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
  ex Poly be INT -valued Polynomial of 10,F_Real st
    for k be positive Nat holds
       k+1 is prime
    iff ex v being natural-valued Function of 10,F_Real st
          v.1 = k & eval(Poly,v) = 0.F_Real
proof
  consider p1 be INT -valued Polynomial of 17,F_Real such that
A1: vars p1 c= {0}\/(17\8) and
A2: 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(p1,x) = 0.F_Real by Th82;
  set N=16,IN=idseq N,E=9,IE=idseq E;
  IN|E =IE by FINSEQ_3:51;
  then consider f be FinSequence such that
A3: IN = IE^f by FINSEQ_1:80;
  set R=f^IE,Z= id {0},RZ=R+*Z;
A4: rng IE misses rng f & f is one-to-one by A3,FINSEQ_3:91;
A5: rng IN = rng IE \/rng f = rng R by A3,FINSEQ_1:31;
A6: R is one-to-one by A4,FINSEQ_3:91;
A7: N= len IN = len IE+len f=len R & len IE = E by A3,FINSEQ_1:22;
  then
A8: dom R = Seg N by FINSEQ_1:def 3;
A9: rng Z = {0} = dom Z;
  not 0 in Seg N by FINSEQ_1:1;
  then rng Z misses rng R by A5,ZFMISC_1:50;
  then
A10: RZ is one-to-one by A6,FUNCT_4:92;
A11: {0}\/Seg N = N+1 by AFINSQ_1:4;
  then
A12: dom RZ = N+1 by A8,A9,FUNCT_4:def 1;
  rng RZ c= rng Z \/rng R by FUNCT_4:17;
  then reconsider RZ as Function of N+1,N+1 by A5,A11,A12,FUNCT_2:2;
  card (N+1) = card (N+1);
  then RZ is onto by A10,FINSEQ_4:63;
  then reconsider RZ as Permutation of 17 by A10;
A13: not 0 in Seg N by FINSEQ_1:1;
A14: {0} c= {0}\/Seg 7 by XBOOLE_1:7;
A15: 16+1 = {0}\/Seg N & 7+1 = {0}\/Seg 7 by AFINSQ_1:4;
  then
A16: 17\8 = ({0}\ ({0}\/Seg 7)) \/ (Seg N \ ({0}\/Seg 7)) by XBOOLE_1:42
  .= {}\/(Seg N\ ({0}\/Seg 7)) by A14
  .= {}\/(Seg N\{0}\Seg 7) by XBOOLE_1:41
  .= Seg N\Seg 7 by A13,ZFMISC_1:57;
A17: not 0 in Seg 7 by FINSEQ_1:1;
  then {0}\Seg 7= {0} & {0} misses Seg 7 by ZFMISC_1:59,50;
  then {0}\/(17\8) = 17\Seg 7 by A15,A16,XBOOLE_1:42;
  then
A18: RZ.: (vars p1) c= RZ.: (17\Seg 7 ) by A1,RELAT_1:123;
A19: Z = 0 .--> 0 = {0}-->0 by FUNCT_4:96,FUNCOP_1:def 9;
  then
A20:RZ.:Seg 7 = R.: Seg 7 by A17,ZFMISC_1:50,BOOLMARK:3;
A21: R.: Seg 7 =(R|7).:Seg 7 by RELAT_1:129
  .=f.:dom f by A7,FINSEQ_1:def 3
  .=rng f by RELAT_1:113;
  RZ.:17= RZ.:dom RZ by A11,A8,A9,FUNCT_4:def 1
  .= rng RZ by RELAT_1:113
  .= 17 by FUNCT_2:def 3;
  then
A22: RZ.:(vars p1) c= 17 \rng f by A18,FUNCT_1:64,A20,A21;
A23: Segm 17 \ rng f c= Segm 10
  proof
    let y be object such that
A24:  y in Segm 17 \ rng f;
    assume
A25:  not y in Segm 10;
    reconsider y as Nat by A24;
A26:  y >=9+1 by A25,NAT_1:44;
    then
A27:  y > 9 by NAT_1:13;
    then reconsider y1=y-9 as Nat by NAT_1:21;
    y < N+1 by A24,NAT_1:44;
    then
A28:  y <= N by NAT_1:13;
    then 9-9 < y-9 <= N-9 by A27,XREAL_1:6;
    then 1<= y1 <= len f by NAT_1:14,A7;
    then
A29:  y1 in dom f by FINSEQ_3:25;
    then
A30:  f.y1 = IN.(y1+E) by A7,A3,FINSEQ_1:def 7;
    1<=y by A26,NAT_1:14;
    then y in Seg N by A28;
    then f.y1 =y by A30,FINSEQ_2:49;
    then y in rng f by A29,FUNCT_1:def 3;
    hence thesis by A24,XBOOLE_0:def 5;
  end;
A31: rng RZ = 17 by FUNCT_2:def 3;
A32: RZ.0=0 by A19,FUNCT_4:113;
A33: 0 in Segm 17 by NAT_1:44;
  then
A34: RZ".0 = 0 by A32,A12,FUNCT_1:32;
A35: for i st 1<=i<=9 holds RZ".i = i+7 & RZ.(i+7) = i
  proof
    let i such that
A36:  1<=i<=9;
A37:  i in dom IE & i in Seg 9 by A36;
    not i+7 in dom Z;
    then
A38:  RZ.(i+7) = R.(i+7) by FUNCT_4:11
    .= IE.i by A37,A7,FINSEQ_1:def 7
    .= i by A37,FINSEQ_2:49;
    i+7<=9+7 by A36,XREAL_1:6;
    then i+7 < 16+1 by NAT_1:13;
    then i+7 in Segm 17 by NAT_1:44;
    hence thesis by A38,A12,FUNCT_1:32;
  end;
  set P2 = p1 permuted_by RZ;
A39: rng P2 = rng p1 c= INT by HILB10_2:26,RELAT_1:def 19;
  then reconsider p2=P2 as INT -valued Polynomial of 10+7,F_Real
  by RELAT_1:def 19;
  vars p2 c= RZ.:(vars p1) by Th84;
  then vars p2 c= Segm 10 by A23,A22;
  then consider p3 be Polynomial of 10,F_Real such that
A40:vars p3 c= 10 & rng p3 c= rng p2 and
  for b be bag of 10+7 holds b|10 in Support p3 &
  (for i st i >= 10 holds b.i=0) iff b in Support p2 and
  for b be bag of 10+7 st b in Support p2 holds
  p3.(b|10) = p2.b and
A41:for x being Function of (10+7), F_Real,
    y being Function of 10, F_Real st x|10=y
    holds eval(p2,x) = eval(p3,y) by Th83;
A42: for xk be Nat st xk > 0 holds
  xk+1 is prime iff ex x being INT -valued Function of 10, F_Real st
  x.0 is Nat &x.1 = xk & x.2 is positive Nat &
  x.3 is positive Nat & x.4 is positive Nat & x.5 is positive Nat &
  x.6 is positive Nat & x.7 is Nat & x.8 is Nat & x.9 is Nat &
  eval(p3,x) = 0.F_Real
  proof
    let xk be Nat such that
A43:  xk > 0;
    thus xk+1 is prime implies
    ex x being INT -valued Function of 10, F_Real st
    x.0 is Nat & x.1 = xk & x.2 is positive Nat & x.3 is positive Nat &
    x.4 is positive Nat & x.5 is positive Nat & x.6 is positive Nat &
    x.7 is Nat & x.8 is Nat & x.9 is Nat & eval(p3,x) = 0.F_Real
    proof
      assume xk+1 is prime;
      then consider x be INT -valued Function of 17, F_Real such that
A44:  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 and
A45:  eval(p1,x) = 0.F_Real by A2,A43;
      reconsider xRZ=x*(RZ") as INT -valued Function of 10+7, F_Real;
A46:  dom xRZ = 17 & Segm 10 c= Segm 17 by NAT_1:39,PARTFUN1:def 2;
      then dom (xRZ|10)=10 & rng (xRZ|10)c= rng xRZ c= the carrier of F_Real
      by RELAT_1:62,70;
      then reconsider y = xRZ|10 as INT -valued Function of 10,F_Real
      by FUNCT_2:2;
      take y;
      0 in dom xRZ by A46,NAT_1:44;
      then
A47:  (RZ".0) in dom x by FUNCT_1:11;
      0 in Segm 10 by NAT_1:44;
      then y.0 = xRZ.0 by FUNCT_1:49
      .= x .(RZ".0) by A46,NAT_1:44,FUNCT_1:12
      .= x/.0 by A47,A34,PARTFUN1:def 6;
      hence y.0 is Nat by A44;
A48:  for i st 1<=i<=9 holds y.i = x/.(i+7)
      proof
        let i such that
A49:    1<=i<=9;
A50:    i < 17 by A49,XXREAL_0:2;
        then i in dom xRZ by A46,NAT_1:44;
        then
A51:    i+7 = (RZ".i) in dom x by A49,A35,FUNCT_1:11;
        i<9+1 by A49,NAT_1:13;
        then i in Segm 10 by NAT_1:44;
        hence y.i = xRZ.i by FUNCT_1:49
        .= x.(RZ".i) by A50,A46,NAT_1:44,FUNCT_1:12
        .= x/.(i+7) by A51,PARTFUN1:def 6;
      end;
      then y.1 = x/.(1+7);
      hence y.1 = xk by A44;
A52:  y.2 = x/.(2+7) & y.3 = x/.(3+7) &
      y.4 = x/.(4+7) & y.5 = x/.(5+7) &
      y.6 = x/.(6+7) & y.7 = x/.(7+7) &
      y.8 = x/.(8+7) & y.9 = x/.(9+7) by A48;
      0.F_Real = eval(p2,xRZ) by HILB10_2:25,A45
      .= eval(p3,y) by A41;
      hence thesis by A52,A44;
    end;
    given x being INT -valued Function of 10, F_Real such that
A53: x.0 is Nat & x.1 = xk & x.2 is positive Nat &
    x.3 is positive Nat & x.4 is positive Nat & x.5 is positive Nat &
    x.6 is positive Nat & x.7 is Nat & x.8 is Nat & x.9 is Nat &
    eval(p3,x) = 0.F_Real;
    reconsider X7 = 7-->0 as XFinSequence of REAL;
    reconsider xx7=@x^X7 as 10+7 -element XFinSequence of REAL;
    reconsider XX7= @xx7 as INT -valued Function of 17, F_Real;
A54: len (@x) = 10 by CARD_1:def 7;
A55: (@x^X7)|10 = @x|10 by A54,AFINSQ_1:58
    .= @x;
A56:dom XX7 = 17 by FUNCT_2:def 1;
    XX7*RZ*RZ" = XX7*(RZ*RZ") by RELAT_1:36
    .=XX7*(id 17) by A31,FUNCT_1:39
    .= XX7 by A56,RELAT_1:51;
    then
A57: eval(P2,XX7) = eval(p1,XX7*RZ) by HILB10_2:25;
    ex y being INT -valued Function of 17, F_Real st
    y/.8 = xk & y/.9 is positive Nat & y/.10 is positive Nat &
    y/.11 is positive Nat & y/.12 is positive Nat & y/.13 is positive Nat &
    y/.14 is Nat & y/.15 is Nat & y/.16 is Nat & y/.0 is Nat &
    eval(p1,y) = 0.F_Real
    proof
      take y = XX7*RZ;
A58:  dom y = 17 by FUNCT_2:def 1;
A59:  for i st 1<=i<=9 holds y/.(7+i) = x.i
      proof
        let i such that
A60:    1<=i<=9;
        i+7 <= 9+7 by A60,XREAL_1:6;
        then i+7 < 16+1 by NAT_1:13;
        then
A61:    i+7 in Segm 17 by NAT_1:44;
        then
A62:    y/.(7+i) = y.(7+i) by A58,PARTFUN1:def 6
        .= XX7.(RZ.(7+i)) by A61,A58,FUNCT_1:12;
A63:    i = RZ.(7+i) in dom XX7 by A60,A35,A61,A58,FUNCT_1:11;
        i <9+1 by A60,NAT_1:13;
        then i in Segm 10 by NAT_1:44;
        hence thesis by A63,A62,A54,AFINSQ_1:def 3;
      end;
A64:  0 in Segm 17 by NAT_1:44;
A65:  y/.0 = y.0 by A33,A58,PARTFUN1:def 6
      .= XX7.(RZ.0) by A64,A58,FUNCT_1:12;
A66:  0 in Segm 10 by NAT_1:44;
      x.1 = y/.(1+7)& x.2 = y/.(2+7) & x.3 = y/.(3+7) &
      x.4 = y/.(4+7) & x.5 = y/.(5+7) & x.6 = y/.(6+7) & x.7 = y/.(7+7) &
      x.8 = y/.(8+7) & x.9 = y/.(9+7) by A59;
      hence thesis by A53,A66,A65,A32,A54,AFINSQ_1:def 3,A55,A41,A57;
    end;
    hence thesis by A2,A43;
  end;
  rng p3 c= INT by A39,A40;
  then reconsider p3 as INT -valued Polynomial of 10,F_Real by RELAT_1:def 19;
  set EB = EmptyBag 10,O = 1_(10,F_Real);
  set P2 = Monom(1.F_Real,EB +*(2,1)) + O;
  set P3 = Monom(1.F_Real,EB +*(3,1)) + O;
  set P4 = Monom(1.F_Real,EB +*(4,1)) + O;
  set P5 = Monom(1.F_Real,EB +*(5,1)) + O;
  set P6 = Monom(1.F_Real,EB +*(6,1)) + O;
  reconsider Z2 = Subst(p3,2,P2) as INT -valued Polynomial of 10,F_Real;
  reconsider Z3 = Subst(Z2,3,P3) as INT -valued Polynomial of 10,F_Real;
  reconsider Z4 = Subst(Z3,4,P4) as INT -valued Polynomial of 10,F_Real;
  reconsider Z5 = Subst(Z4,5,P5) as INT -valued Polynomial of 10,F_Real;
  reconsider Z6 = Subst(Z5,6,P6) as INT -valued Polynomial of 10,F_Real;
  take Z6;
  let k be positive Nat;
A67:vars O = {} by Th38;
A68: 6 in Segm 10 by NAT_1:44;
A69: 5 in Segm 10 by NAT_1:44;
A70: vars Monom(1.F_Real,EB +*(5,1)) \/vars O c= {5}\/{} by A67,Th48;
  vars P5 c= vars Monom(1.F_Real,EB +*(5,1)) \/vars O by Th41;
  then vars P5 c= {5} by A70;
  then not 6 in vars P5 by TARSKI:def 1;
  then
A71: 6 in 10\vars P5 by A68,XBOOLE_0:def 5;
A72: 4 in Segm 10 by NAT_1:44;
A73: vars Monom(1.F_Real,EB +*(4,1)) \/vars O c= {4}\/{} by A67,Th48;
  vars P4 c= vars Monom(1.F_Real,EB +*(4,1)) \/vars O by Th41;
  then vars P4 c= {4} by A73;
  then not 5 in vars P4 & not 6 in vars P4 by TARSKI:def 1;
  then
A74: 5 in 10\vars P4 & 6 in 10\vars P4 by A69,A68,XBOOLE_0:def 5;
A75: 3 in Segm 10 by NAT_1:44;
A76: vars Monom(1.F_Real,EB +*(3,1)) \/vars O c= {3}\/{} by A67,Th48;
  vars P3 c= vars Monom(1.F_Real,EB +*(3,1)) \/vars O by Th41;
  then vars P3 c= {3} by A76;
  then not 4 in vars P3 & not 5 in vars P3 & not 6 in vars P3 by TARSKI:def 1;
  then
A77: 4 in 10\vars P3 & 5 in 10\vars P3 & 6 in 10\vars P3
    by A72,A69,A68,XBOOLE_0:def 5;
A78: 2 in Segm 10 by NAT_1:44;
A79: vars Monom(1.F_Real,EB +*(2,1)) \/vars O c= {2}\/{} by A67,Th48;
  vars P2 c= vars Monom(1.F_Real,EB +*(2,1)) \/vars O by Th41;
  then vars P2 c= {2} by A79;
  then not 3 in vars P2 & not 4 in vars P2 & not 5 in vars P2 &
  not 6 in vars P2 by TARSKI:def 1;
  then
A80: 3 in 10\vars P2 & 4 in 10\vars P2 & 5 in 10\vars P2 &
    6 in 10\vars P2 by A75,A72,A69,A68,XBOOLE_0:def 5;
  thus k+1 is prime
  implies ex v being natural-valued Function of 10,F_Real st
  v.1 = k & eval(Z6,v) = 0.F_Real
  proof
    assume k+1 is prime;
    then consider x be INT -valued Function of 10, F_Real such that
A81:x.0 is Nat &
    x.1 = k & x.2 is positive Nat & x.3 is positive Nat &
    x.4 is positive Nat & x.5 is positive Nat & x.6 is positive Nat &
    x.7 is Nat & x.8 is Nat & x.9 is Nat and
A82:eval(p3,x) = 0.F_Real by A42;
    reconsider x0 =x.0,x1=x.1,x7=x.7,x8=x.8,x9=x.9 as Element of NAT
    by A81,ORDINAL1:def 12;
    reconsider x2=x.2-1,x3=x.3-1,x4=x.4-1,x5=x.5-1,x6=x.6-1
    as Element of NAT by A81,NAT_1:20;
    set v = <%x0%>^<%x1%>^<%x2%>^<%x3%>^<%x4%>^<%x5%>^<%x6%>^<%x7%>^<%x8%>;
    set V=v^<%x9%>;
    reconsider VV=@V as natural-valued Function of 10,F_Real;
    take VV;
A83: len v = 9 by AFINSQ_1:50;
A84: len V =Segm 10 by CARD_1:def 7;
A85: 2 in Segm 9 & 3 in Segm 9 & 4 in Segm 9 & 5 in Segm 9 & 6 in Segm 9
    by NAT_1:44;
    set VV6 = VV+*(6,eval(P6,VV));
A86: eval(Z6,VV) =eval(Z5,VV6) by A68,Th37;
A87: eval(P6,VV) = eval(Monom(1.F_Real,EB +*(6,1)),VV) +
    eval(O,VV) by POLYNOM2:23
    .= 1.F_Real * (VV/.6) + eval(O,VV) by A68,Th27
    .= VV/.6+1 by POLYNOM2:21
    .= VV.6 +1 by A84,NAT_1:44,PARTFUN1:def 6
    .= v.6+1 by A85,A83,AFINSQ_1:def 3
    .= x6+1 by AFINSQ_1:50;
    set VV5 = VV6+*(5,eval(P5,VV));
    eval(P5,VV6) = eval(P5,VV) by Th53,A71;
    then
A88: eval(Z5,VV6) =eval(Z4,VV5) by A69,Th37;
A89: eval(P5,VV) = eval(Monom(1.F_Real,EB +*(5,1)),VV) +
    eval(O,VV) by POLYNOM2:23
    .= 1.F_Real * (VV/.5) + eval(O,VV) by A69,Th27
    .= VV/.5+1 by POLYNOM2:21
    .= VV.5 +1 by A84,NAT_1:44,PARTFUN1:def 6
    .= v.5+1 by A85,A83,AFINSQ_1:def 3
    .= x5+1 by AFINSQ_1:50;
    set VV4 = VV5+*(4,eval(P4,VV));
    eval(P4,VV5) = eval(P4,VV6) by Th53,A74
    .=eval(P4,VV) by Th53,A74;
    then
A90: eval(Z4,VV5) =eval(Z3,VV4) by A72,Th37;
A91: eval(P4,VV) = eval(Monom(1.F_Real,EB +*(4,1)),VV) +
    eval(O,VV) by POLYNOM2:23
    .= 1.F_Real * (VV/.4) + eval(O,VV) by A72,Th27
    .= VV/.4+1 by POLYNOM2:21
    .= VV.4 +1 by A84,NAT_1:44,PARTFUN1:def 6
    .= v.4+1 by A85,A83,AFINSQ_1:def 3
    .= x4+1 by AFINSQ_1:50;
    set VV3 = VV4+*(3,eval(P3,VV));
    eval(P3,VV4) = eval(P3,VV5) by Th53,A77
    .= eval(P3,VV6) by Th53,A77
    .=eval(P3,VV) by Th53,A77;
    then
A92: eval(Z3,VV4) =eval(Z2,VV3) by A75,Th37;
A93: eval(P3,VV) = eval(Monom(1.F_Real,EB +*(3,1)),VV) +
    eval(O,VV) by POLYNOM2:23
    .= 1.F_Real * (VV/.3) + eval(O,VV) by A75,Th27
    .= VV/.3+1 by POLYNOM2:21
    .= VV.3 +1 by A84,NAT_1:44,PARTFUN1:def 6
    .= v.3+1 by A85,A83,AFINSQ_1:def 3
    .= x3+1 by AFINSQ_1:50;
    set VV2 = VV3+*(2,eval(P2,VV));
A94: eval(P2,VV3) = eval(P2,VV4) by Th53,A80
    .= eval(P2,VV5) by Th53,A80
    .= eval(P2,VV6) by Th53,A80
    .=eval(P2,VV) by Th53,A80;
A95: eval(P2,VV) = eval(Monom(1.F_Real,EB +*(2,1)),VV) +
    eval(O,VV) by POLYNOM2:23
    .= 1.F_Real * (VV/.2) + eval(O,VV) by A78,Th27
    .= VV/.2+1 by POLYNOM2:21
    .= VV.2 +1 by A84,NAT_1:44,PARTFUN1:def 6
    .= v.2+1 by A85,A83,AFINSQ_1:def 3
    .= x2+1 by AFINSQ_1:50;
A96:Segm 10 =dom VV2 by FUNCT_2:def 1;
    1 in Segm 9 by NAT_1:44;
    then VV.1 = v.1 by A83,AFINSQ_1:def 3
    .= x1 by AFINSQ_1:50;
    hence VV.1 = k by A81;
    for y be object st y in dom VV2 holds x.y = VV2.y
    proof
      let y be object;
      assume
A97:  y in dom VV2;
      then reconsider y as Nat by A84;
      y <9+1 by A97,A84,NAT_1:44;
      then y <= 9 by NAT_1:13;
      then y =0 or...or y=9;
      then per cases;
      suppose
A98:    y=0 or y = 1 or y=7 or y=8 or y =9;
        then
A99:   VV2.y = VV3.y by FUNCT_7:32
        .= VV4.y by A98,FUNCT_7:32
        .= VV5.y by A98,FUNCT_7:32
        .= VV6.y by A98,FUNCT_7:32
        .= VV.y by A98,FUNCT_7:32;
        VV.y = x.y
        proof
          per cases;
          suppose y =9;
            hence thesis by A83,AFINSQ_1:36;
          end;
          suppose
A100:       y <>9;
            then y in Segm 9 by A98,NAT_1:44;
            hence VV.y = v.y by A83,AFINSQ_1:def 3
            .= x.y by A98,A100,AFINSQ_1:50;
          end;
        end;
        hence thesis by A99;
      end;
      suppose
A101:   y =2 or...or y=6;
A102:   dom VV6 = 10 & dom VV5 = 10 by FUNCT_2:def 1;
A103:   dom VV4 = 10 & dom VV3 = 10 by FUNCT_2:def 1;
        per cases by A101;
        suppose
A104:     y=6;
          then VV2.y = VV3.y by FUNCT_7:32
          .= VV4.y by A104,FUNCT_7:32
          .= VV5.y by A104,FUNCT_7:32
          .= VV6.y by A104,FUNCT_7:32
          .= eval(P6,VV) by A104,A84,NAT_1:44,FUNCT_7:31;
          hence thesis by A87,A104;
        end;
        suppose
A105:     y=5;
          then VV2.y = VV3.y by FUNCT_7:32
          .= VV4.y by A105,FUNCT_7:32
          .= VV5.y by A105,FUNCT_7:32
          .= eval(P5,VV) by A102,A105,A69,FUNCT_7:31;
          hence thesis by A89,A105;
        end;
        suppose
A106:     y=4;
          then VV2.y = VV3.y by FUNCT_7:32
          .= VV4.y by A106,FUNCT_7:32
          .= eval(P4,VV) by A102,A106,A72,FUNCT_7:31;
          hence thesis by A91,A106;
        end;
        suppose
A107:     y=3;
          then VV2.y = VV3.y by FUNCT_7:32
          .= eval(P3,VV) by A103,A107,A75,FUNCT_7:31;
          hence thesis by A93,A107;
        end;
        suppose y=2;
          hence thesis by A95,A103,A78,FUNCT_7:31;
        end;
      end;
    end;
    then x = VV2 by A96;
    hence thesis by A82,A88,A86,A90,A92,A94,A78,Th37;
  end;
  given VV be natural-valued Function of 10,F_Real such that
A108:VV.1 = k & eval(Z6,VV) = 0.F_Real;
A109: dom VV =Segm 10 by FUNCT_2:def 1;
  set VV6 = VV+*(6,eval(P6,VV));
A110: eval(Z6,VV) =eval(Z5,VV6) by A68,Th37;
A111: eval(P6,VV) = eval(Monom(1.F_Real,EB +*(6,1)),VV) +
  eval(O,VV) by POLYNOM2:23
  .= 1.F_Real * (VV/.6) + eval(O,VV) by A68,Th27
  .= VV/.6+1 by POLYNOM2:21
  .= VV.6 +1 by A109,NAT_1:44,PARTFUN1:def 6;
  then reconsider e6=eval(P6,VV) as Element of INT by INT_1:def 1;
  set VV5 = VV6+*(5,eval(P5,VV));
  eval(P5,VV6) = eval(P5,VV) by Th53,A71;
  then
A112: eval(Z5,VV6) =eval(Z4,VV5) by A69,Th37;
A113: eval(P5,VV) = eval(Monom(1.F_Real,EB +*(5,1)),VV) +
  eval(O,VV) by POLYNOM2:23
  .= 1.F_Real * (VV/.5) + eval(O,VV) by A69,Th27
  .= VV/.5+1 by POLYNOM2:21
  .= VV.5 +1 by A109,NAT_1:44,PARTFUN1:def 6;
  then reconsider e5=eval(P5,VV) as Element of INT by INT_1:def 1;
  set VV4 = VV5+*(4,eval(P4,VV));
  eval(P4,VV5) = eval(P4,VV6) by Th53,A74
  .=eval(P4,VV) by Th53,A74;
  then
A114: eval(Z4,VV5) =eval(Z3,VV4) by A72,Th37;
A115: eval(P4,VV) = eval(Monom(1.F_Real,EB +*(4,1)),VV) +
  eval(O,VV) by POLYNOM2:23
  .= 1.F_Real * (VV/.4) + eval(O,VV) by A72,Th27
  .= VV/.4+1 by POLYNOM2:21
  .= VV.4 +1 by A109,NAT_1:44,PARTFUN1:def 6;
  then reconsider e4=eval(P4,VV) as Element of INT by INT_1:def 1;
  set VV3 = VV4+*(3,eval(P3,VV));
  eval(P3,VV4) = eval(P3,VV5) by Th53,A77
  .= eval(P3,VV6) by Th53,A77
  .=eval(P3,VV) by Th53,A77;
  then
A116: eval(Z3,VV4) =eval(Z2,VV3) by A75,Th37;
A117: eval(P3,VV) = eval(Monom(1.F_Real,EB +*(3,1)),VV) +
  eval(O,VV) by POLYNOM2:23
  .= 1.F_Real * (VV/.3) + eval(O,VV) by A75,Th27
  .= VV/.3+1 by POLYNOM2:21
  .= VV.3 +1 by A109,NAT_1:44,PARTFUN1:def 6;
  then reconsider e3=eval(P3,VV) as Element of INT by INT_1:def 1;
  set VV2 = VV3+*(2,eval(P2,VV));
  eval(P2,VV3) = eval(P2,VV4) by Th53,A80
  .= eval(P2,VV5) by Th53,A80
  .= eval(P2,VV6) by Th53,A80
  .=eval(P2,VV) by Th53,A80;
  then
A118: eval(Z2,VV3) =eval(p3,VV2) by A78,Th37;
A119: eval(P2,VV) = eval(Monom(1.F_Real,EB +*(2,1)),VV) +
  eval(O,VV) by POLYNOM2:23
  .= 1.F_Real * (VV/.2) + eval(O,VV) by A78,Th27
  .= VV/.2+1 by POLYNOM2:21
  .= VV.2 +1 by A109,NAT_1:44,PARTFUN1:def 6;
  then reconsider e2=eval(P2,VV) as Element of INT by INT_1:def 1;
  VV2= VV+*(6,e6)+*(5,e5)+*(4,e4)+*(3,e3)+*(2,e2);
  then reconsider VV2 as INT -valued Function of 10, F_Real;
A120:for y be Nat st y=0 or y = 1 or y=7 or y=8 or y =9
  holds VV2.y = VV.y
  proof
    let y be Nat;
    assume
A121: y=0 or y = 1 or y=7 or y=8 or y =9;
    hence VV2.y = VV3.y by FUNCT_7:32
    .= VV4.y by A121,FUNCT_7:32
    .= VV5.y by A121,FUNCT_7:32
    .= VV6.y by A121,FUNCT_7:32
    .= VV.y by A121,FUNCT_7:32;
  end;
  VV.0 is Nat & VV.7 is Nat &VV.8 is Nat & VV.9 is Nat;
  then
A122:VV2.0 is Nat & VV2.7 is Nat & VV2.8 is Nat & VV2.9 is Nat by A120;
A123:VV2.1 = k by A120,A108;
A124:VV2.6 = VV3.6 by FUNCT_7:32
  .= VV4.6 by FUNCT_7:32
  .= VV5.6 by FUNCT_7:32
  .= VV6.6 by FUNCT_7:32
  .= eval(P6,VV) by A109,NAT_1:44,FUNCT_7:31;
A125:dom VV6 = 10 & dom VV5 = 10 by FUNCT_2:def 1;
A126:dom VV4 = 10 & dom VV3 = 10 by FUNCT_2:def 1;
A127:VV2.5 = VV3.5 by FUNCT_7:32
  .= VV4.5 by FUNCT_7:32
  .= VV5.5 by FUNCT_7:32
  .= eval(P5,VV) by A125,A69,FUNCT_7:31;
A128: VV2.4 = VV3.4 by FUNCT_7:32
  .= VV4.4 by FUNCT_7:32
  .= eval(P4,VV) by A125,A72,FUNCT_7:31;
A129: VV2.3 = VV3.3 by FUNCT_7:32
  .= eval(P3,VV) by A126,A75,FUNCT_7:31;
  VV2.2 = VV.2+1 by A119,A126,A78,FUNCT_7:31;
  hence thesis by A42,A122,A123,A124,A127,A113,A128,A129,A117,A108,A115,
    A118,A116,A114,A112,A110,A111;
end;
