
theorem mpol5:
for F being Field,
    E being FieldExtension of F
for a being F-algebraic Element of E
for i,j being Element of NAT st i < j & j < deg MinPoly(a,F) holds a|^i <> a|^j
proof
let F be Field, E be FieldExtension of F; let a be F-algebraic Element of E;
let i,j be Element of NAT;
assume AS: i < j & j < deg MinPoly(a,F);
set ma = MinPoly(a,F);
reconsider n = deg ma as Element of NAT;
j >= 0 + 1 by AS,INT_1:7; then
X: ma is non linear by AS,FIELD_5:def 1;
j - i <= j by XREAL_1:43; then
j - i < n by AS,XXREAL_0:2; then
Y3: n - n < n - (j - i) by XREAL_1:15; then
0 + 1 <= i + n - j by INT_1:7; then
Y7: 0 + 1 - 1 <= i + n - j - 1 by XREAL_1:9;
i - j < 0 by AS,XREAL_1:49; then
Y4: n + (i - j) < n + 0 by XREAL_1:8;
reconsider k = i + n - j as Element of NAT by Y3,INT_1:3;
Y8: (n + (i - j)) - 1 < n - 1 by Y4,XREAL_1:9;
Y9: now assume i + n - j >= n; then
    i + n - j - n >= n - n by XREAL_1:9; then
    i - j + j >= 0 + j by XREAL_1:6;
    hence contradiction by AS;
    end;
reconsider h = n - j as Element of NAT by AS,INT_1:5;
set p = 0_.(F) +* (k,n) --> (1.F,-1.F);
now let x be object;
  assume x in {k,n}; then
  per cases by TARSKI:def 2;
  suppose x = k; hence x in NAT; end;
  suppose x = n; hence x in NAT; end;
  end; then
A: {k,n} c= NAT;
B: dom((k,n) --> (1.F,-1.F)) = {k,n} & dom(0_.(F)) = NAT
   by FUNCT_2:def 1; then
dom p = NAT \/ {k,n} by FUNCT_4:def 1 .= NAT by A,XBOOLE_1:12; then
dom p = NAT & rng p c= the carrier of F; then
reconsider p as sequence of F by FUNCT_2:2;
reconsider p as Element of the carrier of Polynom-Ring F by POLYNOM3:def 10;
k in dom((k,n) --> (1.F,-1.F)) by TARSKI:def 2,B; then
p.k = ((k,n) --> (1.F,-1.F)).k by FUNCT_4:13 .= 1.F by Y4,FUNCT_4:63; then
p <> 0_.(F); then
reconsider p as non zero Element of the carrier of Polynom-Ring F
   by UPROOTS:def 5;
reconsider p1 = p, ma1 = ma as non zero Polynomial of F;
reconsider q1 = p1 + ma1 as Polynomial of F;
now assume K: p1 + ma1 = 0_.(F);
  K1: -p1 = -p1 + 0_.(F)
         .= (p1 - p1) + ma1 by K,POLYNOM3:26
         .= 0_.(F) + ma1 by POLYNOM3:29;
  set q = 0_.(F) +* (n+i-j-1,n-1) --> (-1.F,1.F);
  now let x be object;
    assume x in {n+i-j-1,n-1}; then
    per cases by TARSKI:def 2;
    suppose x = n+i-j-1; hence x in NAT by Y7,INT_1:3; end;
    suppose x = n-1; hence x in NAT by AS,INT_1:3; end;
    end; then
  Z2: {n+i-j-1,n-1} c= NAT;
  Z3: dom((n+i-j-1,n-1) --> (-1.F,1.F)) = {n+i-j-1,n-1} & dom(0_.(F)) = NAT
      by FUNCT_2:def 1; then
  dom q = NAT \/ {n+i-j-1,n-1} by FUNCT_4:def 1 .= NAT by Z2,XBOOLE_1:12; then
  dom q = NAT & rng q c= the carrier of F; then
  reconsider q as sequence of F by FUNCT_2:2;
  reconsider q as Polynomial of F;
  K: <%0.F,1.F%> *' q = -p1
     proof
     now let u be Element of NAT;
     per cases;
     suppose K2: u = 0; then
        K3: not u in dom((k,n) --> (1.F,-1.F)) by AS,Y3;
        K4: 0.F = (0_.(F)).u
               .= p1.u by K3,FUNCT_4:11;
        thus (<%0.F,1.F%> *' q).u
              = -(p1.u) by K4,K2,thE2
             .= (-p1).u by BHSP_1:44;
        end;
     suppose u <> 0;
        then consider u1 being Nat such that K2: u = u1 + 1 by NAT_1:6;
        K5: (<%0.F,1.F%> *' q).u = q.u1 by K2,thE1;
        per cases;
        suppose K6: u = k;
          k in dom((k,n) --> (1.F,-1.F)) by TARSKI:def 2,B; then
          K7: p.k = ((k,n) --> (1.F,-1.F)).k by FUNCT_4:13
                 .= 1.F by Y4,FUNCT_4:63;
          K8: u1 in dom((n+i-j-1,n-1)-->(-1.F,1.F)) by K2,K6,Z3,TARSKI:def 2;
          (n+i-j-1,n-1) --> (-1.F,1.F).u1 = -1.F by K2,K6,Y8,FUNCT_4:63;
          hence (<%0.F,1.F%> *' q).u
             = -1.F by K8,K5,FUNCT_4:13
            .= (-p1).u by K6,K7,BHSP_1:44;
          end;
        suppose K6: u = n;
          n in dom((k,n) --> (1.F,-1.F)) by TARSKI:def 2,B; then
          K7: p.n = ((k,n) --> (1.F,-1.F)).n by FUNCT_4:13
                 .= -1.F by FUNCT_4:63;
          K8: u1 in dom((n+i-j-1,n-1)-->(-1.F,1.F)) by K2,K6,Z3,TARSKI:def 2;
          (n+i-j-1,n-1) --> (-1.F,1.F).u1 = 1.F by K2,K6,FUNCT_4:63;
          hence (<%0.F,1.F%> *' q).u
              = -(p1.u) by K6,K7,K8,K5,FUNCT_4:13
             .= (-p1).u by BHSP_1:44;
          end;
        suppose K6: u <> k & u <> n;
          then not u in dom((k,n) --> (1.F,-1.F)) by TARSKI:def 2; then
          K7: p.u = (0_.(F)).u by FUNCT_4:11 .= 0.F;
          u1 <> n+i-j-1 & u1 <> n - 1 by K2,K6; then
          not u1 in dom((n+i-j-1,n-1) --> (-1.F,1.F)) by TARSKI:def 2;
          then q.u1 = (0_.(F)).u1 by FUNCT_4:11
                   .= 0.F by ORDINAL1:def 12,FUNCOP_1:7;
          hence (<%0.F,1.F%> *' q).u
              = -(p1.u) by K7,K2,thE1
             .= (-p1).u by BHSP_1:44;
          end;
        end;
      end;
      hence thesis;
      end;
  eval(<%0.F,1.F%>,0.F) = 0.F + 0.F by POLYNOM5:47;
  then <%0.F,1.F%> is with_roots by POLYNOM5:def 7,POLYNOM5:def 8;
  hence contradiction by K,K1,X;
  end;
then reconsider q1 as non zero Polynomial of F by UPROOTS:def 5;
reconsider q = q1 as Element of the carrier of Polynom-Ring F
          by POLYNOM3:def 10;
X: F is Subring of E by FIELD_4:def 1;
now assume A0: a|^i = a|^j;
  A1: a|^k = a|^(i+h) .= a|^i * a|^h by BINOM:10
          .= a|^(j+h) by A0,BINOM:10 .= a|^n;
  A: deg q < deg ma
     proof
     D: now assume B0: deg q1 = deg ma1;
        B2: deg ma1 + 1 = (len ma1 - 1) + 1 &
            deg q1 + 1 = (len q1 - 1) + 1 by HURWITZ:def 2;
        B3: len ma1 -' 1 = n by B2,XREAL_0:def 2;
        n in dom((k,n) --> (1.F,-1.F)) by TARSKI:def 2,B; then
        p.n = ((k,n) --> (1.F,-1.F)).n by FUNCT_4:13
           .= -1.F by FUNCT_4:63;
        then q1.n = -1.F + ma1.n by NORMSP_1:def 2
                 .= -1.F + LC ma1 by B3,RATFUNC1:def 6
                 .= -1.F + 1.F by RATFUNC1:def 7;
        hence contradiction by B2,B0,ALGSEQ_1:10,RLVECT_1:5;
        end;
     now assume G: deg p1 > n;
       B3: deg p1 + 1 = (len p1 - 1) + 1 by HURWITZ:def 2;
       not deg p1 in dom((k,n) --> (1.F,-1.F)) by G,Y9,TARSKI:def 2; then
       p1.(deg p1) = (0_.(F)).(deg p1) by FUNCT_4:11 .= 0.F;
       hence contradiction by B3,ALGSEQ_1:10;
       end;
     then max(deg p1, deg ma1) = deg ma1 by XXREAL_0:def 10;
     then deg q <= deg ma1 by HURWITZ:22;
     hence thesis by D,XXREAL_0:1;
     end;
  reconsider pE = anpoly(1.E,k) + anpoly(-1.E,n) as
            Element of the carrier of (Polynom-Ring E) by POLYNOM3:def 10;
  p = anpoly(1.E,k) + anpoly(-1.E,n)
    proof
    set g = anpoly(1.E,k) + anpoly(-1.E,n);
    now let u be Element of NAT;
    per cases;
    suppose H: u = k;
      k in dom((k,n) --> (1.F,-1.F)) by TARSKI:def 2,B; then
      H1: p.k = ((k,n) --> (1.F,-1.F)).k by FUNCT_4:13
             .= 1.F by Y4,FUNCT_4:63;
      thus g.u = anpoly(1.E,k).u + anpoly(-1.E,n).u by NORMSP_1:def 2
              .= anpoly(1.E,k).u + 0.E by Y4,H,POLYDIFF:25
              .= 1.E + 0.E by H,POLYDIFF:24
              .= p.u by H1,H,X,C0SP1:def 3;
      end;
    suppose H: u = n;
      n in dom((k,n) --> (1.F,-1.F)) by TARSKI:def 2,B; then
      H1: p.n = ((k,n) --> (1.F,-1.F)).n by FUNCT_4:13
             .= -1.F by FUNCT_4:63;
      H2: 1.E = 1.F by X,C0SP1:def 3;
      thus g.u = anpoly(1.E,k).u + anpoly(-1.E,n).u by NORMSP_1:def 2
              .= 0.E + anpoly(-1.E,n).u by Y4,H,POLYDIFF:25
              .= 0.E + -1.E by H,POLYDIFF:24
              .= p.u by H2,H1,H,X,Th19;
      end;
    suppose H: u <> k & u <> n;
      then not u in dom((k,n) --> (1.F,-1.F)) by TARSKI:def 2; then
      H1: p.u = (0_.(F)).u by FUNCT_4:11 .= 0.F;
      thus g.u = anpoly(1.E,k).u + anpoly(-1.E,n).u by NORMSP_1:def 2
              .= 0.E + anpoly(-1.E,n).u by H,POLYDIFF:25
              .= 0.E + 0.E by H,POLYDIFF:25
              .= p.u by H1,X,C0SP1:def 3;
      end;
    end;
    hence thesis;
    end; then
  B: Ext_eval(p,a)
       = eval(pE,a) by FIELD_4:26
      .= eval(anpoly(1.E,k),a) + eval(anpoly(-1.E,n),a) by POLYNOM4:19
      .= 1.E * a|^k + eval(anpoly(-1.E,n),a) by Y3,FIELD_1:6
      .= 1.E * a|^k + (-1.E) * a|^k by A1,AS,FIELD_1:6
      .= (1.E + (-1.E)) * a|^k by VECTSP_1:def 3
      .= 0.E * a|^k by RLVECT_1:5;
  Ext_eval(q,a) = Ext_eval(p,a) + Ext_eval(ma,a) by X,ALGNUM_1:15
               .= 0.E + 0.E by B,mpol3;
  hence contradiction by mpol4,A,RING_5:13;
  end;
hence thesis;
end;
