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 Th76:
  for M be Jpolynom of 4,F_Complex
    ex K2 be INT -valued Polynomial of 6,F_Real st
      (for f be Function of 6,F_Real st f.5 <>0
       holds eval(K2,f) = (power F_Real).(f/.5,8)*
         eval(Jsqrt M, @ <%- (f.0) + (f.4/f.5),f.1,f.2,f.3%>)) &
            for f be INT -valued Function of 6,F_Real st f.5 <>0 &
                  eval(K2,f) = 0 holds f.5 divides f.4
proof
  let M be Jpolynom of 4,F_Complex;
  set p = Jsqrt M; set R=F_Real;
  consider q be Polynomial of 4+2,R such that
A1:rng q c= rng p \/ {0.R} and
A2:for b be bag of 4+2 holds b in Support q iff b|4 in Support p &
                 for i st i >= 4 holds b.i=0 and
A3:for b be bag of 4+2 st b in Support q holds
           q.b = p.(b|4) and
A4:for x being Function of 4, R,
         y being Function of (4+2), R st y|4=x
       holds eval(p,x) = eval(q,y) by Th75;
  rng q c= INT by A1,INT_1:def 2;
  then reconsider q as INT -valued Polynomial of 6,R by RELAT_1:def 19;
  set Yb = (EmptyBag 6) +*(0,1),Y=Monom(-1.R,Yb),
  Zb = (EmptyBag 6) +*(4,1),Z=Monom(1.R,Zb), YZ=Y+Z;
  set Sup=SgmX(BagOrder 6, Support q);
  BagOrder 6  linearly_orders Support q by POLYNOM2:18;
  then
A5: rng Sup = Support q by PRE_POLY:def 2;
  consider S be FinSequence of Polynom-Ring (6, R) such that
A6: Subst(q,0,YZ) = Sum S & len Sup = len S and
A7: for i st i in dom S holds
  S.i = q.(Sup/.i) * Subst(Sup/.i,0, YZ) by Def4;
A8:dom S = dom Sup by A6,FINSEQ_3:29;
  4-'1 = 4-1 by XREAL_1:233;
  then
A9: 2|^(4-'1) = 2|^(2+1)
  .= 2*(2|^(1+1)) by NEWTON:6
  .= 2*(2 * 2|^1) by NEWTON:6
  .= 8;
  set E6=EmptyBag 6;
  set MAX = (EmptyBag 4) +* (0,8), MAX8 = E6+* (0,8);
A10: M.MAX = 1.F_Complex by Def10,A9;
A11: MAX in Bags 4 = dom p by FUNCT_2:def 1,PRE_POLY:def 12;
  2(#)MAX +* (0,MAX.0) = MAX
  proof
A12: dom (2(#)MAX) = 4 = dom (2(#)MAX +* (0,MAX.0)) by PARTFUN1:def 2;
    for x st x in 4 holds MAX.x = (2(#)MAX +* (0,MAX.0)).x
    proof
      let x such that
A13:  x in 4;
      per cases;
      suppose x=0;
        hence thesis by A12,A13,FUNCT_7:31;
      end;
      suppose
A14:    x<>0;
        then
A15:    MAX.x = (EmptyBag 4).x by FUNCT_7:32
        .= 0;
        2*(MAX.x) = (2(#)MAX).x by A12,A13,VALUED_1:def 5
        .= (2(#)MAX +* (0,MAX.0)).x by A14,FUNCT_7:32;
        hence thesis by A15;
      end;
    end;
    hence thesis by A12,PARTFUN1:def 2;
  end;
  then (JsqrtSeries M).MAX = M.MAX by Def12;
  then
A16: p.MAX = 1.F_Complex by Def13,A10
  .= 1.R by COMPLEX1:def 4,COMPLFLD:def 1;
  then p.MAX<>0.R;
  then
A17: MAX in Support p by A11,POLYNOM1:def 3;
A18: dom E6 = 6 & dom (EmptyBag 4) = 4 by PARTFUN1:def 2;
A19: 4 in Segm 6 & 0 in Segm 6 & 5 in Segm 6 by NAT_1:44;
A20: MAX8.0 = 8 by A18,A19,FUNCT_7:31;
A21: Segm 4 c= Segm 6 by NAT_1:39;
A22: dom (MAX8|4) = 4 by PARTFUN1:def 2,A21;
  for x st x in 4 holds (MAX8|4).x = MAX.x
  proof
    let x such that
A23: x in 4;
A24: (MAX8|4).x = MAX8.x by A23,FUNCT_1:49;
    per cases;
    suppose x =0;
      hence thesis by A18,A20,A24,A23,FUNCT_7:31;
    end;
    suppose
A25:  x<>0;
      then (MAX8|4).x = E6.x =(EmptyBag 4).x by A24,FUNCT_7:32;
      hence thesis by A25,FUNCT_7:32;
      end;
    end;
    then
A26: MAX8|4 = MAX by PARTFUN1:def 2,A22;
A27:for i st i >= 4 holds MAX8.i=0
    proof
      let i such that
A28:  i >=4;
      MAX8.i = E6.i by A28,FUNCT_7:32;
      hence thesis;
    end;
    MAX8 in Support q by A2,A26,A17,A27;
    then consider I be object such that
A29:I in dom Sup & Sup.I = MAX8 by A5,FUNCT_1:def 3;
A30: q.MAX8 = 1.R by A16,A17,A26,A27,A3,A2;
    reconsider I as Nat by A29;
A31: Sup/.I = Sup.I & S/.I = S.I by A8,A29,PARTFUN1:def 6;
    defpred P[Nat] means (YZ `^ $1).(E6 +* (4,$1)) = 1.R;
A32: YZ `^ 0 = 1_(6,R) by Th28;
    E6.4 = 0;
    then E6 +* (4,0) = E6 by FUNCT_7:35;
    then
A33: P[0] by A32,POLYNOM1:25;
A34: P[i] implies P[i+1]
    proof
      assume
A35:  P[i];
A36:  (YZ `^ (i+1)) = (YZ `^ i) *' YZ by Th29
      .= Y *' (YZ `^ i) + Z *'(YZ `^ i) by POLYNOM1:26;
A37:  (E6 +* (4,i+1)) = (E6 +* (4,i))+ Zb by Th22;
      then
A38:  Zb divides (E6 +* (4,i+1)) by PRE_POLY:50;
      (E6 +* (4,i+1)) -' Zb = (E6 +* (4,i)) by A37,PRE_POLY:48;
      then
A39:  (Zb *' (YZ `^ i)).(E6 +* (4,i+1)) = (YZ `^ i).(E6 +* (4,i)) = 1.R
      by A35,A38,POLYRED:def 1;
      Z *' (YZ `^ i) = (1.R) * (Zb *' (YZ `^ i)) by POLYRED:22;
      then
A40:  Z *' (YZ `^ i).(E6 +* (4,i+1)) = (1.R) *1.R
      by A39, POLYNOM7:def 9;
A41:  (E6 +* (4,i+1)).0 = E6.0 by FUNCT_7:32
      .=0;
      Yb.0 = 1 by A18,A19,FUNCT_7:31;
      then not Yb divides (E6 +* (4,i+1)) by A41,PRE_POLY:def 11;
      then
A42:  (Yb *' (YZ `^ i)).(E6 +* (4,i+1)) = 0.R by POLYRED:def 1;
      Y *' (YZ `^ i) = (-1.R) * (Yb *' (YZ `^ i)) by POLYRED:22;
      then Y *' (YZ `^ i).(E6 +* (4,i+1)) = (-1.R)*0.R by A42, POLYNOM7:def 9;
      then (YZ `^ (i+1)).(E6 +* (4,i+1)) = 0.R+1.R by A36,A40,POLYNOM1:15;
      hence thesis;
    end;
A43: P[i] from NAT_1:sch 2(A33,A34);
    set Z8 = E6+* (4,8);
    E6 +* (0,8) +*(0,0) = E6 +* (0,0) & E6.0 = 0 by FUNCT_7:34;
    then Subst(MAX8,0, Z8) = E6 + E6+*(4,8) by FUNCT_7:35
    .= Z8 by PRE_POLY:53;
    then
A44:Subst(Sup/.I,0, YZ).Z8 = (YZ `^ (MAX8.0)).Z8 by Th33,A29,A31
    .= 1_R by A20,A43;
A45: for i st i in dom S
    for b be bag of 6 st b in Support (q.(Sup/.i) * Subst(Sup/.i,0, YZ)) &
    b.4 >= 8 holds i= I & b = Z8
    proof
  let i such that
A46: i in dom S; set Supi = Sup/.i;
  let b be bag of 6 such that
A47: b in Support (q.(Supi) * Subst(Supi,0, YZ))
  & b.4 >= 8;
A48: Supi =Sup.i in Support q by A46,A8,A5,PARTFUN1:def 6,FUNCT_1:def 3;
  then
A49: q.(Sup/.i) <>0.R by POLYNOM1:def 3;
A50:q.(Sup/.i) = p.((Sup/.i)|4) by A48,A3;
A51:4-'0 =4-0;
  (Supi)|4 =(0,4)-cut Supi by HILB10_2:3;
  then reconsider Supi4 = (Supi)|4 as bag of 4 by A51;
A52: ( 2(#) (Supi4) +* (0,(Supi4.0))) in Bags 4 = dom M
    by PRE_POLY:def 12,FUNCT_2:def 1;
  set 2Supi4 =( 2(#) (Supi4) +* (0,(Supi4.0)));
  p = JsqrtSeries M by Def13;
  then p.Supi4 = M.( 2(#) (Supi4) +* (0,(Supi4.0))) by Def12;
  then (2(#) (Supi4) +* (0,(Supi4.0))) in Support M
  by A52,A49,A50,COMPLFLD:7,POLYNOM1:def 3;
  then
A53: degree ( 2(#) (Supi4) +* (0,(Supi4.0))) = 2|^(4-'1) by Def10;
  consider g be FinSequence of REAL such that
A54: Sum 2Supi4 = Sum g & g = 2Supi4 * canFS(support 2Supi4)
  by UPROOTS:def 3;
A55: for i st i in dom g holds 0 <= g.i by A54;
A56: 2Supi4.0 <= 8
  proof
A57: rng canFS(support 2Supi4) = support 2Supi4 by FUNCT_2:def 3;
    per cases;
    suppose
A58:  0 in support 2Supi4;
      then
A59:  0 in rng canFS(support 2Supi4) c= dom 2Supi4 by PRE_POLY:37,A57;
      consider d be object such that
A60:  d in dom canFS(support 2Supi4) &
      (canFS(support 2Supi4)).d = 0 by A58,A57,FUNCT_1:def 3;
      reconsider d as Nat by A60;
      d in dom g & g.d = 2Supi4.0 by A59,A60,A54,FUNCT_1:12,11;
      hence thesis by A54,A53,A9,A55,MATRPROB:5;
    end;
    suppose not 0 in support 2Supi4;
      hence thesis by PRE_POLY:def 7;
    end;
  end;
A61: 0 in Segm 4 = dom (2(#) (Supi4)) by NAT_1:44,PARTFUN1:def 2;
  then
A62: 2Supi4.0 = Supi4.0 by FUNCT_7:31
  .= Supi.0 by A61,FUNCT_1:49;
  defpred Q[Nat] means for b be bag of 6 st (YZ `^ $1).b <> 0.R holds
    b.4 <= $1;
A63:Q[0]
  proof
    let b be bag of 6 such that A64:(YZ `^ 0).b <> 0.R;
    b = E6 by A64,POLYNOM1:25,A32;
    hence thesis;
  end;
A65:for i st Q[i] holds Q[i+1]
  proof
    let i;
    assume
A66: Q[i];
    let b be bag of 6 such that
A67: (YZ `^ (i+1)).b <> 0.R;
    (YZ `^ (i+1)) = (YZ `^ i) *' YZ by Th29
    .= Y *' (YZ `^ i) + Z *'(YZ `^ i) by POLYNOM1:26;
    then (YZ `^ (i+1)).b = (Y *' (YZ `^ i)).b + (Z *'(YZ `^ i)).b
      by POLYNOM1:15;
    then per cases by A67;
    suppose
A68:  (Z *' (YZ `^ i)).b<>0.R;
      Z *' (YZ `^ i) = (1.R) * (Zb *' (YZ `^ i)) by POLYRED:22;
      then
A69:  (Z *' (YZ `^ i)).b = (1.R) * ((Zb *' (YZ `^ i)).b) by POLYNOM7:def 9;
      then
A70:  Zb divides b by A68,POLYRED:def 1;
      then (Zb *' (YZ `^ i)).b = (YZ `^ i).(b-'Zb) by POLYRED:def 1;
      then (b-'Zb).4 <= i by A66,A69,A68;
      then
A71:  (b.4)-'(Zb.4) <= i by PRE_POLY:def 6;
      Zb.4 = 1 by A18,A19,FUNCT_7:31;
      then b.4 -1 <= i by A71,A70,PRE_POLY:def 11,XREAL_1:233;
      then b.4 -1 + 1 <= i+1 by XREAL_1:6;
      hence thesis;
    end;
    suppose
A72:  (Y *'(YZ `^ i)).b<>0.R;
      Y *' (YZ `^ i) = (-1.R) * (Yb *' (YZ `^ i)) by POLYRED:22;
      then Y *' (YZ `^ i).b = (-1.R) *( (Yb *' (YZ `^ i)).b) by POLYNOM7:def 9;
      then
A73:  (Yb *' (YZ `^ i)).b<>0.R by A72;
      then
A74:  Yb divides b by POLYRED:def 1;
      then (Yb *' (YZ `^ i)).b = (YZ `^ i).(b-'Yb) by POLYRED:def 1;
      then (b-'Yb).4 <= i by A66,A73;
      then
A75:  (b.4)-'(Yb.4) <= i by PRE_POLY:def 6;
A76:  Yb.4 = E6.4 by FUNCT_7:32;
      (b.4)-'(Yb.4) = (b.4)- (Yb.4) by XREAL_1:233,A74,PRE_POLY:def 11;
      hence thesis by A75,A76,NAT_1:13;
    end;
  end;
A77: Q[j] from NAT_1:sch 2(A63,A65);
  (q.(Supi) * Subst(Supi,0, YZ)).b <>0.R by A47,POLYNOM1:def 3;
  then (q.(Supi)) * (Subst(Supi,0, YZ).b) <>0.R by POLYNOM7:def 9;
  then
A78:  Subst(Supi,0, YZ).b <>0.R;
  then consider s be bag of 6 such that
A79:  b = Subst(Supi,0,s) by Def3;
  (Supi+*(0,0)).4 = Supi.4=0 by A48,A2,FUNCT_7:32;
  then
A80: (Supi+*(0,0) + s).4 = 0+(s.4) by PRE_POLY:def 5;
  (YZ `^ (Supi.0) ).s<>0.R by A78,A79,Th33;
  then
A81: s.4 <= Supi.0 by A77;
  then
A82: 8 <= Supi.0 by A80,A79,A47,XXREAL_0:2;
  then
A83: Supi.0 = 8 by A62,A56,XXREAL_0:1;
A84: dom Supi = Segm 6 = dom MAX8 by PARTFUN1:def 2;
  for x st x in dom Supi holds Supi.x = MAX8.x
  proof
    let x such that
A85:x in dom Supi;
    reconsider x as Nat by A84,A85;
    per cases;
    suppose x=0;
      hence thesis by A82,A62,A56,XXREAL_0:1,A20;
    end;
    suppose
A86:  x<>0 & x <4;
      then
A87:  x in Segm 4 by NAT_1:44;
      2Supi4.x = ((EmptyBag 4)+* (0,8)).x by A83,A62,Th19,A9,A53;
      then
A88:  2Supi4.x = (EmptyBag 4).x by A86,FUNCT_7:32;
      2Supi4.x = (2(#) (Supi4)).x by A86,FUNCT_7:32
      .= 2* (Supi4.x) by VALUED_1:6;
      then Supi.x = 0 =E6.x = MAX8.x by A86,A87,A88,FUNCT_1:49,FUNCT_7:32;
      hence thesis;
    end;
    suppose
A89:  x >=4;
      then MAX8.x = E6.x by FUNCT_7:32;
      hence thesis by A89,A48,A2;
    end;
  end;
  then
A90: Supi = MAX8 by A84;
  Sup is one-to-one by POLYNOM2:18,PRE_POLY:10;
  hence i=I by A29,A46,A8,A48,A90;
A91: (YZ `^ 8 ).s<>0.R by A78,A79,Th33,A83;
  defpred R[Nat] means for b be bag of 6 st (YZ `^ $1).b <> 0.R holds
  degree b = $1;
A92:R[0] by UPROOTS:11,POLYNOM1:25,A32;
A93:for i st R[i] holds R[i+1]
  proof
    let i;
    assume
A94: R[i];
    let b be bag of 6 such that
A95: (YZ `^ (i+1)).b <> 0.R;
    (YZ `^ (i+1)) = (YZ `^ i) *' YZ by Th29
    .= Y *' (YZ `^ i) + Z *'(YZ `^ i) by POLYNOM1:26;
    then (YZ `^ (i+1)).b = (Y *' (YZ `^ i)).b+(Z *'(YZ`^ i)).b by POLYNOM1:15;
    then per cases by A95;
    suppose
A96:  (Y *' (YZ `^ i)).b<>0.R;
      Y *' (YZ `^ i) = (-1.R) * (Yb *' (YZ `^ i)) by POLYRED:22;
      then (Y *' (YZ `^ i)).b =(-1.R) *((Yb *' (YZ `^ i)).b) by POLYNOM7:def 9;
      then
A97:  (Yb *' (YZ `^ i)).b<>0.R by A96;
      then
A98:  Yb divides b by POLYRED:def 1;
      then (Yb *' (YZ `^ i)).b = (YZ `^ i).(b-'Yb) by POLYRED:def 1;
      then
A99:  degree (b-'Yb) = i by A94,A97;
      reconsider z=0 as Element of 6 by A19;
      reconsider Z={z} as Subset of 6;
      Yb = ({z},1)-bag by Th12;
      then
A100: degree Yb = card Z by UPROOTS:13
      .= 1 by CARD_1:30;
      b = (b-'Yb) +Yb by A98,PRE_POLY:47;
      hence thesis by A99,A100,UPROOTS:15;
    end;
    suppose
A101: (Z *'(YZ `^ i)).b<>0.R;
      Z *' (YZ `^ i) = (1.R) * (Zb *' (YZ `^ i)) by POLYRED:22;
      then
A102: Z *' (YZ `^ i).b = (1.R) *( (Zb *' (YZ `^ i)).b) by POLYNOM7:def 9;
      then
A103: Zb divides b by A101,POLYRED:def 1;
      then (Zb *' (YZ `^ i)).b = (YZ `^ i).(b-'Zb) by POLYRED:def 1;
      then
A104: degree (b-'Zb) = i by A94,A102,A101;
      reconsider z=4 as Element of 6 by A19;
      reconsider Z={z} as Subset of 6;
      Zb = ({z},1)-bag by Th12;
      then
A105: degree Zb = card Z by UPROOTS:13
      .= 1 by CARD_1:30;
      b = (b-'Zb) +Zb by A103,PRE_POLY:47;
      hence thesis by A104,A105,UPROOTS:15;
    end;
  end;
  R[j] from NAT_1:sch 2(A92,A93);
  then
A106: degree s = 8 by A91;
  s.4 = 8 by A81,A83,A47,A80,A79,XXREAL_0:1;
  then
A107: s = Z8 by A106,Th19;
  b = (E6+*(0,E6.0)) +s by A90,A79,FUNCT_7:34;
  then b = E6 +s by FUNCT_7:35;
  hence thesis by A107,PRE_POLY:53;
end;
A108: for i st i in dom S
  for b be bag of 6 st b in Support (q.(Sup/.i) * Subst(Sup/.i,0, YZ)) holds
    b.5=0
  proof
    let i such that
A109: i in dom S; set Supi = Sup/.i;
    let b be bag of 6 such that
A110: b in Support (q.(Supi) * Subst(Supi,0, YZ));
A111: Supi =Sup.i in Support q
    by A5,A109,A8,PARTFUN1:def 6,FUNCT_1:def 3;
A112:4-'0 =4-0;
    (Supi)|4 =(0,4)-cut Supi by HILB10_2:3;
    then reconsider Supi4 = (Supi)|4 as bag of 4 by A112;
    set 2Supi4 =( 2(#) (Supi4) +* (0,(Supi4.0)));
    defpred Q[Nat] means for b be bag of 6 st (YZ `^ $1).b<>0.R holds b.5 = 0;
A113:Q[0]
    proof
      let b be bag of 6 such that
A114: (YZ `^ 0).b <> 0.R;
      b = E6 by A114,POLYNOM1:25,A32;
      hence thesis;
    end;
A115:for i st Q[i] holds Q[i+1]
    proof
      let i;
      assume
A116: Q[i];
      let b be bag of 6 such that
A117: (YZ `^ (i+1)).b <> 0.R;
      (YZ `^ (i+1)) = (YZ `^ i) *' YZ by Th29
      .= Y *' (YZ `^ i) + Z *'(YZ `^ i) by POLYNOM1:26;
      then (YZ `^ (i+1)).b = (Y *' (YZ `^ i)).b+(Z *'(YZ `^ i)).b
      by POLYNOM1:15;
      then per cases by A117;
      suppose
A118:   (Z *' (YZ `^ i)).b<>0.R;
        Z *' (YZ `^ i) = (1.R) * (Zb *' (YZ `^ i)) by POLYRED:22;
        then
A119:   (Z *' (YZ `^ i)).b = (1.R) * ((Zb *' (YZ `^ i)).b) by POLYNOM7:def 9;
        then
A120:   Zb divides b by A118,POLYRED:def 1;
        then (Zb *' (YZ `^ i)).b = (YZ `^ i).(b-'Zb) by POLYRED:def 1;
        then
A121:   (b-'Zb).5 =0 by A116,A119,A118;
A122:   Zb.5 = E6.5 by FUNCT_7:32;
        (b.5)-'(Zb.5) = (b.5)- (Zb.5) by XREAL_1:233,A120,PRE_POLY:def 11;
        hence thesis by A121,A122,PRE_POLY:def 6;
      end;
      suppose
A123:   (Y *'(YZ `^ i)).b<>0.R;
        Y *' (YZ `^ i) = (-1.R) * (Yb *' (YZ `^ i)) by POLYRED:22;
        then Y *'(YZ `^ i).b =(-1.R) *( (Yb *' (YZ `^ i)).b) by POLYNOM7:def 9;
        then
A124:   (Yb *' (YZ `^ i)).b<>0.R by A123;
        then
A125:   Yb divides b by POLYRED:def 1;
        then (Yb *' (YZ `^ i)).b = (YZ `^ i).(b-'Yb) by POLYRED:def 1;
        then
A126:   (b-'Yb).5 = 0 by A116,A124;
A127:   Yb.5 = E6.5 by FUNCT_7:32;
        (b.5)-'(Yb.5) = (b.5)- (Yb.5) by XREAL_1:233,A125,PRE_POLY:def 11;
        hence thesis by A126,A127,PRE_POLY:def 6;
      end;
    end;
A128: Q[j] from NAT_1:sch 2(A113,A115);
    (q.(Supi) * Subst(Supi,0, YZ)).b <>0.R by A110,POLYNOM1:def 3;
    then (q.(Supi)) * (Subst(Supi,0, YZ).b) <>0.R by POLYNOM7:def 9;
    then
A129: Subst(Supi,0, YZ).b <>0.R;
    then consider s be bag of 6 such that
A130: b = Subst(Supi,0,s) by Def3;
    (Supi+*(0,0)).5 = Supi.5=0 by A111,A2,FUNCT_7:32;
    then
A131: (Supi+*(0,0) + s).5 = 0+(s.5) by PRE_POLY:def 5;
    (YZ `^ (Supi.0) ).s<>0.R by A129,A130,Th33;
    hence thesis by A131,A130,A128;
  end;
  set PR=Polynom-Ring (6, R);
  defpred W[Nat] means for i be Nat st $1=i & i <= len S holds
  for w be Polynomial of 6,R st w = Sum (S|i) holds
  (I <= i implies w.Z8 = 1_R) & (i < I implies w.Z8 = 0.R) &
  (for b be bag of 6 st b in Support w & b <> Z8 holds b.4 < 8) &
  (for b be bag of 6 st b in Support w holds b.5 =0);
A132: W[0]
  proof
    let i be Nat;
    assume
A133: 0=i & i<=len S;
    let w be Polynomial of 6,R such that
A134:w = Sum (S | i);
    thus I <= i implies w.Z8 = 1_R by A133,A29,FINSEQ_3:25;
    S | i = <*>(the carrier of PR) by A133;
    then
A135: w = 0.PR by A134,RLVECT_1:43
    .= 0_(6,R) by POLYNOM1:def 11;
    hence i < I implies w.Z8 = 0.R by POLYNOM1:22;
    thus for b be bag of 6 st b in Support w & b <> Z8 holds b.4 < 8
    proof
      let b be bag of 6;
      assume b in Support w;
      then w.b <> 0.R by POLYNOM1:def 3;
      hence thesis by A135,POLYNOM1:22;
    end;
    thus  for b be bag of 6 st b in Support w holds b.5 =0
    proof
      let b be bag of 6;
      assume b in Support w;
      then w.b <> 0.R by POLYNOM1:def 3;
      hence thesis by A135,POLYNOM1:22;
    end;
  end;
A136: Z8.4 = 8 by A19,A18,FUNCT_7:31;
A137: W[n] implies W[n+1]
  proof
    assume
A138:W[n];
    let n1 be Nat such that
A139:n1=n+1 & n1 <= len S;
    let w be Polynomial of 6,R such that
A140:w = Sum (S | n1);
    reconsider w1=Sum(S|n),Sn1=S/.n1 as Polynomial of 6,R by POLYNOM1:def 11;
A141: n1 in dom S by A139,FINSEQ_3:25,NAT_1:11;
    then
A142: S|n1 = (S|n)^<*S.n1*> & S.n1 =S/.n1
    by A139,FINSEQ_5:10,PARTFUN1:def 6;
    then
A143: w = Sum(S|n) + Sum(<*S/.n1*>) by A140,RLVECT_1:41
    .= Sum(S|n) + S/.n1 by RLVECT_1:44
    .= w1+ Sn1 by POLYNOM1:def 11;
A144: Sn1 = q.(Sup/.n1) * Subst(Sup/.n1,0, YZ) by A142,A7,A141;
A145: n < len S by A139,NAT_1:13;
A146: Z8 in Bags 6 = dom Sn1 by PRE_POLY:def 12,FUNCT_2:def 1;
    thus I <= n1 implies w.Z8 = 1_R
    proof
      assume
A147: I <= n1;
A148: w.Z8 = w1.Z8+ Sn1.Z8 by A143,POLYNOM1:15;
      per cases by A147,XXREAL_0:1;
      suppose
A149:   I=n1;
        then n <I by A139,NAT_1:13;
        then
A150:   w1.Z8 = 0.R by A145,A138;
        Sn1.Z8 = (1.R * Subst(Sup/.I,0, YZ)).Z8 by A141,A149,A29,A31,A30,A7
        .= 1.R * 1_R by A44,POLYNOM7:def 9;
        hence thesis by A148,A150;
      end;
      suppose
A151:   I <n1;
        then
A152:   I <= n by A139,NAT_1:13;
        assume w.Z8 <> 1_R;
        then Sn1.Z8<>0.R by A148,A152,A145,A138;
        then Z8 in Support Sn1 by A146,POLYNOM1:def 3;
        hence thesis by A45,A144,A141,A136,A151;
      end;
    end;
    thus n1 < I implies w.Z8 = 0.R
    proof
      assume
A153: n1 <I & w.Z8 <> 0.R;
      then n < I by NAT_1:13,A139;
      then w1.Z8 =0.R by A145,A138;
      then w.Z8 = 0.R + Sn1.Z8 by A143,POLYNOM1:15;
      then Z8 in Support Sn1 by A153,A146,POLYNOM1:def 3;
      hence thesis by A153,A45,A144,A141,A136;
    end;
    thus for b be bag of 6 st b in Support w & b<>Z8 holds b.4 < 8
    proof
      let b be bag of 6 such that
A154: b in Support w & b <> Z8;
      Support w c= Support w1 \/ Support Sn1 by A143,POLYNOM1:20;
      then per cases by XBOOLE_0:def 3,A154;
      suppose b in Support w1;
        hence thesis by A145,A154,A138;
      end;
      suppose b in Support Sn1;
        hence thesis by A154,A141,A144,A45;
      end;
    end;
    let b be bag of 6 such that
A155: b in Support w;
    Support w c= Support w1 \/ Support Sn1 by A143,POLYNOM1:20;
    then per cases by XBOOLE_0:def 3,A155;
    suppose b in Support w1;
      hence thesis by A145,A138;
    end;
    suppose b in Support Sn1;
      hence thesis by A141,A144,A108;
    end;
  end;
  set SU =Subst(q,0,YZ);
A156: S|len S=S;
A157: W[n] from NAT_1:sch 2(A132,A137);
A158: I<= len S by A6,A29,FINSEQ_3:25;
A159: for b be bag of 6 st b in Support SU  holds b.4 <= 8
  by A136,A157,A6,A156;
  defpred WW[bag of 6, Element of R] means
     ($1.4+$1.5 = 8 implies $2 = SU. ($1 +*(5,0) )) &
     ($1.4+$1.5 <> 8 implies $2=0.R);
A160:for x being Element of Bags 6 ex y be Element of R st WW[x,y]
  proof
    let x be Element of Bags 6;
    per cases;
    suppose
A161: x.4+x.5 = 8;
      take SU. (x +*(5,0) );
      thus thesis by A161;
    end;
    suppose
A162: x.4+x.5 <> 8;
      take 0.R;
      thus thesis by A162;
    end;
  end;
  consider W being Function of Bags 6,R such that
A163:  for x being Element of Bags 6 holds WW[x,W.x] from FUNCT_2:sch 3(A160);
  set SS =SgmX(BagOrder 6, Support SU);
A164: dom SU = Bags 6 =dom W by FUNCT_2:def 1;
  BagOrder 6 linearly_orders Support SU by POLYNOM2:18;
  then
A165: rng SS = Support SU by PRE_POLY:def 2;
  reconsider SS as one-to-one FinSequence of Bags 6 by POLYNOM2:18,PRE_POLY:10;
  deffunc O(object) = SS/.$1 +*(5,8-'SS/.$1.4);
  consider SW be FinSequence such that
A166: len SW = len SS and
A167: for k st k in dom SW holds SW.k =O(k) from FINSEQ_1:sch 2;
A168: dom SS = dom SW by A166,FINSEQ_3:29;
A169: rng SW c= Support W
  proof
    let y be object;
    assume y in rng SW;
    then consider z be object such that
A170:   z in dom SW & SW.z = y by FUNCT_1:def 3;
    reconsider z as Nat by A170;
A171: O(z) in Bags 6 by PRE_POLY:def 12;
A172: SW.z = O(z) by A170,A167;
A173: 6 = dom (SS/.z) by PARTFUN1:def 2;
A174: O(z).4 = SS/.z.4 by FUNCT_7:32;
A175: O(z).5 = 8-'SS/.z.4 by A19, A173,FUNCT_7:31;
    SS/.z = SS.z in rng SS by A170,A168,FUNCT_1:def 3,PARTFUN1:def 6;
    then O(z).5 = 8 - SS/.z.4 by A159,A165,A175,XREAL_1:233;
    then O(z).4+O(z).5 = 8 by A174;
    then
A176: W.O(z) = SU. (O(z) +*(5,0) ) by A163,A171;
A177: SS/.z = SS.z in rng SS by A170,A168,FUNCT_1:def 3,PARTFUN1:def 6;
    then
A178: SU. (SS/.z)<>0.R by POLYNOM1:def 3,A165;
A179: (SS/.z).5 = 0 by A157,A6,A156,A177,A165;
    SS/.z +*(5,8-'SS/.z.4) +*(5,0) = SS/.z +*(5,0) by FUNCT_7:34
    .= SS/.z by A179,FUNCT_7:35;
    hence thesis by A176,A172,A170, A178,A171,A164,POLYNOM1:def 3;
  end;
A180: Support W c= rng SW
  proof
    let z be object;
    assume
A181:z in Support W;
    then
A182:z in dom W & W.z<>0.R by POLYNOM1:def 3;
    reconsider z as bag of 6 by A181;
A183: (z +*(5,0) ) in Bags 6 by PRE_POLY:def 12;
A184: WW[z,W.z] by A181,A163;
    (z +*(5,0) ) in Support SU by A164,A183,A182,A184,POLYNOM1:def 3;
    then consider i be object such that
A185: i in dom SS & SS.i = (z +*(5,0) ) by A165,FUNCT_1:def 3;
    reconsider i as Nat by A185;
A186: SS.i = SS/.i by PARTFUN1:def 6,A185;
A187: SS/.i.4 = z.4 by A185, A186,FUNCT_7:32;
    SW.i = SS/.i +*(5,8-'SS/.i.4) by A167,A168,A185
    .= (z +*(5,0) ) +*(5,8-'SS/.i.4) by A185,PARTFUN1:def 6
    .= z    +*(5,8-'SS/.i.4) by FUNCT_7:34
    .= z by A184,A181,POLYNOM1:def 3,FUNCT_7:35,A187;
    hence thesis by A185,A168,FUNCT_1:def 3;
  end;
  then
A189: rng SW = Support W by A169,XBOOLE_0:def 10;
A190: SW is one-to-one
  proof
    let i1,i2 be object such that
A191: i1 in dom SW & i2 in dom SW & SW.i1=SW.i2;
A192: SW.i1 = O(i1) & SW.i2 = O(i2) by A191,A167;
A193:  6 = dom (SS/.i1) & 6 = dom (SS/.i2) by PARTFUN1:def 2;
A194: SS/.i1 = SS.i1 in rng SS by A191,A168,FUNCT_1:def 3,PARTFUN1:def 6;
A195: SS/.i2 = SS.i2 in rng SS by A191,A168,FUNCT_1:def 3,PARTFUN1:def 6;
A196: SS/.i1 = SS.i1 & SS/.i2 = SS.i2 by A191,A168,PARTFUN1:def 6;
    for x st x in 6 holds SS/.i1.x = SS/.i2.x
    proof
      let j be object such that
A197: j in 6 & SS/.i1.j <> SS/.i2.j;
A198: j in Segm 6 by A197;
      then reconsider j as Nat;
A199: SS/.i1.5 = 0 = SS/.i2.5 by A195,A194,A157,A6,A156,A165;
      j < 5+1 by A198,NAT_1:44;
      then
A200: j <= 5 & j<>5 by A199,NAT_1:13,A197;
      O(i1).j = (SS/.i1).j by A200,FUNCT_7:32;
      hence thesis by A192,A191,A197,A200,FUNCT_7:32;
    end;
    then
    SS/.i1 = SS/.i2 by A193;
    hence i1=i2 by A196,A191,A168,FUNCT_1:def 4;
  end;
  reconsider SW as one-to-one FinSequence of Bags 6
    by A189,A190,FINSEQ_1:def 4;
  reconsider W as Polynomial of 6,R by A180,POLYNOM1:def 5;
  reconsider RR=R as Field;
A202: dom SU = Bags 6 by FUNCT_2:def 1;
  rng W c= INT
  proof
    let y be object;
    assume
A203: y in rng W & not y in INT;
    then consider x such that
A204: x in dom W & W.x = y by FUNCT_1:def 3;
    reconsider x as bag of 6 by A204;
A205: W.x <>0 by A204,A203,INT_1:def 2;
A206:x +*(5,0) in Bags 6 by PRE_POLY:def 12;
    x.4 + x.5 =8 by A205,A163,A204;
    then W.x = SU. (x +*(5,0)) by A163,A204;
    then W.x in rng SU c= INT by A206,A202,FUNCT_1:def 3,RELAT_1:def 19;
    hence thesis by A204,A203;
  end;
  then
  reconsider W as INT -valued Polynomial of 6,R by RELAT_1:def 19;
  take W;
A207: len SW = card Support W by A189, FINSEQ_4:62;
  reconsider SSU=SU,WW=W as Polynomial of 6,RR;
A208: for f be Function of 6,F_Real, d be Element of F_Real st
  f.5 <>0 & d = f.4/f.5
  holds eval(W,f) = (power F_Real).(f/.5,8)* eval(SU, f+*(4,d))
  proof
    let f be Function of 6,F_Real,divf be Element of F_Real such that
A209: f.5 <>0 and
A210: divf = f.4/f.5;
    reconsider ff=f as Function of 6,RR;
    reconsider divff = divf as Element of RR;
    set ffd = ff +* (4,divff);
    set fd = f +* (4,divf);
    set P = (power RR).(ff/.5,8);
    set p = (power R).(f/.5,8);
    reconsider SSU=SU,WW=W as Polynomial of 6,RR;
    set PSU =P * SSU;
A211: dom SU = Bags 6 =dom PSU by FUNCT_2:def 1;
    dom f = 6 by FUNCT_2:def 1;
    then f/.5=f.5 by A19,PARTFUN1:def 6;
    then
A212:p<>0.R by A209,Th6;
A213:Support SU = Support PSU
    proof
      thus Support SU c= Support PSU
      proof
        let u be object;
        assume
A214:   u in Support SU;
        reconsider u as bag of 6 by A214;
        p*(SU.u) = PSU.u  & SU.u <> 0.R  & u in dom SU
        by A214,POLYNOM1:def 3,POLYNOM7:def 9;
        hence thesis by POLYNOM1:def 3,A211,A212;
      end;
      let u be object;
      assume
A215: u in Support PSU;
      reconsider u as bag of 6 by A215;
      p*(SU.u) = PSU.u  & PSU.u <> 0.R  & u in dom PSU
      by A215,POLYNOM1:def 3,POLYNOM7:def 9;
      then SU.u <>0.R & u in dom SU by A211;
      hence thesis by POLYNOM1:def 3;
    end;
    set SS =SgmX(BagOrder 6, Support SU);
    consider y be FinSequence of RR such that
A216: len y = len SS &
    eval(PSU,ffd) = Sum y and
A217:  for i be Element of NAT st 1 <= i & i <= len y holds
    y/.i = (PSU * SS)/.i * eval((SS/.i),ffd) by A213,POLYNOM2:def 4;
    consider yW being FinSequence of RR such that
A218: len yW = card Support WW & eval(WW,ff) = Sum yW and
A219: for i be Nat st 1 <= i & i <= len yW holds
    yW/.i = (WW * SW)/.i * eval(SW/.i,ff)
    by A180,A169,XBOOLE_0:def 10,HILB10_2:24;
    for j st 1<=j <= len y holds yW.j = y.j
    proof
      let j such that
A220: 1<=j <= len y;
A221: j in NAT by ORDINAL1:def 12;
      set SSj= SS/.j;
A222: j in dom SS & j in dom SW & j in dom y & j in dom yW
      by A220,A216,A166,FINSEQ_3:25,A218,A207;
      dom (PSU * SS) = dom SS by RELAT_1:27,A165,A211;
      then
A223: (PSU * SS)/.j = (PSU * SS).j = PSU. (SS.j) & SS.j = SSj
      by A220,A216,FINSEQ_3:25,PARTFUN1:def 6,FUNCT_1:12;
      dom (WW * SW) = dom SW by A164,A189,RELAT_1:27;
      then
A224: (WW * SW)/.j = (WW * SW).j = WW. (SW.j) & SW.j = SW/.j
      by A220,A216,A166,FINSEQ_3:25,PARTFUN1:def 6,FUNCT_1:12;
A225: SW/.j = SS/.j +*(5,8-' SS/.j.4)
      by A224,A167,A220,A216,A166,FINSEQ_3:25;
A226:  6 = dom (SS/.j) by PARTFUN1:def 2;
A227: O(j) in Bags 6 by PRE_POLY:def 12;
A228: O(j).4 = SS/.j.4 by FUNCT_7:32;
A229: O(j).5 = 8-'SS/.j.4 by A19, A226,FUNCT_7:31;
A230: SS/.j = SS.j in rng SS by A222,FUNCT_1:def 3,PARTFUN1:def 6;
      then
A231: 8 -' SS/.j.4 = 8 - SS/.j.4 by A159,A165,XREAL_1:233;
      then O(j).4+O(j).5 = 8 by A228,A229;
      then
A232: W.O(j) = SU. (O(j) +*(5,0) ) by A163,A227;
A233: SSj.5 = 0 by A230,A157,A6,A156,A165;
A234: (O(j)+*(5,0) ) = SS/.j +*(5,0) by FUNCT_7:34
      .= SS/.j by A233,FUNCT_7:35;
      set SSj5 = SSj  +* (5,0);
A235: SSj +*(5,8-'SSj.4) +* (5,0) = SSj5 by FUNCT_7:34;
A236: dom (SSj ) = 6 =dom SSj5 by PARTFUN1:def 2;
      then SSj +*(5,8-'SSj.4).5 = 8-'SSj.4 by A19,FUNCT_7:31;
      then SW/.j = SSj5  + E6+*(5,8-'SSj.4) by A235,Th15,A225;
      then
A237: eval(SW/.j,ff) = eval( SSj5,ff) * eval(E6+*(5,8-'SSj.4),ff)
      by POLYNOM2:16;
A238: eval(E6+*(5,8-'SSj.4),ff) = (power RR).(ff.5, 8-'SSj.4) by A19,Th14;
A239: eval(SSj5,ffd) = eval(SSj5 +*(4,0) + (E6 +*(4,SSj5.4)),ffd)
      & eval(SSj5,ff) = eval(SSj5 +*(4,0) + (E6 +*(4,SSj5.4)),ff) by Th15;
A240: eval(E6 +*(4,SSj5.4),ffd) = (power RR).(ffd.4, SSj5.4) &
      eval(E6 +*(4,SSj5.4),ff) = (power RR).(ff.4, SSj5.4) by A19,Th14;
A241: dom ff = 6 =dom ffd by PARTFUN1:def 2;
      then
A242: ffd.4 = divff by A19,FUNCT_7:31;
A243: (SSj5 +*(4,0)).4 =0 by A236,A19,FUNCT_7:31;
A244: SSj = SSj5 by A233,FUNCT_7:35;
A245: ff/.5 = ff.5 & ff/.4 = ff.4 by A241,A19,PARTFUN1:def 6;
A246: 8 -' SS/.j.4 in NAT & SS/.j.4 in NAT by ORDINAL1:def 12;
      divff * ff/.5 = divf *f/.5
      .= divf *f.5 by A241,A19,PARTFUN1:def 6
      .= f.4 by XCMPLX_1:87,A209,A210;
      then (power RR).(divff, SSj5.4) * (power RR).(ff/.5, SSj5.4) =
      (power RR).(ff/.4, SSj5.4) by A245,GROUP_1:52;
      then
A247: (power RR).(ff/.4, SSj5.4) * (power RR).(ff/.5, 8-'SSj.4) =
      (power RR).(divff, SSj5.4) * ((power RR).(ff/.5, SSj5.4)
      * (power RR).(ff/.5, 8-'SSj.4)) by GROUP_1:def 3
      .= (power RR).(divff, SSj5.4) * ((power RR).(ff/.5, SSj.4 + (8-'SSj.4)))
      by A246,A244,POLYNOM2:1
      .= (power RR).(divff, SSj5.4) * P by A231;
A248: eval(SW/.j,ff) = eval( SSj5,ff) * ((power RR).(ff/.5, 8-'SSj.4))
      by A241,A19,PARTFUN1:def 6,A237,A238
      .= eval(SSj5 +*(4,0),ff) *eval(E6 +*(4,SSj5.4),ff) *
      ((power RR).(ff/.5, 8-'SSj.4)) by A239,POLYNOM2:16
      .= (eval(SSj5 +*(4,0),ff)* (power RR).(ff/.4, SSj5.4)) *
      (power RR).(ff/.5, 8-'SSj.4) by A241,A19,PARTFUN1:def 6,A240
      .= eval(SSj5 +*(4,0),ff)* ((power RR).(ff/.4, SSj5.4) *
      (power RR).(ff/.5, 8-'SSj.4)) by GROUP_1:def 3
      .= eval(SSj +*(4,0),ffd) * (eval(E6 +*(4,SSj5.4),ffd) * P)
      by A240,A242,A243,Th18,A247,A244
      .= (eval(SSj +*(4,0),ffd) * eval(E6 +*(4,SSj5.4),ffd)) * P
      by GROUP_1:def 3
      .= eval(SSj,ffd) * P by A239,POLYNOM2:16,A244;
A249: (WW * SW)/.j = SU . SSj
      by A234,A224,A167,A220,A216,A166,FINSEQ_3:25,A232;
A250: (PSU * SS)/.j = p * (SU.SSj) by POLYNOM7:def 9,A223;
      y/.j = P * (WW * SW)/.j * eval(SSj,ffd) by A249,A250,A217,A220,A221
      .= (WW * SW)/.j *(P * eval(SSj,ffd)) by GROUP_1:def 3
      .= yW/.j by A216,A218,A219,A166,A207,A220,A248
      .=yW.j by A222,PARTFUN1:def 6;
      hence thesis by A222,PARTFUN1:def 6;
    end;
    then eval(PSU,ffd) = eval(W,f) by A216,A218,A166,A207,FINSEQ_1:14;
    hence thesis by POLYNOM7:29;
  end;
A251: 0 in Segm 6 & 1 in Segm 6 & 2 in Segm 6 & 3 in Segm 6 & 4 in
  Segm 6 & 5 in Segm 6 by NAT_1:44;
  thus
A252:  for f be Function of 6,F_Real st f.5 <>0
  holds eval(W,f) = (power R).(f/.5,8)*
  eval(Jsqrt M, @ <%- (f.0) + (f.4/f.5),f.1,f.2,f.3%>)
  proof
    let f be Function of 6,F_Real such that
A253:f.5 <>0;
    reconsider divf=f.4/f.5 as Element of R by XREAL_0:def 1;
    set fd = f +* (4,divf);
A254: eval(W,f) = (power F_Real).(f/.5,8)* eval(SU, f+*(4,divf))
    by A253,A208;
A255: dom f = 6 =dom fd by FUNCT_2:def 1;
A256:eval(SU,fd) = eval(q, fd+*(0,eval(YZ,fd))) by Th37,A251;
A257: eval(Zb,fd) = power(R).(fd.4,1) by A251,Th14
    .= power(R).(divf,1) by A251,A255,FUNCT_7:31
    .= divf by GROUP_1:50;
A258: fd.0 =f.0 = f/.0 by A251,A255,PARTFUN1:def 6,FUNCT_7:32;
A259:eval(Yb,fd) = power(R).(fd.0,1) by A251,Th14
    .= f/.0 by A258,GROUP_1:50;
A260: eval(YZ,fd) = eval(Y,fd) + eval(Z,fd) by POLYNOM2:23
    .= (-1.R)*eval(Yb,fd) + eval(Z,fd) by POLYNOM7:13
    .= (-1.R)*eval(Yb,fd) + (1.R)*eval(Zb,fd) by POLYNOM7:13
    .= -(f/.0) + divf by A259,A257
    .= -(f.0) + divf by A251,A255,PARTFUN1:def 6;
A261: Segm 4 c= Segm 6 by NAT_1:39;
A262: dom ((fd+*(0,-(f.0) + divf)) |4)=4 by PARTFUN1:def 2,A261;
    for x st x in 4  holds (fd+*(0,-(f.0) + divf)|4).x =
    (@<%- (f.0) + divf,f.1,f.2,f.3%>).x
    proof
      let x such that
A263: x in 4;
A264: 4 = Segm 4;
      then reconsider x as Nat by A263;
      x<4=3+1 by A263,A264,NAT_1:44;
      then x <= 3 by NAT_1:13;
      then x=0 or ... or x=3;
      then per cases;
      suppose
A265:   x=0;
        then (fd+*(0,-(f.0) + divf)).x = <%- (f.0) + divf,f.1,f.2,f.3%>.0
        by A261,A263,A255,FUNCT_7:31;
        hence thesis by A265,A263,FUNCT_1:49;
      end;
      suppose
A266:   x=1 or x=2 or x=3;
        then (fd+*(0,-(f.0) + divf)).x = fd.x by FUNCT_7:32
        .= <%- (f.0) + divf,f.1,f.2,f.3%>.x by A266,FUNCT_7:32;
        hence thesis by A263,FUNCT_1:49;
      end;
    end;
    then fd+*(0,-(f.0) + divf)|4 = (@<%- (f.0) + divf,f.1,f.2,f.3%>)
    by PARTFUN1:def 2,A262;
    hence thesis by A254,A256,A260,A4;
  end;
  let f be INT -valued Function of 6,F_Real such that
A267: f.5 <>0 & eval(W,f) = 0;
  set N=f.5 gcd f.4;
  consider g5,g4 be Integer such that
A268:    f.5 = N*g5 & f.4 = N*g4 & g5,g4 are_coprime by A267,INT_2:23;
  reconsider Nr=N,g5r=g5,g4r=g4 as Element of R by XREAL_0:def 1;
  set g = f +*(4,g4r) +*(5,g5r);
  reconsider g as Function of 6,R;
  reconsider gg = g as Function of 6,RR;
A269: (power F_Real).(f.5,8) = (power F_Real).(Nr*g5r,8) by A268
  .= ((power F_Real).(Nr,8)) * ((power F_Real).(g5r,8)) by GROUP_1:52;
A270: dom f = 6 = dom (f +*(4,g4)) & dom g = 6 =Segm 6 by PARTFUN1:def 2;
  then
A271: g.0 = (f +*(4,g4)).0 = f.0  &
  g.1 = (f +*(4,g4)).1 = f.1 & g.2 = (f +*(4,g4)).2 = f.2  &
  g.3 = (f +*(4,g4)).3 = f.3 & g.4 = (f +*(4,g4)).4 = g4 & g.5 = g5
  by NAT_1:44,FUNCT_7:32,31;
  rng g c= INT
  proof
    let y be object such that
A272:y in rng g;
    consider x such that
A273:x in dom g & g.x=y by A272,FUNCT_1:def 3;
    reconsider x as Nat by A273,A270;
    x < 6=5+1 by A273,A270,NAT_1:44;
    then x <= 5 by NAT_1:13;
    then x=0 or ... or x = 5;
    hence thesis by A271,A273,INT_1:def 2;
  end;
  then
A274: g is INT -valued by RELAT_1:def 19;
A275: (power F_Real).(Nr,8)<>0.R by A267,Th6;
A276: g.5 <>0 by A267,A271,A268;
A277: f/.5 = f.5 & g/.5 = g.5 by A270,NAT_1:44,PARTFUN1:def 6;
  0.R = (power R).(f/.5,8)*
  eval(Jsqrt M, @ <%- (f.0) + (f.4/f.5),f.1,f.2,f.3%>) by A267,A252
  .= (power R).(f/.5,8)*
  eval(Jsqrt M, @<%- (g.0) + (g.4/g.5),g.1,g.2,g.3%>)
  by A268,A267,XCMPLX_1:91,A271
  .= ((power F_Real).(Nr,8)) * ((power F_Real).(g5r,8)) *
  eval(Jsqrt M, @<%- (g.0) + (g.4/g.5),g.1,g.2,g.3%>)
  by A270,NAT_1:44,PARTFUN1:def 6,A269
  .= ((power F_Real).(Nr,8)) * (((power F_Real).(g5r,8)) *
  eval(Jsqrt M, @<%- (g.0) + (g.4/g.5),g.1,g.2,g.3%>));
  then
A278: 0.R = (((power F_Real).(g5r,8)) *
  eval(Jsqrt M, @<%- (g.0) + (g.4/g.5),g.1,g.2,g.3%>)) by A275
  .= (((power F_Real).(g/.5,8)) *
  eval(Jsqrt M, @<%- (g.0) + (g.4/g.5),g.1,g.2,g.3%>))
  by A270,NAT_1:44,FUNCT_7:31,A277
  .= eval(W,g) by A276,A252;
  set R8 = E6+* (4,8);
  R8 in Bags 6 & Z8 in Bags 6 by PRE_POLY:def 12;
  then
A279: WW[R8,W.R8] by A163;
A280: R8.5 = E6.5=0 by FUNCT_7:32;
  R8 +*(5,0) = E6 +*(5,E6.5)+*(4,8) by FUNCT_7:33
  .= Z8 by FUNCT_7:35;
  then
A281: W.R8 = 1_R by A280,A279,A18,A19,FUNCT_7:31,A158,A157,A6,A156;
  set MZ = Monom(1.R,R8);
  set MZZ = Monom(1.RR,R8);
  MZ-MZ = 0_(6,RR) by POLYNOM1:24;
  then
A282: W = MZZ-MZZ + WW by POLYNOM1:23
  .= (MZZ+(-MZZ))+WW by POLYNOM1:def 7
  .= MZZ + (-MZZ+WW) by POLYNOM1:21
  .=MZ + (W -MZ) by POLYNOM1:def 7;
  set S = SgmX(BagOrder 6, Support (WW-MZZ));
  BagOrder 6 linearly_orders Support (WW-MZZ) by POLYNOM2:18;
  then
A283: rng S = Support (WW-MZZ) by PRE_POLY:def 2;
  consider Ry be FinSequence of RR such that
A284:len Ry = len S & eval(WW-MZZ,gg) = Sum Ry and
A285: for i be Element of NAT st 1 <= i & i <= len Ry holds
  Ry/.i = ((WW-MZZ) * S)/.i * eval((S/.i),gg) by POLYNOM2:def 4;
  defpred P[Nat] means for i be Nat st i=$1 <= len S
  ex s be Integer st s*(g.5) = Sum (Ry|i);
  A286:P[0]
  proof
    let i such that
A287: i= 0<= len S;
    take s=0;
    Ry|0 = <*> the carrier of RR;
    then Sum (Ry|i) = 0.RR by A287,RLVECT_1:43;
    hence thesis;
  end;
A288: term MZ = R8 by POLYNOM7:10;
A289: MZ.R8 = coefficient(MZ) by POLYNOM7:10;
A290:P[n] implies P[n+1]
  proof
    assume
A291:P[n];
    let n1 be Nat such that
A292: n1=n+1 <= len S;
    n < len S by A292,NAT_1:13;
    then consider s be Integer such that
A293: s*(g.5) = Sum (Ry|n) by A291;
A294: n1 in dom S by A292,NAT_1:11,FINSEQ_3:25;
    then
A295: S.n1 in rng S & S.n1 = S/.n1 by FUNCT_1:def 3,PARTFUN1:def 6;
    then
A296:(WW-MZZ).(S/.n1) <>0.R by A283,POLYNOM1:def 3;
A297:(WW-MZZ).(S/.n1) = (WW+-MZZ).(S/.n1) by POLYNOM1:def 7
    .= WW.(S/.n1) + (-MZZ).(S/.n1) by POLYNOM1:15
    .= WW.(S/.n1) + -(MZZ.(S/.n1)) by POLYNOM1:17;
A298: S/.n1 <>R8
    proof
      assume S/.n1 =R8;
      then WW.(S/.n1) + -(MZZ.(S/.n1)) = 1_R + -(1_R) by A281,A289,POLYNOM7:9;
      hence thesis by A296,A297;
    end;
    MZ.(S/.n1) = 0.R by A288,A298,POLYNOM7:def 5;
    then
A299: WW.(S/.n1) <> 0.R by A295,A297,A283,POLYNOM1:def 3;
    then
A300: (S/.n1).4 + (S/.n1).5 = 8 by A163;
    then
A301: WW.(S/.n1) = SU.( S/.n1 +*(5,0)) by A163;
A302: dom (S/.n1) = 6 by PARTFUN1:def 2;
A303: (S/.n1 +* (5,0)).4 = (S/.n1).4 by FUNCT_7:32;
A304:  S/.n1 +* (5,0)<>Z8
    proof
      assume
A305: S/.n1  +* (5,0)=Z8;
      consider j be object such that
A306: j in Segm 6 & (S/.n1).j <>R8.j by A302,A298,PARTFUN1:def 2;
      reconsider j as Nat by A306;
A307: R8.5= E6.5 by FUNCT_7:32;
      j < 5+1 by A306,NAT_1:44;
      then j <= 5 & j<>5 by NAT_1:13,A306,A305,A307,A300,A136,A303;
      hence contradiction by A306,A305,FUNCT_7:32;
    end;
    S/.n1  +* (5,0) in Bags 6 by PRE_POLY:def 12;
    then ( S/.n1 +*(5,0)) in Support SU by A164,A299,A301,POLYNOM1:def 3;
    then S/.n1.5 <>0 by A300,A303,A304,A157,A6,A156;
    then reconsider O=(S/.n1.5) -1 as Nat;
A308: S/.n1 = S/.n1 +*(5,O+1) by FUNCT_7:35
    .= S/.n1 +*(5,O) + E6 +*(5,1) by Th22;
A309: eval(E6 +*(5,1),gg) = power(R).(gg.5,1) by A251,Th14
    .= g5r by GROUP_1:50,A271;
    reconsider d1= eval(S/.n1 +*(5,O),gg) as Integer by A274;
    Bags 6 = dom (W-MZ) by FUNCT_2:def 1;
    then dom ((W-MZ) * S) = dom S by A283,RELAT_1:27;
    then ((W-MZ) * S)/.n1 = ((W-MZ) * S).n1 in rng ((W-MZ) * S) c= INT
    by A294,PARTFUN1:def 6,FUNCT_1:def 3,RELAT_1:def 19;
    then reconsider d2 = ((W-MZ) * S)/.n1 as Integer;
    reconsider E1=eval(S/.n1 +*(5,O),g), E2 = eval(E6 +*(5,1),g)
    as Element of REAL;
A310:eval((S/.n1),gg) = eval(S/.n1 +*(5,O),g) * eval(E6 +*(5,1),g)
    by A308,POLYNOM2:16;
A311: Ry/.n1 = ((WW-MZZ) * S)/.n1 * eval((S/.n1),gg)
    by A292,A284,A285,NAT_1:11
    .= d2 * (E1*E2) by BINOP_2:def 11,A310
    .= (d2*d1)* g5 by A309;
    Ry|n1 = (Ry|n) ^ <*Ry/.n1*> by A292,A284,FINSEQ_5:82;
    then Sum (Ry|n1) = Sum (Ry|n) + Sum <*Ry/.n1*> by RLVECT_1:41;
    then Sum (Ry|n1) = (the addF of RR).(Sum (Ry|n),Ry/.n1) by RLVECT_1:44
    .= s*(g.5)+(d2*d1)* g5 by A311,A293, BINOP_2:def 9
    .= (g.5)*(s+d2*d1) by A271;
    hence thesis;
  end;
  P[n] from NAT_1:sch 2(A286,A290);
  then consider s be Integer such that
A312: s*(g.5) = Sum (Ry|len Ry) by A284;
A313: eval(R8,g) = power(R).(g.4,8) by A251,Th14
  .= g4 |^ 8 by NIVEN:7,A271;
A314:  eval(MZ,g) = 1.R * eval(R8,g) by POLYNOM7:13
  .= g4 |^ 8 by A313;
  eval(W,g) = eval(MZ,g) + eval(W-MZ,g) by A282,POLYNOM2:23
  .= s*(g.5) + g4 |^ 8 by A312,A284,A314;
  then
A315: g5*(-s) = g4 |^ 8 by A278,A271;
  defpred H[Nat] means g5 divides g4 |^ $1 implies g5 divides g4;
A316: H[0]
  proof
    A317: g4|^0 = 1 by NEWTON:4;
    assume g5 divides g4 |^ 0;
    then g5 = 1 or g5 = -1 by A317,INT_2:13;
    hence thesis by INT_2:12;
  end;
A318:H[j] implies H[j+1]
  proof
    assume
A319: H[j] & g5 divides g4 |^ (j+1);
    then g5 divides g4* g4 |^ j by NEWTON:6;
    hence thesis by A319,A268,INT_2:25;
  end;
  H[j] from NAT_1:sch 2(A316,A318);
  then g5 divides g4 & g5 divides g5*1 by A315,INT_1:def 3;
  then g5 divides (g5 gcd g4) by INT_2:22;
  then g5 divides 1 by A268,INT_2:def 3;
  then g5 = 1 or g5 = -1 by INT_2:13;
  then f.4 = f.5 *(-g4) or f.4 = f.5 * g4 by A268;
  hence f.5 divides f.4 by INT_1:def 3;
end;
