2 c= VARs
  proof
    vars(Pu*'Pu) c= VARs & vars(S*'S) c= VARs by A30,A12,Th79;
    then vars((Pu*'Pu)*'(S*'S)) c= VARs & vars(T*'T) c= VARs by A32,Th79;
    then vars((Pu*'Pu)*'(S*'S)*'(T*'T)) c= VARs by Th79;
    hence thesis by Th80;
  end;
A87: for x be Function of N,F_Real holds
    eval(V2,x) = Four*(x/.u)^2*(eval(S,x)^2)*(eval(T,x)^2)
  proof
    let x be Function of N,F_Real;
A88: eval(Pu*'Pu,x) = eval(Pu,x) * eval(Pu,x) & eval(Pu,x) = x/.u
    by POLYNOM2:25,A13;
A89: eval(S*'S,x) = eval(S,x) * eval(S,x) &
    eval(T*'T,x) = eval(T,x) * eval(T,x) by POLYNOM2:25;
    then eval((Pu*'Pu)*'(S*'S),x) = (x/.u) * (x/.u) * (eval(S,x) * eval(S,x))
    by A88,POLYNOM2:25;
    then
A90:eval((Pu*'Pu)*'(S*'S)*'(T*'T),x) =
    (x/.u) * (x/.u) * (eval(S,x) * eval(S,x)) * (eval(T,x) * eval(T,x))
    by A89,POLYNOM2:25;
    thus eval(V2,x) = Four*eval((Pu*'Pu)*'(S*'S)*'(T*'T),x) by POLYNOM7:29
    .= Four* ((x/.u)*(x/.u))*(eval(S,x) * eval(S,x))*(eval(T,x)*eval(T,x))
    by A90
    .= Four* (x/.u)^2 * (eval(S,x) * eval(S,x)) * (eval(T,x) * eval(T,x))
    by SQUARE_1:def 1
    .= Four* (x/.u)^2 * (eval(S,x)^2) * (eval(T,x) * eval(T,x))
    by SQUARE_1:def 1
    .= Four*(x/.u)^2 * (eval(S,x)^2) * (eval(T,x)^2) by SQUARE_1:def 1;
  end;
  reconsider V3=(Four*Pf*'Pf-O)*'((Pr-Pm*'S*'T*'U)*'(Pr-Pm*'S*'T*'U))
       as INT -valued Polynomial of N,F_Real;
A91: vars V3 c= VARs
  proof
    vars(Pm*'S) c= VARs by A10,A30,Th79;
    then vars(Pm*'S*'T) c= VARs by A32,Th79;
    then vars(Pm*'S*'T*'U) c= VARs by A23,Th79;
    then vars(Pr-Pm*'S*'T*'U) c= VARs by A14,Th81;
    then
A92: vars((Pr-Pm*'S*'T*'U)*'(Pr-Pm*'S*'T*'U)) c= VARs by Th79;
    vars(Pf*'Pf) c= VARs by A4,Th79;
    then vars(Four*Pf*'Pf) c= VARs by Th80;
    then vars(Four*Pf*'Pf-O) c= VARs by A20,Th81;
    hence thesis by A92,Th79;
  end;
A93: for x be Function of N,F_Real holds
    eval(V3,x) = (Four*(x/.f)^2-1.F_Real)*
    ((x/.r)-(x/.m)*eval(S,x)*eval(T,x)*eval(U,x))^2
  proof
    let x be Function of N,F_Real;
    eval(Pm*'S*'T*'U,x) = eval(Pm*'S*'T,x)*eval(U,x) by POLYNOM2:25
    .= eval(Pm*'S,x)*eval(T,x)*eval(U,x) by POLYNOM2:25
    .= eval(Pm,x)*eval(S,x)*eval(T,x)*eval(U,x) by POLYNOM2:25
    .= (x/.m)*eval(S,x)*eval(T,x)*eval(U,x) by A11;
    then
A94:eval(Pr-Pm*'S*'T*'U,x) =
    eval(Pr,x) - (x/.m)*eval(S,x)*eval(T,x)*eval(U,x) by POLYNOM2:24
    .= x/.r - (x/.m)*eval(S,x)*eval(T,x)*eval(U,x) by A15;
    eval(Pf,x) = x/.f by A5;
    then eval(Pf*'Pf,x) = (x/.f)*(x/.f) by POLYNOM2:25;
    then eval(Four*Pf*'Pf,x) = Four*((x/.f)*(x/.f)) by POLYNOM7:29;
    then
A95: eval(Four*Pf*'Pf-O,x) = Four*((x/.f)*(x/.f))-eval(O,x)
    by POLYNOM2:24
    .= Four*((x/.f)*(x/.f))-1.F_Real by POLYNOM2:21
    .= Four*((x/.f)^2)-1.F_Real by SQUARE_1:def 1;
    eval((Pr-Pm*'S*'T*'U)*'(Pr-Pm*'S*'T*'U),x)
    = eval(Pr-Pm*'S*'T*'U,x) * eval(Pr-Pm*'S*'T*'U,x) by POLYNOM2:25;
    then eval(V3,x) = eval(Four*Pf*'Pf-O,x)* (eval(Pr-Pm*'S*'T*'U,x) *
    eval(Pr-Pm*'S*'T*'U,x)) by POLYNOM2:25
    .=(Four*((x/.f)^2)-1.F_Real) *
    ((x/.r - (x/.m)*eval(S,x)*eval(T,x)*eval(U,x)) *
    (x/.r - (x/.m)*eval(S,x)*eval(T,x)*eval(U,x))) by A95,A94;
    hence thesis by SQUARE_1:def 1;
  end;
  reconsider N1=(M*'S)+(Two*(M*'U*'T))
       as INT -valued Polynomial of N,F_Real;
A96: vars N1 c= VARs
  proof
A97:vars(M*'S) c= VARs by A28,A30,Th79;
    vars(M*'U) c= VARs by A28,A23,Th79;
    then vars(M*'U*'T) c= VARs by A32,Th79;
    then vars(Two*(M*'U*'T)) c= VARs by Th80;
    hence thesis by A97,Th78;
  end;
A98: for x be Function of N,F_Real holds
    eval(N1,x) = (eval(M,x)*eval(S,x))+(Two*eval(M,x)*eval(U,x)*eval(T,x))
  proof
    let x be Function of N,F_Real;
A99: eval(M*'S,x) = eval(M,x) * eval(S,x) by POLYNOM2:25;
    eval(M*'U,x) = eval(M,x) * eval(U,x) by POLYNOM2:25;
    then eval(M*'U*'T,x) = eval(M,x) * eval(U,x) * eval(T,x) by POLYNOM2:25;
    then eval(Two*(M*'U*'T),x) = Two*(eval(M,x) * eval(U,x) * eval(T,x))
      by POLYNOM7:29;
    hence thesis by A99,POLYNOM2:23;
  end;
  reconsider N2=Four*(A*'A*'C*'E*'G*'H)
       as INT -valued Polynomial of N,F_Real;
A100: vars N2 c= VARs
  proof
    vars(A*'A) c= VARs by A38,Th79;
    then vars(A*'A*'C) c= VARs by A42,Th79;
    then vars(A*'A*'C*'E) c= VARs by A46,Th79;
    then vars(A*'A*'C*'E*'G) c= VARs by A51,Th79;
    then vars(A*'A*'C*'E*'G*'H) c= VARs by A53,Th79;
    hence thesis by Th80;
  end;
A101: for x be Function of N,F_Real holds eval(N2,x) =
    Four*(eval(A,x)*eval(A,x)*eval(C,x)*eval(E,x)*eval(G,x)*eval(H,x))
  proof
    let x be Function of N,F_Real;
    eval(A*'A*'C*'E*'G*'H,x) = eval(A*'A*'C*'E*'G,x)*eval(H,x)
    by POLYNOM2:25
    .= eval(A*'A*'C*'E,x)*eval(G,x)*eval(H,x) by POLYNOM2:25
    .= eval(A*'A*'C,x)*eval(E,x)*eval(G,x)*eval(H,x) by POLYNOM2:25
    .= eval(A*'A,x)*eval(C,x)*eval(E,x)*eval(G,x)*eval(H,x) by POLYNOM2:25
    .= eval(A,x)*eval(A,x)*eval(C,x)*eval(E,x)*eval(G,x)*eval(H,x)
    by POLYNOM2:25;
    hence thesis by POLYNOM7:29;
  end;
  reconsider V = V1-V2-V3-O as INT -valued Polynomial of N,F_Real;
  reconsider NN=N1+N2+R+O as INT -valued Polynomial of N,F_Real;
A102: vars V c= VARs
  proof
    vars(V1-V2) c= VARs by A82,A86,Th81;
    then vars(V1-V2-V3) c= VARs by A91,Th81;
    hence thesis by A20,Th81;
  end;
A103: vars NN c= VARs
  proof
    vars(N1+N2) c= VARs by A96,A100,Th78;
    then vars(N1+N2+R) c= VARs by A66,Th78;
    hence thesis by A20,Th78;
  end;
A104: for x be Function of N,F_Real st
     x/.k is positive Nat & x/.f is positive Nat & x/.i is positive Nat &
     x/.j is positive Nat & x/.m is positive Nat & x/.u is positive Nat &
     x/.r is Nat & x/.s is Nat & x/.t is Nat
  holds
   eval(X1,x) is odd Nat & eval(X2,x) is odd Nat & eval(X3,x) is Nat &
   eval(P,x) is positive Nat & eval(R,x) is Nat &
     eval(NN,x) is Nat &
     eval(NN,x)>sqrt eval(X1,x)+2*sqrt eval(X2,x)+4*sqrt eval(X3,x)+eval(R,x)
  proof
    let x be Function of N,F_Real such that
A105: x/.k is positive Nat & x/.f is positive Nat & x/.i is positive Nat &
    x/.j is positive Nat & x/.m is positive Nat & x/.u is positive Nat &
    x/.r is Nat & x/.s is Nat & x/.t is Nat;
    reconsider xk=x/.k, xf=x/.f,xi = x/.i,xj = x/.j,xm = x/.m,
    xu = x/.u as positive Nat by A105;
    reconsider xr=x/.r, xs=x/.s,xt = x/.t as Nat by A105;
A106: eval(W,x) = Hund * x /. f * x/.k * (x/.k+1.F_Real) by A22;
    then
A107: eval(W,x) = Hund * xf * xk * (xk+1.F_Real);
    then reconsider eW=eval(W,x) as Element of NAT by INT_1:3;
A108: eval(U,x) = Hund * ((xu)|^3 ) * eW|^3 + 1.F_Real by A24;
    then reconsider eU=eval(U,x) as positive Nat;
A109: eval(M,x) = Hund * (x /. m) * eval(U,x) * eval(W,x) +
    1.F_Real by A29;
    then eval(M,x) = Hund * xm * eU * eW  + 1.F_Real;
    then reconsider eM=eval(M,x) as positive Nat;
    eval(S,x) = (eval(M,x) - 1.F_Real) * (x/.s) + (x/.k) + 1.F_Real by A31;
    then
A110: eval(S,x) = (eM - 1.F_Real) * xs + xk  + 1.F_Real;
    then reconsider eS=eval(S,x) as Element of NAT by INT_1:3;
A111: eval(T,x) = (eval(M,x)*eval(U,x) - 1.F_Real)*(x/.t)+
    eval(W,x)-(x/.k)+1.F_Real by A33;
A112: 100*xf*xk*(xk+1) >= 1*(xk+1) by NAT_1:14,XREAL_1:64;
    eW > xk by A112,A106,NAT_1:13;
    then eW-xk > 0 by XREAL_1:50;
    then (eM*eU-1)*xt +(eW-xk) >=0+0;
    then reconsider eT=eval(T,x) as Element of NAT by A111,INT_1:3;
A113: eval(Q,x) = Two*eval(M,x)*eval(W,x) - eval(W,x)^2 - 1.F_Real by A35;
    Hund * xm * eU * eW+1 >0+1 by A107,XREAL_1:6;
    then eM >= 1+1 & eW>=1 by A107,A109,NAT_1:8;
    then eM*eW>=(1+1)*1 by XREAL_1:66;
    then
A114: eM*eW - 1 >= 1 by XREAL_1:19;
    100 * xm * eU * eW >= 1*eW by XREAL_1:64,NAT_1:14;
    then eM > eW by A109,NAT_1:13;
    then
A115:eM*eW > eW*eW = eW^2 by A107,XREAL_1:68,SQUARE_1:def 1;
    then
A116: eM*eW+eM*eW > eM*eW+eW^2 by XREAL_1:8;
    then eM*eW+eM*eW -(eW^2+1) > eM*eW+eW^2 -(eW^2+1) by XREAL_1:9;
    then reconsider eQ=eval(Q,x) as positive Nat by A113,A114;
    eval(L,x) = (x/.k + 1.F_Real)*eval(Q,x) by A37;
    then eval(L,x) = (xk + 1.F_Real)*eQ;
    then reconsider eL=eval(L,x) as positive Nat;
A117: eval(A,x) = eval(M,x)*(eval(U,x) + 1.F_Real) by A39;
    then
A118:eval(A,x) = eM*(eU + 1.F_Real);
    eM >=1 & eU+1>=1+1 by XREAL_1:6,NAT_1:14;
    then
A119: eM*(eU + 1.F_Real)>=1*2 by XREAL_1:66;
    reconsider eA=eval(A,x) as positive Nat by A118;
    eval(B,x) = eval(W,x) + 1.F_Real by A41;
    then eval(B,x) = eW + 1.F_Real;
    then reconsider eB=eval(B,x) as positive Nat;
    eval(C,x) = (x/.r) + eval(W,x) + 1.F_Real by A43;
    then eval(C,x) = xr + eW + 1.F_Real;
    then reconsider eC=eval(C,x) as positive Nat;
A120: eval(D,x) = (eA^2 - 1.F_Real)* eC^2+ 1.F_Real by A45;
A121: eA^2=eA*eA <>0 by SQUARE_1:def 1;
    reconsider eD=eval(D,x) as positive Nat by A121,A120;
A122: eval(E,x) = Two *(x/.i)*(eval(C,x)^2)* eval(L,x)*eval(D,x) by A47;
    then
A123:eval(E,x) = Two *(xi)*(eC^2)* eL*eD;
A124:eC*eC=eC^2 by SQUARE_1:def 1;
    then reconsider eE=eval(E,x) as positive Nat by A123;
A125:eval(F,x) = (eA^2 - 1.F_Real)* eE^2+ 1.F_Real by A50;
    reconsider eF=eval(F,x) as positive Nat by A121,A125;
A126: eval(G,x) = eval(A,x)+eval(F,x)*(eval(F,x) - eval(A,x)) by A52;
A127: eE^2=eE*eE>=1 by NAT_1:14,SQUARE_1:def 1;
    eA-1 >=2-1 by A117,A119,XREAL_1:9;
    then
A128: (eA-1) *(eA+1)>= 1*(eA+1) by XREAL_1:64;
    then (eA-1) *(eA+1)*eE^2>= 1*(eA+1)*1 by A127,XREAL_1:66;
    then (eA-1) *(eA+1)*eE^2+1>= eA+1+0 by XREAL_1:7;
    then eF>eA by A121,A125,NAT_1:13;
    then eF-eA>eA-eA by XREAL_1:14;
    then reconsider eG=eval(G,x) as positive Nat by A126;
    eval(H,x) = eval(B,x)+Two*(x/.j - 1.F_Real)*eval(C,x) by A54;
    then
A129:eval(H,x) = eB+Two*(xj - 1.F_Real)*eC;
    reconsider eH=eval(H,x) as positive Nat by A129;
A130:eval(I,x) = (eG^2 - 1.F_Real)* eH^2+ 1.F_Real by A56;
    eG^2=eG*eG <>0 by SQUARE_1:def 1;
    then reconsider eI=eval(I,x) as positive Nat by A130;
A131: eval(X1,x) = (eM^2 - 1.F_Real)* eS^2+ 1.F_Real by A58;
A132: eM^2=eM*eM <>0 by SQUARE_1:def 1;
    then reconsider eX1=eval(X1,x) as Nat by A131;
A133: eM = 2*(50* xm * eU * eW)+1 by A109;
    hence eval(X1,x) is odd Nat by A132,A131;
A134: eval(X2,x) = ((eval(M,x)*eval(U,x))^2 - 1.F_Real)* eval(T,x)^2+
      1.F_Real by A60;
    then
A135: eval(X2,x) = ((eM*eU)^2 - 1.F_Real)* eT^2+ 1.F_Real;
A136: (eM*eU)^2=(eM*eU)*(eM*eU) <>0 by SQUARE_1:def 1;
    then reconsider eX2=eval(X2,x) as Nat by A135;
    eU = 2*(50 * (xu|^3 ) * eW|^3) + 1.F_Real by A108;
    hence eval(X2,x) is odd Nat by A133,A136,A135;
A137: eval(X3,x) = eval(D,x) *eval(F,x) * eval(I,x) by A63;
    then
A138: eval(X3,x) = eD*eF*eI;
    then reconsider eX3 = eval(X3,x) as Nat;
    thus eval(X3,x) is Nat by A138;
    eval(P,x) = eval(F,x) * eval(L,x) by A65;
    then eval(P,x) = eF * eL;
    hence eval(P,x) is positive Nat;
A139: eval(R,x) =
    (eval(H,x)-eval(C,x))*eval(L,x) + eval(F,x)*(x/.f+1.F_Real)*eval(Q,x) +
    eval(F,x)*(x/.k+1.F_Real) *
    ((eval(W,x)^2-1.F_Real)*eval(S,x)*x/.u-eval(W,x)^2*(x/.u)^2 +1.F_Real)
    by A72;
    then eval(R,x) =
    (eH-eC)*eL + eF*(xf+1)*eQ + eF*(xk+1) * ((eW^2-1)*eS*xu-eW^2*(xu)^2 +1);
    then reconsider eR = eval(R,x) as Integer;
    xk+1>=1 by NAT_1:14;
    then
A140: eF*(xk+1)>= eF *1 by XREAL_1:66;
    eE>=1 by NAT_1:14;
    then
A141: eE^2 = eE*eE>= eE *1 by XREAL_1:66,SQUARE_1:def 1;
A142: eA*eA=eA^2 by SQUARE_1:def 1;
    eA^2-1 >=1 by A142,A128,NAT_1:14;
    then (eA^2-1)*eE^2 +1>  eE by NAT_1:13,A141,XREAL_1:66;
    then
A143: eF*(xk+1) > eE by A125,A140,XXREAL_0:2;
    Two *xi*eC*eD >=1 by NAT_1:14;
    then Two *xi*eC*eD *(eC*eL) >= 1*(eC*eL) by XREAL_1:66;
    then eF*(xk+1) > eC*eL by A124,A143,A122,XXREAL_0:2;
    then eF*(xk+1) - eC*eL > 0 by XREAL_1:50;
    then
A144: eF*(xk+1) - eC*eL + (eH*eL + eF*(xk+1) * (eW^2-1)*eS*xu) >= 0+0
    by A107,A115;
    eM*eW+eM*eW >= eM*eW+eW^2+1 by A116,NAT_1:13;
    then
A145: eM*eW+eM*eW -(eW^2+1) >= eM*eW+eW^2 +1 -(eW^2+1) by XREAL_1:6;
    Hund * xf * xk >=1 by NAT_1:14;
    then
A146: eW >= (xk+1)*1 by A106,XREAL_1:66;
    then eM*eW >= (xk+1)*eM by XREAL_1:66;
    then
A147: eQ >= (xk+1)*eM by A145,A113,XXREAL_0:2;
A148: xu >=1 by NAT_1:14;
A149: 100 * xm  >=1 by NAT_1:14;
    xu|^(2+1) = xu|^2 *xu by NEWTON:6
    .= xu^2 *xu by NEWTON:81;
    then
A150: xu|^3 >= xu^2*1 by A148,XREAL_1:66;
    eW>=1 by A107,NAT_1:14;
    then eW|^3 >= eW by PREPOWER:12;
    then xu|^3 * eW|^3 >= xu^2*eW by A150,XREAL_1:66;
    then 100*(xu|^3 * eW|^3) >= 1*(xu^2*eW) by XREAL_1:66;
    then eU > xu^2*eW by A108,NAT_1:13;
    then eU*eW >= xu^2*eW*eW by XREAL_1:66;
    then 100 * xm * (eU * eW) >= 1*(xu^2*eW*eW) by A149,XREAL_1:66;
    then eM > xu^2*eW*eW by A109,NAT_1:13;
    then (xk+1)*eM>= (xk+1) * (xu^2*eW*eW) by XREAL_1:66;
    then eQ >= (xk+1) * (xu^2*eW*eW) & xf+1 >=1 by A147,XXREAL_0:2,NAT_1:14;
    then (xf+1)*eQ >= (xk+1) * (xu^2*eW*eW)*1 by XREAL_1:66;
    then eF*((xf+1)*eQ) >= eF*((xk+1) * eW^2*(xu)^2)
    by A115,XREAL_1:66;
    then eF*(xf+1)*eQ - eF*(xk+1) * eW^2*(xu)^2 >=0 by XREAL_1:48;
    then
A151: (eH-eC)*eL + eF*(xk+1) + eF*(xk+1) * (eW^2-1)*eS*xu +
    (eF*(xf+1)*eQ - eF*(xk+1) * eW^2*(xu)^2) >=0 by A144;
    hence eval(R,x) is Nat by A139;
A152: eval(N1,x) = (eval(M,x)*eval(S,x))+
    (Two*eval(M,x)*eval(U,x)*eval(T,x)) by A98;
    then eval(N1,x) = eM*eS+(2*eM*eU*eT);
    then reconsider eN1 = eval(N1,x) as Nat;
A153:eval(N2,x) = Four*(eval(A,x)*eval(A,x)*eval(C,x)*
    eval(E,x)*eval(G,x)*eval(H,x)) by A101;
    then eval(N2,x) = 4* (eA *eA*eC *eE*eH *eG);
    then reconsider eN2 = eval(N2,x) as Nat;
    eW - (xk+1)*1 >=0 by A146,XREAL_1:48;
    then eW - xk-1 +2 >= 0+2 by XREAL_1:6;
    then (eM*eU - 1)*xt+(eW-xk+1) >=0+2 by XREAL_1:7;
    then
A154: eT^2=eT*eT>=1 by A111,NAT_1:14,SQUARE_1:def 1;
    then (eM*eU)^2 * eT^2 - (eT^2-1) <= (eM*eU)^2 * eT^2 -0
    by XREAL_1:6;
    then eX2 <= (eM*eU*eT)*(eM*eU*eT) by A134,A154,A136;
    then eX2 <= (eM*eU*eT)^2 by SQUARE_1:def 1;
    then sqrt eX2 <= sqrt ((eM*eU*eT)^2) = eM*eU*eT by SQUARE_1:22,26;
    then
A155: 2* sqrt eX2 <= 2*(eM*eU*eT) by XREAL_1:64;
A156: eS^2=eS*eS>=1 by A110,NAT_1:14,SQUARE_1:def 1;
    then eM^2 * eS^2 - (eS^2-1) <= (eM)^2 * eS^2 -0 by XREAL_1:6;
    then
A157: eX1 <= (eM)^2 * eS^2 & eM*eM=eM^2 by A131,SQUARE_1:def 1;
    eX1 <= (eM*eS)*(eM*eS) by A156,A157;
    then eX1 <= (eM*eS)^2 by SQUARE_1:def 1;
    then sqrt eX1 <= sqrt ((eM*eS)^2) = eM*eS by SQUARE_1:22,26;
    then
A158: sqrt eX1 + 2* sqrt eX2 <= eN1 by A152,A155,XREAL_1:7;
    eC^2=eC*eC>=1 by NAT_1:14,SQUARE_1:def 1;
    then
A159: eA^2 *eC^2 - (eC^2-1) <= eA^2 *eC^2 -0 by XREAL_1:6;
A160: eE^2=eE*eE>=1 by NAT_1:14,SQUARE_1:def 1;
    then eA^2 *eE^2 - (eE^2-1) <= eA^2 *eE^2 -0 by XREAL_1:6;
    then
A161:eD*eF <= eA^2 *eC^2*(eA^2 *eE^2) by A120,A125,A159,XREAL_1:66;
A162: eH^2=eH*eH>=1 by NAT_1:14,SQUARE_1:def 1;
    then eG^2 *eH^2 - (eH^2-1) <= eG^2 *eH^2 -0 by XREAL_1:6;
    then eD*eF*eI <= eA^2 *eC^2*(eA^2 *eE^2)*(eG^2 *eH^2)
    by A130,A161,XREAL_1:66;
    then eX3 <= eA*eA *(eC*eC)*(eA*eA *(eE*eE))*((eG*eG) *(eH*eH))
    by A137,A124,A121,A162,A160,SQUARE_1:def 1;
    then eX3 <= (eA *eA*eC *eE*eH *eG)*(eA *eA*eC *eE*eH *eG);
    then eX3 <= (eA *eA*eC *eE*eH *eG)^2 by SQUARE_1:def 1;
    then sqrt eX3 <= sqrt (eA *eA*eC *eE*eH *eG)^2 = eA *eA*eC *eE*eH *eG
    by SQUARE_1:22,26;
    then
A163: 4* sqrt eX3 <= eN2 by A153,XREAL_1:64;
A164: eval(NN,x) = eval(N1+N2+R,x)+eval(O,x) by POLYNOM2:23
    .= eval(N1+N2+R,x)+1.F_Real by POLYNOM2:21
    .= eval(N1+N2,x)+eval(R,x)+1.F_Real by POLYNOM2:23
    .= eval(N1,x)+eval(N2,x) +eval(R,x)+1.F_Real by POLYNOM2:23
    .= eN1+eN2+eR+1;
A165: eN1+eN2 +eR+1 > eN1+eN2 +eR+0 by XREAL_1:6;
    eN1+eN2 >= sqrt eX1 + 2* sqrt eX2 +4* sqrt eX3 by A158,A163,XREAL_1:7;
    then eN1+eN2 +eR >= sqrt eX1 + 2* sqrt eX2 +4* sqrt eX3 +eR by XREAL_1:6;
    hence thesis by A151,A139,A164,A165,XXREAL_0:2;
  end;
  consider K3 be INT -valued Polynomial of 8,F_Real such that
  A166: for x1,x2,x3,P,R,N be Nat,V be Integer st
    x1 is odd & x2 is odd & P > 0 &
    N > sqrt x1 + 2*sqrt x2 + 4*sqrt x3 +R
  holds
  (x1 is square & x2 is square  & x3 is square & P divides R & V>=0)
  iff
  ex z be Nat st for f be Function of 8,F_Real
  st f = <%z,x1,4*x2,16*x3%>^<%R,P,N,V%> holds eval(K3,f) = 0 by Th77;
  consider Z be Polynomial of 8+9,F_Real such that
A167:  rng Z c= rng K3 \/ {0.F_Real} and
A168: for b be bag of 8+9 holds b in Support Z iff b|8 in Support K3 &
  for i st i >= 8 holds b.i=0 and
  for b be bag of 8+9 st b in Support Z holds
  Z.b = K3.(b|8) and
A169: for x being Function of 8, F_Real,
          y being Function of (8+9), F_Real st y|8=x
    holds eval(K3,x) = eval(Z,y) by Th75;
  rng Z c= INT by A167,INT_1:def 2;
  then reconsider Z as INT -valued Polynomial of N,F_Real
  by RELAT_1:def 19;
  reconsider Z1 = Subst(Z,1,X1) as INT -valued Polynomial of N,F_Real;
  reconsider Z2 = Subst(Z1,2,Four*X2) as INT -valued Polynomial of N,F_Real;
  reconsider Z3 =Subst(Z2,3,(Four*Four)*X3)
  as INT -valued Polynomial of N,F_Real;
  reconsider Z4 = Subst(Z3,4,R) as INT -valued Polynomial of N,F_Real;
  reconsider Z5 = Subst(Z4,5,P) as INT -valued Polynomial of N,F_Real;
  reconsider Z6 = Subst(Z5,6,NN) as INT -valued Polynomial of N,F_Real;
  reconsider Z7 = Subst(Z6,7,V) as INT -valued Polynomial of N,F_Real;
  take Z7;
A170: for xk be Nat st xk > 0 holds
  xk+1 is prime iff ex x being INT -valued Function of N, F_Real st
  x/.k = xk & x/.f is positive Nat & x/.i is positive Nat &
  x/.j is positive Nat & x/.m is positive Nat & x/.u is positive Nat &
  x/.r is Nat & x/.s is Nat & x/.t is Nat & x/.0 is Nat &
  eval(Z7,x) = 0.F_Real
  proof
    let k1 be Nat such that
A171:k1 > 0;
    thus k1+1 is prime implies ex x being INT -valued Function of N, F_Real st
    x/.k = k1 & x/.f is positive Nat & x/.i is positive Nat &
    x/.j is positive Nat & x/.m is positive Nat & x/.u is positive Nat &
    x/.r is Nat & x/.s is Nat & x/.t is Nat & x/.0 is Nat &
    eval(Z7,x) = 0.F_Real
    proof
      assume k1+1 is prime;
      then consider f1,i1,j1,m1,u1 be positive Nat,r1,s1,t1 be Nat,
      A1,B1,C1,D1,E1,F1,G1,H1,I1,L1,W1,U1,M1,S1,T1,Q1 be Integer such that
A172: D1*F1*I1 is square & (M1^2-1)*S1^2 +1 is square &
      ((M1*U1)^2 -1)*T1^2 + 1 is square and
A173:  (4*f1^2 -1)*(r1-m1*S1*T1*U1)^2 + 4*u1^2*S1^2*T1^2 <
      8*f1*u1*S1*T1*(r1-m1*S1*T1*U1) and
A174: F1*L1 divides (H1-C1)*L1 + F1*(f1+1)*Q1 + F1*(k1+1)
      *((W1^2-1)*S1*u1-W1^2*u1^2 +1) and
A175: A1 = M1*(U1+1) and
A176: B1 = W1+1 and
A177: C1 = r1 + W1 + 1 and
A178: D1 = (A1^2-1)*C1^2+1 and
A179: E1 = 2*i1*C1^2*L1*D1 and
A180: F1= (A1^2 -1) *E1^2+1 and
A181: G1 = A1+F1*(F1-A1) and
A182: H1 = B1+2*(j1-1)*C1 and
A183: I1 = (G1^2-1)*H1^2+1 and
A184: L1 = (k1+1)*Q1 and
A185: W1 = 100*f1*k1*(k1+1) and
A186: U1 = 100 * (u1|^3)*(W1|^3)+1 and
A187: M1 = 100 * m1 * U1 *W1+1 and
A188: S1 = (M1-1)*s1+k1+1 and
A189: T1 = (M1*U1-1)*t1 +W1-k1+1 and
A190: Q1 = 2*M1*W1-W1^2-1 by A171, HILB10_8:23;
      set zz = <%0%>^<%0%>^<%0%>^<%0%>^<%0%>^<%0%>^<%0%>^<%0%>;
      set vv=<%k1%>^<%f1%>^<%i1%>^<%j1%>^<%m1%>^<%u1%>^<%r1%>^<%s1%>^<%t1%>;
      set x = @(zz^vv);
      reconsider x as INT -valued Function of N, F_Real;
A191: len zz = 8 by AFINSQ_1:49;
A192: len vv = 9 =Segm 9 by AFINSQ_1:50;
A193: dom x = N = Segm N by FUNCT_2:def 1;
A194: 0 in dom vv & 1 in dom vv & 2 in dom vv & 3 in dom vv & 4 in dom vv
      &5 in dom vv & 6 in dom vv & 7 in dom vv & 8 in dom vv by A192,NAT_1:44;
A195: x/.k = x.(8+0) by A193,NAT_1:44,PARTFUN1:def 6
      .= vv.0 by A194,A191,AFINSQ_1:def 3
      .= k1 by AFINSQ_1:50;
A196: x/.f = x.(8+1) by A193,NAT_1:44,PARTFUN1:def 6
      .= vv.1 by A194,A191,AFINSQ_1:def 3
      .= f1 by AFINSQ_1:50;
A197: x/.i = x.(8+2) by A193,NAT_1:44,PARTFUN1:def 6
      .= vv.2 by A194,A191,AFINSQ_1:def 3
      .= i1 by AFINSQ_1:50;
A198: x/.j = x.(8+3) by A193,NAT_1:44,PARTFUN1:def 6
      .= vv.3 by A194,A191,AFINSQ_1:def 3
      .= j1 by AFINSQ_1:50;
A199: x/.m = x.(8+4) by A193,NAT_1:44,PARTFUN1:def 6
      .= vv.4 by A194,A191,AFINSQ_1:def 3
      .= m1 by AFINSQ_1:50;
A200: x/.u = x.(8+5) by A193,NAT_1:44,PARTFUN1:def 6
      .= vv.5 by A194,A191,AFINSQ_1:def 3
      .= u1 by AFINSQ_1:50;
A201: x/.r = x.(8+6) by A193,NAT_1:44,PARTFUN1:def 6
      .= vv.6 by A194,A191,AFINSQ_1:def 3
      .= r1 by AFINSQ_1:50;
A202: x/.s = x.(8+7) by A193,NAT_1:44,PARTFUN1:def 6
      .= vv.7 by A194,A191,AFINSQ_1:def 3
      .= s1 by AFINSQ_1:50;
A203: x/.t = x.(8+8) by A193,NAT_1:44,PARTFUN1:def 6
      .= vv.8 by A194,A191,AFINSQ_1:def 3
      .= t1 by AFINSQ_1:50;
A204:  eval(W,x) = Hund * x /. f * x/.k * (x/.k+1.F_Real) by A22;
A205: U1=eval(U,x) by A186,A200,A204,A185,A195,A196,A24;
A206: eval(M,x) = Hund * (x /.m) *eval(U,x)*eval(W,x) +1.F_Real by A29;
      then
A207: M1 = eval(M,x)
      by A187,A199,A186,A200,A204,A185,A195,A196,A24;
A208: eval(S,x) = (eval(M,x) - 1.F_Real)*(x/.s)+(x/.k)+1.F_Real by A31;
      then
A209: S1 = eval(S,x) by A188,A202,A195,A206,
      A187,A199,A186,A200,A204,A185,A196,A24;
A210: eval(T,x) = (eval(M,x)*eval(U,x) - 1.F_Real)*(x/.t)+
      eval(W,x)-(x/.k)+1.F_Real by A33;
A211: eval(Q,x) = Two*eval(M,x)*eval(W,x) - eval(W,x)^2-1.F_Real by A35;
      then
A212: Q1 = eval(Q,x) by A190,A206,
      A187,A199,A186,A200,A204,A185,A195,A196,A24;
A213: eval(L,x) = (x/.k + 1.F_Real)*eval(Q,x) by A37;
A214: eval(A,x) = eval(M,x)*(eval(U,x) + 1.F_Real) by A39;
A215: eval(B,x) = eval(W,x) + 1.F_Real by A41;
A216: eval(C,x) = (x/.r) + eval(W,x) + 1.F_Real by A43;
A217: D1 = eval(D,x) by A45,A178,A214,A175,A206,A187,A199,A204,A185,A195,
      A196,A205,A216,A177,A201;
A218: eval(E,x) = Two *(x/.i)*(eval(C,x)^2)* eval(L,x)*eval(D,x) by A47;
A219: F1 = eval(F,x) by A50,A180,A214,A175,A206,A187,A199,A204,A185,A195,
      A196,A205,A218,A179,A197,A213,A184,A211,A190,A216,A177,A201,A217;
A220: eval(G,x) = eval(A,x)+eval(F,x)*(eval(F,x) - eval(A,x)) by A52;
A221: eval(H,x) = eval(B,x)+Two*(x/.j - 1.F_Real)*eval(C,x) by A54;
A222: I1 = eval(I,x) by A56,A183,A220,A181,A214,A175,A206,A187,A199,A204,A185,
      A195,A196,A205,A219,A221,A182,A215,A176,A198,A216,A177,A201;
      reconsider X11=eval(X1,x),X21=eval(X2,x) as odd Nat
      by A104,A171,A195,A196,A197,A198,A199,A200,A201,A202,A203;
      reconsider X31= eval(X3,x),R1=eval(R,x),NN1=eval(NN,x) as Nat
      by A171,A195,A196,A197,A198,A199,A200,A201,A202,A203,A104;
      reconsider P1=eval(P,x) as positive Nat
      by A171,A195,A196,A197,A198,A199,A200,A201,A202,A203,A104;
A223: X11 = (M1^2 - 1)* S1^2+ 1 by A58,A208,A207,A188,A202,A195;
A224: X21 = ((M1*U1)^2 - 1)* T1^2+ 1 by A60,A206,A205,A210,A189,A204,
      A185,A195,A196,A203,A187,A199;
A225: eval(X3,x) = eval(D,x) *eval(F,x) * eval(I,x) by A63;
A226: eval(P,x) = eval(F,x) * eval(L,x) by A65;
A227:eval(R,x) =
      (eval(H,x)-eval(C,x))*eval(L,x) + eval(F,x)*(x/.f+1.F_Real)*eval(Q,x) +
      eval(F,x)*(x/.k+1.F_Real) *
      ((eval(W,x)^2-1.F_Real)*eval(S,x)*x/.u-eval(W,x)^2*(x/.u)^2 +1.F_Real)
      by A72;
      reconsider V11 = eval(V1,x),V21 = eval(V2,x),V31 = eval(V3,x)
      as Integer;
A228: eval(V1,x) = Eight*((x/.f)*(x/.u)*eval(S,x)*eval(T,x)*
      ((x/.r)-(x/.m)*eval(S,x)*eval(T,x)*eval(U,x))) by A84;
A229: eval(V3,x) = (Four*(x/.f)^2-1.F_Real)*
      ((x/.r)-(x/.m)*eval(S,x)*eval(T,x)*eval(U,x))^2 by A93;
      reconsider VV=V11-V21-V31-1 as Integer;
A230: NN1 > sqrt X11 + 2* sqrt X21 +4* sqrt X31 + R1
      by A171,A195,A196,A197,A198,A199,A200,A201,A202,A203,A104;
A231: eval(V,x) = eval(V1-V2-V3,x)-eval(O,x) by POLYNOM2:24
      .= eval(V1-V2-V3,x)-1.F_Real by POLYNOM2:21
      .= eval(V1-V2,x)-eval(V3,x)-1.F_Real by POLYNOM2:24
      .= eval(V1,x)-eval(V2,x)-eval(V3,x)-1.F_Real by POLYNOM2:24;
      V31 + V21 < V11 & VV+1 = V11 -(V31 + V21) by A173,A228,A87,A200,A208,
      A210,A189,A206,A204,A185,A195,A196,A203,A187,A199,A229,A201,A205,
      A188,A202;
      then
A232: 0 < VV+1 by XREAL_1:50;
      consider z1 be Nat such that
A233:  for f be Function of 8,F_Real
        st f = <%z1,X11,4*X21,16*X31%>^<%R1,P1,NN1,VV%> holds
      eval(K3,f) = 0 by A166,A230,A172,A223,A224,A225,A217,A213,A184,A195,
      A200,A204,A185,A196,A216,A177,A201,A222,A232,A174,A226,
      A227,A221,A182,A215,A198,A219,A212,A209,A176;
      set yy = <%z1%>^<%0%>^<%0%>^<%0%>^<%0%>^<%0%>^<%0%>^<%0%>;
      set p = @(yy^vv);
      reconsider p as INT -valued Function of N, F_Real;
A234: N=Segm N;
      then
A235: 0 in N & 1 in N & 2 in N & 3 in N & 4 in N & 5 in N & 6 in N &
      7 in N by NAT_1:44;
      set p7 = p+*(7,eval(V,p));
A236: eval(Z7,p) = eval(Z6,p7) by Th37,A234,NAT_1:44;
      set p6 = p7+*(6,eval(NN,p));
      not 7 in vars NN by A1,A103;
      then 7 in N\vars NN by A235,XBOOLE_0:def 5;
      then eval(NN,p7) = eval(NN,p) by Th53;
      then
A237: eval(Z6,p7) = eval(Z5,p6) by Th37,A234,NAT_1:44;
      set p5 = p6+*(5,eval(P,p));
      not 6 in vars P & not 7 in vars P by A64,A1;
      then
A238: 6 in N\vars P & 7 in N\vars P by A235,XBOOLE_0:def 5;
      then eval(P,p6) = eval(P,p7) by Th53
      .= eval(P,p) by A238,Th53;
      then
A239:eval(Z5,p6)= eval(Z4,p5) by Th37,A234,NAT_1:44;
      set p4 = p5+*(4,eval(R,p));
      not 5 in vars R & not 6 in vars R & not 7 in vars R by A66,A1;
      then
A240: 5 in N\vars R & 6 in N\vars R & 7 in N\vars R by A235,XBOOLE_0:def 5;
      then eval(R,p5) = eval(R,p6) by Th53
      .= eval(R,p7) by A240,Th53
      .= eval(R,p) by A240,Th53;
      then
A241: eval(Z4,p5)= eval(Z3,p4) by Th37,A234,NAT_1:44;
A242: vars ((Four*Four)*X3) c= VARs by Th80,A62;
      set p3 = p4+*(3,eval((Four*Four)*X3,p));
      not 4 in vars ((Four*Four)*X3) & not 5 in vars ((Four*Four)*X3) &
      not 6 in vars ((Four*Four)*X3) & not 7 in vars ((Four*Four)*X3)
      by A1,A242;
      then
A243: 4 in N\vars ((Four*Four)*X3) & 5 in N\vars ((Four*Four)*X3) &
      6 in N\vars ((Four*Four)*X3) & 7 in N\vars ((Four*Four)*X3)
      by A235,XBOOLE_0:def 5;
      then eval((Four*Four)*X3,p4) = eval((Four*Four)*X3,p5) by Th53
      .= eval((Four*Four)*X3,p6) by A243,Th53
      .= eval((Four*Four)*X3,p7) by A243,Th53
      .= eval((Four*Four)*X3,p) by A243,Th53;
      then
A244: eval(Z3,p4)= eval(Z2,p3) by Th37,A234,NAT_1:44;
A245: vars (Four*X2) c= VARs by Th80,A59;
      set p2 = p3+*(2,eval(Four*X2,p));
      not 3 in vars (Four*X2) & not 4 in vars (Four*X2) &
      not 5 in vars (Four*X2) &
      not 6 in vars (Four*X2) & not 7 in vars (Four*X2) by A1,A245;
      then A246: 3 in N\vars (Four*X2) & 4 in N\vars (Four*X2) &
      5 in N\vars (Four*X2) &
      6 in N\vars (Four*X2) & 7 in N\vars (Four*X2) by A235,XBOOLE_0:def 5;
      then eval(Four*X2,p3) = eval(Four*X2,p4) by Th53
      .= eval(Four*X2,p5) by A246,Th53
      .= eval(Four*X2,p6) by A246,Th53
      .= eval(Four*X2,p7) by A246,Th53
      .= eval(Four*X2,p) by A246,Th53;
      then
A247: eval(Z2,p3)= eval(Z1,p2) by Th37,A234,NAT_1:44;
      set p1 = p2+*(1,eval(X1,p));
      not 2 in vars X1 & not 3 in vars X1 & not 4 in vars X1 &
      not 5 in vars X1 & not 6 in vars X1 & not 7 in vars X1 by A1,A57;
      then
A248: 2 in N\vars X1 & 3 in N\vars X1 & 4 in N\vars X1 &
      5 in N\vars X1 &
      6 in N\vars X1 & 7 in N\vars X1 by A235,XBOOLE_0:def 5;
      then eval(X1,p2) = eval(X1,p3) by Th53
      .= eval(X1,p4) by A248,Th53
      .= eval(X1,p5) by A248,Th53
      .= eval(X1,p6) by A248,Th53
      .= eval(X1,p7) by A248,Th53
      .= eval(X1,p) by A248,Th53;
      then
A249: eval(Z1,p2)= eval(Z,p1) by Th37,A234,NAT_1:44;
A250: len yy = 8 by AFINSQ_1:49;
A251:  dom p1 = N & Segm 8 c= Segm N by NAT_1:39,PARTFUN1:def 2;
      then
A252: dom (p1|8) =8  & rng (p1|8) c= rng p1 c= the carrier of F_Real
      by RELAT_1:62,70;
      then reconsider p8 = p1|8 as Function of 8, F_Real by FUNCT_2:2;
      set ZR= <%z1,X11,4*X21,16*X31%>^<%R1,P1,NN1,VV%>;
A253: ZR=
      (<%z1%>^<%X11%>^<%4*X21%>^<%16*X31%>)^<%R1,P1,NN1,VV%> by AFINSQ_1:def 14
      .= (<%z1%>^<%X11%>^<%4*X21%>^<%16*X31%>)^(<%R1%>^<%P1%>^<%NN1%>^<%VV%>)
      by AFINSQ_1:def 14
      .= (<%z1%>^<%X11%>^<%4*X21%>^<%16*X31%>)^(<%R1%>^<%P1%>^<%NN1%>)^<%VV%>
      by AFINSQ_1:27
      .= (<%z1%>^<%X11%>^<%4*X21%>^<%16*X31%>)^(<%R1%>^<%P1%>)^<%NN1%>^<%VV%>
      by AFINSQ_1:27
      .= (<%z1%>^<%X11%>^<%4*X21%>^<%16*X31%>)^<%R1%>^<%P1%>^<%NN1%>^<%VV%>
      by AFINSQ_1:27;
      then
A254: dom ZR = 8 by AFINSQ_1:49;
A255: dom p1 =N & dom p2=N & dom p3 =N & dom p4=N
      & dom p5 =N & dom p6=N & dom p7 =N & dom p = N by FUNCT_2:def 1;
A256: dom (x+*(0,z1)) = dom x = N = dom p by FUNCT_2:def 1,FUNCT_7:30;
      for ii be object st ii in dom p holds p.ii = (x+*(0,z1)).ii
      proof
        let ii be object such that
A257:   ii in dom p;
        reconsider ii as Nat by A257;
        per cases;
        suppose
A258:     ii=0;
A259:     0 in Segm 8 by NAT_1:44;
          p.0 = yy.0 by A259,A250,AFINSQ_1:def 3
          .= z1 by AFINSQ_1:49;
          hence thesis by A258,A256,A257,FUNCT_7:31;
        end;
        suppose
A260:     ii<>0;
          per cases;
          suppose ii < Segm 8;
            then
A261:       ii in 8 =Segm(7+1) by NAT_1:44;
A262:       x.ii = zz.ii by A191,A261,AFINSQ_1:def 3;
A263:       p.ii = yy.ii by A261,A250,AFINSQ_1:def 3;
            ii = 0 or ... or ii=7 by A261,NAT_1:61;
            then yy.ii = 0 = zz.ii by A260,AFINSQ_1:49;
            hence thesis by A260,FUNCT_7:32,A262,A263;
          end;
          suppose ii>=8;
            then reconsider jj=ii-8 as Nat by NAT_1:21;
            ii in Segm N by A257;
            then jj+8 < 8+9 by NAT_1:44;
            then
A264:       jj in dom vv by A192,NAT_1:44,XREAL_1:6;
A265:       p.ii = (yy^vv).(8+jj)
            .= vv.jj by A264,A250,AFINSQ_1:def 3;
            x.ii = (zz^vv).(8+jj)
            .= vv.jj by A264,A191,AFINSQ_1:def 3;
            hence thesis by A260,FUNCT_7:32,A265;
          end;
        end;
      end;
      then
A266: x+*(0,z1) = p by A256;
A267: z1 in the carrier of F_Real by XREAL_0:def 1;
      0 in Segm 8 by NAT_1:44;
      then
A268: p.0 = yy.0 by A250,AFINSQ_1:def 3
      .= z1 by AFINSQ_1:49;
      for ii be object st ii in dom ZR holds ZR.ii=p8.ii
      proof
        let ii be object such that
A269:   ii in dom ZR;
        reconsider ii as Nat by A269;
A270:   p8.ii = p1.ii & ii in dom p1 by A251,A269,A254,FUNCT_1:49;
        ii in Segm(7+1) by A269,A253,AFINSQ_1:49;
        then ii=0 or ... or ii=7 by NAT_1:61;
        then per cases;
        suppose
A271:     ii=0;
          p1.0 = p2.0 = p3.0 = p4.0 by FUNCT_7:32;
          then p1.0 = p5.0 = p6.0 = p7.0 = p.0 by FUNCT_7:32;
          hence thesis by A253,AFINSQ_1:49,A268,A271,A270;
        end;
        suppose
A272:     ii=1;
          not 0 in vars X1 by A1,A57;
          then
A273:     0 in N\vars X1 by A235,XBOOLE_0:def 5;
          p1.1 = eval(X1,p) by A255,A251,A269,A254,A272,FUNCT_7:31
          .= eval(X1,x) by A273,A267,Th53,A266;
          hence thesis by A270,A272,A253,AFINSQ_1:49;
        end;
        suppose
A274:     ii=2;
          not 0 in vars X2 by A1,A59;
          then
A275:     0 in N\vars X2 by A235,XBOOLE_0:def 5;
          p1.2 = p2.2 by FUNCT_7:32
          .= eval(Four*X2,p) by A255,A251,
          A269,A254,A274,FUNCT_7:31
          .= Four*eval(X2,p) by POLYNOM7:29
          .= Four* eval(X2,x) by A275,A267,Th53,A266;
          hence thesis by A270,A274,A253,AFINSQ_1:49;
        end;
        suppose
A276:     ii=3;
          not 0 in vars X3 by A1,A62;
          then
A277:     0 in N\vars X3 by A235,XBOOLE_0:def 5;
          p1.3 = p2.3 by FUNCT_7:32
          .= p3.3 by FUNCT_7:32
          .= eval((Four*Four)*X3,p) by A255,A251,
          A269,A254,A276,FUNCT_7:31
          .= (Four*Four)*eval(X3,p) by POLYNOM7:29
          .= (Four*Four)* eval(X3,x) by A277,A267,Th53,A266;
          hence thesis by A270,A276,A253,AFINSQ_1:49;
        end;
        suppose
A278:     ii=4;
          not 0 in vars R by A1,A66;
          then
A279:     0 in N\vars R by A235,XBOOLE_0:def 5;
          p1.4 = p2.4 by FUNCT_7:32
          .= p3.4 by FUNCT_7:32
          .= p4.4 by FUNCT_7:32
          .= eval(R,p) by A255,A251,A269,A254,A278,FUNCT_7:31
          .= eval(R,x) by A279,A267,Th53,A266;
          hence thesis by A270,A278,A253,AFINSQ_1:49;
        end;
        suppose
A280:     ii=5;
          not 0 in vars P by A1,A64;
          then
A281:     0 in N\vars P by A235,XBOOLE_0:def 5;
          p1.5 = p2.5 by FUNCT_7:32
          .= p3.5 by FUNCT_7:32
          .= p4.5 by FUNCT_7:32
          .= p5.5 by FUNCT_7:32
          .= eval(P,p) by A255,A251,A269,A254,A280,FUNCT_7:31
          .= eval(P,x) by A281,A267,Th53,A266;
          hence thesis by A270,A280,A253,AFINSQ_1:49;
        end;
        suppose
A282:     ii=6;
          not 0 in vars NN by A1,A103;
          then
A283:     0 in N\vars NN by A235,XBOOLE_0:def 5;
          p1.6 = p2.6 by FUNCT_7:32
          .= p3.6 by FUNCT_7:32
          .= p4.6 by FUNCT_7:32
          .= p5.6 by FUNCT_7:32
          .= p6.6 by FUNCT_7:32
          .= eval(NN,p) by A255,A251,A269,A254,A282,FUNCT_7:31
          .= eval(NN,x) by A283,A267,Th53,A266;
          hence thesis by A270,A282,A253,AFINSQ_1:49;
        end;
        suppose
A284:     ii=7;
          not 0 in vars V by A1,A102;
          then
A285:     0 in N\vars V by A235,XBOOLE_0:def 5;
          p1.7 = p2.7 by FUNCT_7:32
          .= p3.7 by FUNCT_7:32
          .= p4.7 by FUNCT_7:32
          .= p5.7 by FUNCT_7:32
          .= p6.7 by FUNCT_7:32
          .= p7.7 by FUNCT_7:32
          .= eval(V,p) by A255,A251,A269,A254,A284,FUNCT_7:31
          .= eval(V,x) by A285,A267,Th53,A266;
          hence thesis by A231,A270,A284,A253,AFINSQ_1:49;
        end;
      end;
      then p8 = ZR by A254,A252;
      then
A286: eval(K3,p8) = 0 by A233;
      take p;
      p/.k = (yy^vv).(8+0) by A255,A193,NAT_1:44,PARTFUN1:def 6
      .= vv.0 by A194,A250,AFINSQ_1:def 3;
      hence p/.k = k1 by AFINSQ_1:50;
A287: dom p = N by FUNCT_2:def 1;
      p/.f = (yy^vv).(8+1) by A287,A193,NAT_1:44,PARTFUN1:def 6
      .= vv.1 by A194,A250,AFINSQ_1:def 3;
      hence p/.f is positive Nat by AFINSQ_1:50;
      p/.i = (yy^vv).(8+2) by A287,A193,NAT_1:44,PARTFUN1:def 6
      .= vv.2 by A194,A250,AFINSQ_1:def 3;
      hence p/.i is positive Nat by AFINSQ_1:50;
      p/.j = (yy^vv).(8+3) by A287,A193,NAT_1:44,PARTFUN1:def 6
      .= vv.3 by A194,A250,AFINSQ_1:def 3;
      hence p/.j is positive Nat by AFINSQ_1:50;
      p/.m = (yy^vv).(8+4) by A287,A193,NAT_1:44,PARTFUN1:def 6
      .= vv.4 by A194,A250,AFINSQ_1:def 3;
      hence p/.m is positive Nat by AFINSQ_1:50;
      p/.u = (yy^vv).(8+5) by A287,A193,NAT_1:44,PARTFUN1:def 6
      .= vv.5 by A194,A250,AFINSQ_1:def 3;
      hence p/.u is positive Nat by AFINSQ_1:50;
      p/.r = (yy^vv).(8+6) by A287,A193,NAT_1:44,PARTFUN1:def 6
      .= vv.6 by A194,A250,AFINSQ_1:def 3;
      hence p/.r is Nat by AFINSQ_1:50;
      p/.s = (yy^vv).(8+7) by A287,A193,NAT_1:44,PARTFUN1:def 6
      .= vv.7 by A194,A250,AFINSQ_1:def 3;
      hence p/.s is Nat by AFINSQ_1:50;
      p/.t = (yy^vv).(8+8) by A287,A193,NAT_1:44,PARTFUN1:def 6
      .= vv.8 by A194,A250,AFINSQ_1:def 3;
      hence p/.t is Nat by AFINSQ_1:50;
      thus p/.0 is Nat by A268,A287,A193,NAT_1:44,PARTFUN1:def 6;
      thus thesis by A236,A237,A239,A241,A244,A247,A249,A169,A286;
    end;
    given x be INT -valued Function of N, F_Real such that
A288: x/.k = k1 and
A289:  x/.f is positive Nat &
    x/.i is positive Nat &
    x/.j is positive Nat &
    x/.m is positive Nat &
    x/.u is positive Nat &
    x/.r is Nat & x/.s is Nat & x/.t is Nat & x/.0 is Nat &
    eval(Z7,x) = 0.F_Real;
    reconsider f1=x/.f,i1=x/.i,j1=x/.j,m1=x/.m,u1=x/.u as positive Nat by A289;
    reconsider r1=x/.r,s1=x/.s,t1=x/.t,z1=x/.0 as Nat by A289;
A290: eval(W,x) = Hund * x /. f * x/.k * (x/.k+1.F_Real) by A22;
    reconsider W1=eval(W,x) as Integer;
A291: eval(U,x)= Hund * (u1|^3 ) * W1|^3 + 1 by A24;
    reconsider U1=eval(U,x) as Integer;
A292: eval(M,x) = Hund * (x /. m) * eval(U,x) * eval(W,x)+1.F_Real by A29;
    reconsider M1=eval(M,x) as Integer;
    eval(S,x) = (eval(M,x) - 1.F_Real) * (x/.s) + (x/.k) + 1.F_Real by A31;
    then
A293: eval(S,x) = (M1 - 1) * s1 + k1  + 1 by A288;
    reconsider S1=eval(S,x) as Integer;
    eval(T,x) = (eval(M,x)*eval(U,x) - 1.F_Real)*(x/.t)+
    eval(W,x)-(x/.k)+1.F_Real by A33;
    then
A294: eval(T,x) = (M1*U1 - 1)*(t1)+ W1-k1 +1 by A288;
    reconsider T1 = eval(T,x) as Integer;
A295: eval(Q,x) = Two*eval(M,x)*eval(W,x) - eval(W,x)^2 - 1.F_Real by A35;
    reconsider Q1 = eval(Q,x) as Integer;
A296: eval(L,x) = (x/.k + 1.F_Real)*eval(Q,x) by A37;
    reconsider L1 = eval(L,x) as Integer;
A297: eval(A,x) = eval(M,x)*(eval(U,x) + 1.F_Real) by A39;
    reconsider A1 = eval(A,x) as Integer;
A298:eval(B,x) = eval(W,x) + 1.F_Real by A41;
    reconsider B1 = eval(B,x) as Integer;
A299: eval(C,x) = (x/.r) + eval(W,x) + 1.F_Real by A43;
    reconsider C1 = eval(C,x) as Integer;
A300: eval(D,x) = (A1^2 - 1)* C1^2+ 1 by A45;
    reconsider D1 = eval(D,x) as Integer;
    eval(E,x) = Two *(x/.i)*(eval(C,x)^2)* eval(L,x)*eval(D,x) by A47;
    then
A301:eval(E,x) = 2 *i1*(C1^2)* L1*D1;
    reconsider E1 = eval(E,x) as Integer;
A302:eval(F,x) = (A1^2 - 1)* E1^2+ 1 by A50;
    reconsider  F1 = eval(F,x) as Integer;
A303:  eval(G,x) = eval(A,x)+eval(F,x)*(eval(F,x) - eval(A,x)) by A52;
    reconsider G1 = eval(G,x) as Integer;
    eval(H,x) = eval(B,x)+Two*(x/.j - 1.F_Real)*eval(C,x) by A54;
    then
A304: eval(H,x) = B1+2*(j1 - 1)*C1;
    reconsider H1 = eval(H,x) as Integer;
A305:eval(I,x) = (G1^2 - 1)* H1^2+ 1 by A56;
    reconsider I1 = eval(I,x) as Integer;
    reconsider X11=eval(X1,x),X21=eval(X2,x) as odd Nat by A104,A171,A289,A288;
    reconsider X31= eval(X3,x),R1=eval(R,x),NN1=eval(NN,x) as Nat
    by A171,A289,A288,A104;
    reconsider P1=eval(P,x) as positive Nat by A171,A289,A288,A104;
    reconsider V11 = eval(V1,x),V21 = eval(V2,x),V31 = eval(V3,x)
    as Integer;
    reconsider VV=V11-V21-V31-1 as Integer;
A306:X11 = (M1^2 - 1)* S1^2+ 1 by A58;
A307: eval(X2,x) = ((eval(M,x)*eval(U,x))^2 - 1.F_Real)* eval(T,x)^2+
    1.F_Real by A60;
A308:  eval(X3,x) = eval(D,x) *eval(F,x) * eval(I,x) by A63;
A309:  eval(P,x) = eval(F,x) * eval(L,x) by A65;
A310:  eval(R,x) =
    (eval(H,x)-eval(C,x))*eval(L,x) + eval(F,x)*(x/.f+1.F_Real)*eval(Q,x) +
    eval(F,x)*(x/.k+1.F_Real) *
    ((eval(W,x)^2-1.F_Real)*eval(S,x)*x/.u-eval(W,x)^2*(x/.u)^2 +1.F_Real)
    by A72;
    reconsider V11 = eval(V1,x),V21 = eval(V2,x),V31 = eval(V3,x)
    as Integer;
A311: eval(V1,x) = Eight*((x/.f)*(x/.u)*eval(S,x)*eval(T,x)*
    ((x/.r)-(x/.m)*eval(S,x)*eval(T,x)*eval(U,x))) by A84;
A312: eval(V3,x) = (Four*(x/.f)^2-1.F_Real)*
    ((x/.r)-(x/.m)*eval(S,x)*eval(T,x)*eval(U,x))^2 by A93;
    reconsider VV=V11-V21-V31-1 as Integer;
A313: eval(V,x) = eval(V1-V2-V3,x)-eval(O,x) by POLYNOM2:24
    .= eval(V1-V2-V3,x)-1.F_Real by POLYNOM2:21
    .= eval(V1-V2,x)-eval(V3,x)-1.F_Real by POLYNOM2:24
    .= eval(V1,x)-eval(V2,x)-eval(V3,x)-1.F_Real by POLYNOM2:24;
A314: NN1 > sqrt X11 + 2*sqrt X21+4* sqrt X31 + R1 by A171,A289,A288,A104;
A315: N=Segm N;
    then
A316: 0 in N & 1 in N & 2 in N & 3 in N & 4 in N & 5 in N & 6 in N & 7 in N
    by NAT_1:44;
    set x7 = x+*(7,eval(V,x));
A317:eval(Z7,x) = eval(Z6,x7) by Th37,A315,NAT_1:44;
    set x6 = x7+*(6,eval(NN,x));
    not 7 in vars NN by A1,A103;
    then 7 in N\vars NN by A316,XBOOLE_0:def 5;
    then eval(NN,x7) = eval(NN,x) by Th53;
    then
A318: eval(Z6,x7) = eval(Z5,x6) by Th37,A315,NAT_1:44;
    set x5 = x6+*(5,eval(P,x));
    not 6 in vars P & not 7 in vars P by A1,A64;
    then
A319: 6 in N\vars P & 7 in N\vars P by A316,XBOOLE_0:def 5;
    then eval(P,x6) = eval(P,x7) by Th53
    .= eval(P,x) by A319,Th53;
    then
A320: eval(Z5,x6)= eval(Z4,x5) by Th37,A315,NAT_1:44;
    set x4 = x5+*(4,eval(R,x));
    not 5 in vars R & not 6 in vars R & not 7 in vars R by A1,A66;
    then
A321: 5 in N\vars R & 6 in N\vars R & 7 in N\vars R
    by A316,XBOOLE_0:def 5;
    then eval(R,x5) = eval(R,x6) by Th53
    .= eval(R,x7) by A321,Th53
    .= eval(R,x) by A321,Th53;
    then
A322:eval(Z4,x5)= eval(Z3,x4) by Th37,A315,NAT_1:44;
A323: vars ((Four*Four)*X3) c= VARs by Th80,A62;
    set x3 = x4+*(3,eval((Four*Four)*X3,x));
    not 4 in vars ((Four*Four)*X3) & not 5 in vars ((Four*Four)*X3) &
    not 6 in vars ((Four*Four)*X3) & not 7 in vars ((Four*Four)*X3)
    by A1,A323;
    then
A324: 4 in N\vars ((Four*Four)*X3) & 5 in N\vars ((Four*Four)*X3) &
    6 in N\vars ((Four*Four)*X3) & 7 in N\vars ((Four*Four)*X3)
    by A316,XBOOLE_0:def 5;
    then eval((Four*Four)*X3,x4) = eval((Four*Four)*X3,x5) by Th53
    .= eval((Four*Four)*X3,x6) by A324,Th53
    .= eval((Four*Four)*X3,x7) by A324,Th53
    .= eval((Four*Four)*X3,x) by A324,Th53;
    then
A325:eval(Z3,x4)= eval(Z2,x3) by Th37,A315,NAT_1:44;
A326: vars (Four*X2) c= VARs by Th80,A59;
    set x2 = x3+*(2,eval(Four*X2,x));
    not 3 in vars (Four*X2) & not 4 in vars (Four*X2) &
    not 5 in vars (Four*X2) &
    not 6 in vars (Four*X2) & not 7 in vars (Four*X2)
    by A1,A326;
    then
A327: 3 in N\vars (Four*X2) & 4 in N\vars (Four*X2) &
    5 in N\vars (Four*X2) &
    6 in N\vars (Four*X2) & 7 in N\vars (Four*X2) by A316,XBOOLE_0:def 5;
    then eval(Four*X2,x3) = eval(Four*X2,x4) by Th53
    .= eval(Four*X2,x5) by A327,Th53
    .= eval(Four*X2,x6) by A327,Th53
    .= eval(Four*X2,x7) by A327,Th53
    .= eval(Four*X2,x) by A327,Th53;
    then
A328: eval(Z2,x3)= eval(Z1,x2) by Th37,A315,NAT_1:44;
    set x1 = x2+*(1,eval(X1,x));
    not 2 in vars X1 & not 3 in vars X1 & not 4 in vars X1 &
    not 5 in vars X1 & not 6 in vars X1 & not 7 in vars X1 by A1,A57;
    then
A329: 2 in N\vars X1 & 3 in N\vars X1 & 4 in N\vars X1 &
    5 in N\vars X1 &
    6 in N\vars X1 & 7 in N\vars X1 by A316,XBOOLE_0:def 5;
    then eval(X1,x2) = eval(X1,x3) by Th53
    .= eval(X1,x4) by A329,Th53
    .= eval(X1,x5) by A329,Th53
    .= eval(X1,x6) by A329,Th53
    .= eval(X1,x7) by A329,Th53
    .= eval(X1,x) by A329,Th53;
    then
A330: eval(Z1,x2)= eval(Z,x1) by Th37,A315,NAT_1:44;
    dom x1 = N & Segm 8 c= Segm N by NAT_1:39,PARTFUN1:def 2;
    then dom (x1|8) =8  & rng (x1|8) c= rng x1 c= the carrier of F_Real
    by RELAT_1:62,70;
    then reconsider x8 = x1|8 as Function of 8, F_Real by FUNCT_2:2;
A331: for f be Function of 8,F_Real st
    f= <%z1,X11,4*X21,16*X31%>^<%R1,P1,NN1,VV%>
    holds 0 = eval(K3,f)
    proof
      let ff be Function of 8,F_Real such that
A332: ff= <%z1,X11,4*X21,16*X31%>^<%R1,P1,NN1,VV%>;
A333: ff =
      (<%z1%>^<%X11%>^<%4*X21%>^<%16*X31%>)^<%R1,P1,NN1,VV%>
      by A332,AFINSQ_1:def 14
      .= (<%z1%>^<%X11%>^<%4*X21%>^<%16*X31%>)^(<%R1%>^<%P1%>^<%NN1%>^<%VV%>)
      by AFINSQ_1:def 14
      .= (<%z1%>^<%X11%>^<%4*X21%>^<%16*X31%>)^(<%R1%>^<%P1%>^<%NN1%>)^<%VV%>
      by AFINSQ_1:27
      .= (<%z1%>^<%X11%>^<%4*X21%>^<%16*X31%>)^(<%R1%>^<%P1%>)^<%NN1%>^<%VV%>
      by AFINSQ_1:27
      .= (<%z1%>^<%X11%>^<%4*X21%>^<%16*X31%>)^<%R1%>^<%P1%>^<%NN1%>^<%VV%>
      by AFINSQ_1:27;
      then
A334: dom ff = Segm 8 by AFINSQ_1:49;
A335: dom x = N & dom x1 =N & dom x2=N & dom x3 =N & dom x4=N
      & dom x5 =N & dom x6=N & dom x7 =N by FUNCT_2:def 1;
A336: 8 = dom x8 by FUNCT_2:def 1;
      for ii be object st ii in dom ff holds ff.ii = x8.ii
      proof
        let ii be object such that
A337:   ii in dom ff;
        reconsider ii as Nat by A337,A333;
A338:   x8.ii = x1.ii & ii in dom x1 by A337,FUNCT_1:49,A336,RELAT_1:57;
        ii in Segm(7+1) by A337;
        then ii=0 or ... or ii=7 by NAT_1:61;
        then
        per cases;
        suppose
A339:     ii=0;
          then
A340:     ff.ii = z1 by A333,AFINSQ_1:49;
          x1.0 = x2.0 = x3.0 = x4.0 by FUNCT_7:32;
          then x1.0 = x5.0 = x6.0 = x7.0 = x.0 by FUNCT_7:32;
          hence thesis by A340,A339,A338,A335,PARTFUN1:def 6;
        end;
        suppose
A341:     ii=1;
          x1.1 = eval(X1,x) by A335,A338,A341,FUNCT_7:31;
          hence thesis by A338,A341,A333,AFINSQ_1:49;
        end;
        suppose
A342:     ii=2;
          x1.2 = x2.2 by FUNCT_7:32
          .= eval(Four*X2,x) by A335,A338,A342,FUNCT_7:31
          .= Four*eval(X2,x) by POLYNOM7:29;
          hence thesis by A338,A342,A333,AFINSQ_1:49;
        end;
        suppose
A343:     ii=3;
          x1.3 = x2.3 by FUNCT_7:32
          .= x3.3 by FUNCT_7:32
          .= eval((Four*Four)*X3,x) by A335,A338,A343,FUNCT_7:31
          .= (Four*Four)*eval(X3,x) by POLYNOM7:29;
          hence thesis by A338,A343,A333,AFINSQ_1:49;
        end;
        suppose
A344:     ii=4;
          x1.4 = x2.4 by FUNCT_7:32
          .= x3.4 by FUNCT_7:32
          .= x4.4 by FUNCT_7:32
          .= eval(R,x) by A335,A338,A344,FUNCT_7:31;
          hence thesis by A338,A344,A333,AFINSQ_1:49;
        end;
        suppose
A345:     ii=5;
          x1.5 = x2.5 by FUNCT_7:32
          .= x3.5 by FUNCT_7:32
          .= x4.5 by FUNCT_7:32
          .= x5.5 by FUNCT_7:32
          .= eval(P,x) by A335,A338,A345,FUNCT_7:31;
          hence thesis by A338,A345,A333,AFINSQ_1:49;
        end;
        suppose
A346:     ii=6;
          x1.6 = x2.6 by FUNCT_7:32
          .= x3.6 by FUNCT_7:32
          .= x4.6 by FUNCT_7:32
          .= x5.6 by FUNCT_7:32
          .= x6.6 by FUNCT_7:32
          .= eval(NN,x) by A335,A338,A346,FUNCT_7:31;
          hence thesis by A338,A346,A333,AFINSQ_1:49;
        end;
        suppose
A347:     ii=7;
          x1.7 = x2.7 by FUNCT_7:32
          .= x3.7 by FUNCT_7:32
          .= x4.7 by FUNCT_7:32
          .= x5.7 by FUNCT_7:32
          .= x6.7 by FUNCT_7:32
          .= x7.7 by FUNCT_7:32
          .= eval(V,x) by A335,A338,A347,FUNCT_7:31;
          hence thesis by A313,A338,A347,A333,AFINSQ_1:49;
        end;
      end;
      then ff=x8 by A334;
      hence thesis by A289,A169,A317,A318,A320,A322,A325,A328,A330;
    end;
    then
A348: X11 is square & X21 is square  & X31 is square & P1 divides R1 &
      VV>=0 by A166,A314;
    V11-V21-V31-1+1 > 0 by A331,A166,A314;
    then V11-(V21+V31)+ (V21+V31) > 0+(V21+V31) by XREAL_1:8;
    then (4*f1^2 -1)*(r1-m1*S1*T1*U1)^2 + 4*u1^2*S1^2*T1^2 <
    8*f1*u1*S1*T1*(r1-m1*S1*T1*U1) by A311,A87,A312;
    hence thesis by A171,HILB10_8:23,A297,A298,A299,A300,A301,A302,A303,
      A304,A305,A296,A290,A288,A291,A292,A293,A294,A295,A348,A308,
      A307,A306,A309,A310;
  end;
A349: vars Z c= 8
  proof
    let y be object;
    assume y in vars Z;
    then consider b be bag of N such that
A350: b in Support Z & b.y <> 0 by Def5;
    y in dom b = N =Segm N by PARTFUN1:def 2,A350,FUNCT_1:def 2;
    then reconsider y as Nat;
    y in Segm 8 by A168,A350,NAT_1:44;
    hence thesis;
  end;
  vars Z\{1} c= 8\{1} by A349,XBOOLE_1:33;
  then
A351: (vars Z\{1})\/vars X1 c= 8\(Seg 1)\/VARs by A57,XBOOLE_1:13,FINSEQ_1:2;
  vars Z1 c= (vars Z\ {1})\/vars X1 by Th47;
  then
A352:vars Z1 c= 8\Seg 1 \/ VARs by A351;
A353: vars (Four*X2) c= VARs by Th80,A59;
  not 2 in VARs by A1;
  then VARs\{2} = VARs by ZFMISC_1:57;
  then (8\Seg 1 \/ VARs)\{2} = (8\Seg 1\{2}) \/ VARs by XBOOLE_1:42
  .= (8\(Seg 1\/{1+1})) \/ VARs by XBOOLE_1:41
  .= (8\(Seg 2)) \/ VARs by FINSEQ_1:9;
  then vars Z1\{2} c= (8\Seg 2) \/ VARs by A352,XBOOLE_1:33;
  then
A354: (vars Z1\{2})\/vars (Four*X2) c= (8\Seg 2) \/ VARs \/VARs
  = (8\Seg 2) \/ (VARs \/VARs) by A353,XBOOLE_1:4,XBOOLE_1:13;
  vars Z2 c= (vars Z1\{2})\/vars (Four*X2) by Th47;
  then
A355: vars Z2 c= (8\Seg 2) \/ VARs by A354;
A356: vars ((Four*Four)*X3) c= VARs by Th80,A62;
  not 3 in VARs by A1;
  then VARs\{3} = VARs by ZFMISC_1:57;
  then (8\Seg 2 \/ VARs)\{3} = (8\Seg 2\{3}) \/ VARs by XBOOLE_1:42
  .= (8\(Seg 2\/{2+1})) \/ VARs by XBOOLE_1:41
  .= (8\(Seg 3)) \/ VARs by FINSEQ_1:9;
  then vars Z2\{3} c= (8\Seg 3) \/ VARs by A355,XBOOLE_1:33;
  then
A357: (vars Z2\{3})\/vars ((Four*Four)*X3) c= (8\Seg 3) \/ VARs \/VARs
  = (8\Seg 3) \/ (VARs \/VARs) by A356,XBOOLE_1:4,13;
  vars Z3 c= (vars Z2\{3})\/vars ((Four*Four)*X3) by Th47;
  then
A358: vars Z3 c= (8\Seg 3) \/ VARs by A357;
  not 4 in VARs by A1;
  then VARs\{4} = VARs by ZFMISC_1:57;
  then(8\Seg 3 \/ VARs)\{4} = (8\Seg 3\{4}) \/ VARs by XBOOLE_1:42
  .= (8\(Seg 3\/{3+1})) \/ VARs by XBOOLE_1:41
  .= (8\(Seg 4)) \/ VARs by FINSEQ_1:9;
  then vars Z3\{4} c= (8\Seg 4) \/ VARs by A358,XBOOLE_1:33;
  then
A359: (vars Z3\{4})\/(vars R) c= (8\Seg 4) \/ VARs \/VARs
  = (8\Seg 4) \/ (VARs \/VARs) by A66,XBOOLE_1:4,13;
  vars Z4 c= (vars Z3\{4})\/vars R by Th47;
  then
A360: vars Z4 c= (8\Seg 4) \/ VARs by A359;
  not 5 in VARs by A1;
  then VARs\{5} = VARs by ZFMISC_1:57;
  then (8\Seg 4 \/ VARs)\{5} = (8\Seg 4\{5}) \/ VARs by XBOOLE_1:42
  .= (8\(Seg 4\/{4+1})) \/ VARs by XBOOLE_1:41
  .= (8\(Seg 5)) \/ VARs by FINSEQ_1:9;
  then vars Z4\{5} c= (8\Seg 5) \/ VARs by A360,XBOOLE_1:33;
  then
A361: (vars Z4\{5})\/(vars P) c= (8\Seg 5) \/ VARs \/VARs
  = (8\Seg 5) \/ (VARs \/VARs) by A64,XBOOLE_1:4,13;
  vars Z5 c= (vars Z4\{5})\/vars P by Th47;
  then
A362: vars Z5 c= (8\Seg 5) \/ VARs by A361;
  not 6 in VARs by A1;
  then VARs\{6} = VARs by ZFMISC_1:57;
  then (8\Seg 5 \/ VARs)\{6} = (8\Seg 5\{6}) \/ VARs by XBOOLE_1:42
  .= (8\(Seg 5\/{5+1})) \/ VARs by XBOOLE_1:41
  .= (8\(Seg 6)) \/ VARs by FINSEQ_1:9;
  then vars Z5\{6} c= (8\Seg 6) \/ VARs by A362,XBOOLE_1:33;
  then
A363: (vars Z5\{6})\/(vars NN) c= (8\Seg 6) \/ VARs \/VARs
  = (8\Seg 6) \/ (VARs \/VARs) by A103,XBOOLE_1:4,13;
  vars Z6 c= (vars Z5\{6})\/vars NN by Th47;
  then
A364: vars Z6 c= (8\Seg 6) \/ VARs by A363;
  not 7 in VARs by A1;
  then VARs\{7} = VARs by ZFMISC_1:57;
  then (8\Seg 6 \/ VARs)\{7} = (8\Seg 6\{7}) \/ VARs by XBOOLE_1:42
  .= (8\(Seg 6\/{6+1})) \/ VARs by XBOOLE_1:41
  .= (8\(Seg 7)) \/ VARs by FINSEQ_1:9;
  then vars Z6\{7} c= (8\Seg 7) \/ VARs by A364,XBOOLE_1:33;
  then
A365: (vars Z6\{7})\/(vars V) c= (8\Seg 7) \/ VARs \/VARs
  = (8\Seg 7) \/ (VARs \/VARs) by A102,XBOOLE_1:4,13;
A366: vars Z7 c= (vars Z6\{7})\/vars V by Th47;
A367:not 0 in Seg 7 by FINSEQ_1:1;
  7+1 = {0} \/ Seg 7 by AFINSQ_1:4;
  then 8\Seg 7 = ({0} \Seg 7)\/(Seg 7 \Seg 7) by XBOOLE_1:42
  .= {0} by A367,ZFMISC_1:59;
  hence vars Z7 c= {0} \/ VARs by A366,A365;
  let xk be Nat;
  thus thesis by A170;
end;
