reserve i,j,n,n1,n2,m,k,u for Nat,
        r,r1,r2 for Real,
        x,y for Integer,
        a,b for non trivial Nat;

theorem Th12:
  ex Fy,Fx be FinSequence of NAT st
    Sum Fy = Py(a,n) & len Fy = [\ (n+1)/2 /] &
    (for i st 1 <= i <= (n+1)/2 holds
      Fy.i = (n choose (2*i-'1)) * a |^ (n+1-'2*i) * (a^2-'1) |^ (i-'1))&
      a|^n + Sum Fx = Px(a,n) & len Fx = [\ n/2 /] &
    for i st 1 <= i <= n/2 holds
      Fx.i = (n choose (2*i)) * a |^ (n-'2*i) * (a^2-'1) |^ i
proof
  set A = a^2-'1;
  defpred P[Nat] means
      ex Fy,Fx be FinSequence of NAT st
        Sum Fy = Py(a,$1) & len Fy = [\ ($1+1)/2 /] &
       (for i be Nat st 1 <= i <= ($1+1)/2 holds
         Fy.i = ($1 choose (2*i-'1)) * a |^ ($1+1-'2*i) * A |^ (i-'1))&
         a|^$1 + Sum Fx = Px(a,$1) & len Fx = [\ $1/2 /] &
       for i be Nat st 1 <= i <= $1/2 holds
         Fx.i = ($1 choose (2*i)) * a |^ ($1-'2*i) * A |^ i;
A1:P[0]
  proof
    set F = <*>NAT;
 A2:  len F = 0;
 A3:  0 = [\(0*2+1)/2/] by Lm1;
 A4:  for i be Nat st 1 <= i <= (0+1)/2 holds
    F.i = (0 choose (2*i-'1)) * a |^ (0+1-'2*i) * A |^ (i-'1) by XXREAL_0:2;
 A5:  Sum F = Py(a,0) by Th6;
    a|^0 = 1 by NEWTON:4;
    then
A6:   a|^0 + Sum F = Px(a,0) by Th6;
    for i be Nat st 1 <= i <= 0/2 holds
    F.i = (0 choose (2*i)) * a |^ (0-'2*i) * A |^ i;
    hence thesis by A2,A3,A4,A5,A6;
  end;
A7: for n be Nat st P[n] holds P[n+1]
  proof
    let n;
    set n1=n+1;
    assume P[n];
    then consider Fy,Fx be FinSequence of NAT such that
A8:   Sum Fy = Py(a,n) & len Fy = [\ n1/2 /] and
A9:   for i be Nat st 1 <= i <= n1/2 holds
        Fy.i = (n choose (2*i-'1)) * a |^ (n1-'2*i) * A |^ (i-'1)and
A10:  a|^n + Sum Fx = Px(a,n) & len Fx = [\ n/2 /] and
A11:  for i be Nat st 1 <= i <= n/2 holds
        Fx.i = (n choose (2*i)) * a |^ (n-'2*i) * A |^ i;
    set Fxa = <*a|^n*>^Fx;
    set Fx0=Fx^<*0*>;
    set Fy0=Fy^<*0*>;
A12:  Px(a,n1) = (a|^n + Sum Fx)*a + (Sum Fy)*A by Th9,A8,A10
      .= (a|^n *a) + (Sum Fx)*a + (Sum Fy)*A
      .= a|^n1 + (Sum Fx)*a + (Sum Fy)*A by NEWTON:6
      .= a|^n1 + Sum (a*Fx) + (Sum Fy)*A by RVSUM_1:87;
A13:  (Sum Fy)*A = Sum (A*Fy) by RVSUM_1:87;
A14:  Sum Fy0 = Sum Fy+0 by RVSUM_1:74;
A15:  Sum Fx0 = Sum Fx+0 by RVSUM_1:74;
A16:  Sum (a*Fx0) = a * Sum Fx0 by RVSUM_1:87;
A17:  Py(a,n1) = Px(a,n) + Py(a,n) * a by Th9
      .= Sum Fxa + (Sum Fy) *a by A8,A10,RVSUM_1:76;
    per cases;
    suppose
A18:    n is odd;
      then consider k such that
A19:    n = 2*k+1 by ABIAN:9;
      set k1=k+1;
A20:    [\ n1/2 /] = k1  & [\ n/2 /] = k by A19,Lm1;
      then
A21:    len Fxa =k1 & len (a*Fy) = k1 by A10,FINSEQ_5:8,A8,RVSUM_1:117;
      rng Fxa c= REAL & rng (a*Fy) c= REAL;
      then Fxa is FinSequence of REAL & a*Fy is FinSequence of REAL
        by FINSEQ_1:def 4;
      then reconsider FxA=Fxa,aFy=a*Fy as Element of k1-tuples_on REAL
        by A21,FINSEQ_2:92;
      rng (Fxa + a*Fy) c= NAT by VALUED_0:def 6;
      then reconsider FY = Fxa + a*Fy as FinSequence of NAT by FINSEQ_1:def 4;
A22:    len (a*Fx0) = len Fx0 = k1 & len (A*Fy) = k1
        by A20,A8,A10,FINSEQ_2:16,RVSUM_1:117;
      reconsider AFx0=a*Fx0,AFy=A*Fy as Element of k1-tuples_on REAL
        by A22,FINSEQ_2:92;
      rng (a*Fx0 + A*Fy) c= NAT by VALUED_0:def 6;
      then reconsider FX = a*Fx0 + A*Fy as FinSequence of NAT
        by FINSEQ_1:def 4;
      take FY,FX;
      thus Sum FY = Sum (FxA) + Sum aFy by RVSUM_1:89
        .= Py(a,n1) by A17,RVSUM_1:87;
A23:    n1+1 = 2*k1+1 by A19;
      len (FxA+aFy) = k1 by CARD_1:def 7;
      hence len FY = [\(n1+1)/2/] by A23,Lm1;
      thus for i be Nat st 1 <= i <= (n1+1)/2 holds
      FY.i = (n1 choose (2*i-'1)) * a |^ ((n1+1)-'2*i) * A |^ (i-'1)
      proof
        let i be Nat such that
A24:    1 <= i <= (n1+1)/2;
        set i1=i-'1;
A25:    i1 +1 = i by A24,XREAL_1:235;
        2*1 <=2*i by A24,XREAL_1:66;
        then
A26:    2*i-'1 = 2*i-1 by XREAL_1:233,XXREAL_0:2;
A27:    2*i <= n1+1 by A24,XREAL_1:83;
        then
A28:    2*i < n1+1 by A18,XXREAL_0:1;
        then
A29:    2*i <= n1 by NAT_1:13;
A30:    i <= n1/2 by A28,NAT_1:13,XREAL_1:77;
A31:    n1-'2*i = n1-2*i by A29,XREAL_1:233;
A32:    n1+1-'2*i = n1+1-2*i by XREAL_1:233,A27;
        then
A33:    n1+1-'2*i = 1+ (n1-'2*i) by A31;
        Fy.i = (n choose (2*i-'1)) * a |^ (n1-'2*i) * A |^ i1 by A24,A9,A30;
        then
A34:    (a*Fy).i = a*((n choose (2*i-'1)) * a |^ (n1-'2*i) * A |^ i1)
             by VALUED_1:6
          .= (n choose (2*i-'1)) * (a * a |^ (n1-'2*i)) * A |^ i1
          .= (n choose (2*i1+1)) * a |^ (n1+1-'2*i) * A |^ i1
             by A26,A25,A33,NEWTON:6;
        2*i <= n1+1 by A24,XREAL_1:83;
        then 2*i1+1+1 <= n1+1 by A25;
        then 2*i1+1 <= n1 by XREAL_1:8;
        then
A35:    2*i1 < n1 by NAT_1:13;
        then
A36:    2*i1 <=n by NAT_1:13;
        then
A37:    n-'2*i1 = n-2*i1 by XREAL_1:233;
A38:    Fxa.i = (n choose (2*i1)) * a |^ (n-'2*i1) * A |^ i1
        proof
          per cases;
          suppose
A39:        i > 1;
            then
A40:        i1 >=1 by A25,NAT_1:13;
            i1 <= n/2 by XREAL_1:77,A35,NAT_1:13;
            then
A41:          Fx.i1 = (n choose (2*i1)) * a |^ (n-'2*i1) * A |^ i1
              by A11,A39,A25,NAT_1:13;
            2*i1< 2*k+1 by A36,A19,XXREAL_0:1;
            then i1 <= k by XREAL_1:68,NAT_1:13;
            then i1 in dom Fx by A40,A20,A10,FINSEQ_3:25;
            hence thesis by A41,A25,FINSEQ_3:103;
          end;
          suppose i <=1;
            then
A42:          i=1 by A24,XXREAL_0:1;
A43:          n choose (2*i1) = 1 by A42,A25,NEWTON:19;
            n-'2*i1 = n-0 by A42,A25,XREAL_1:233;
            hence thesis by A43,NEWTON:4,A42,A25;
          end;
        end;
        thus FY.i = (FxA.i) + (aFy.i) by RVSUM_1:11
           .= ((n choose (2*i1))+(n choose (2*i1+1))) * a |^ (n1+1-'2*i) *
              A |^ i1 by A38,A34,A37,A32,A25
           .= (n1 choose (2*i-'1)) * a |^ (n1+1-'2*i) * A |^ i1
              by A26,A25,NEWTON:22;
      end;
      Sum FX = Sum AFx0 + Sum AFy by RVSUM_1:89
        .= Sum (a*Fx) + Sum (A*Fy) by RVSUM_1:87,A15,A16;
      hence a|^n1 + Sum FX = Px(a,n1) by A12,A13;
      len (AFx0 + AFy) = k1 by CARD_1:def 7;
      hence len FX = [\ n1/2 /]  by A19;
      thus for i be Nat st 1 <= i <= n1/2 holds
      FX.i = (n1 choose (2*i)) * a |^ (n1-'2*i) * A |^ i
      proof
        let i be Nat such that
A44:    1 <= i <= n1/2;
        set i1=i-'1;
        i1 +1 = i by A44,XREAL_1:235;
        then
A45:    A* A |^ i1 = A |^ i by NEWTON:6;
        Fy.i = (n choose (2*i-'1)) * a |^ (n1-'2*i) * A |^ i1 by A9,A44;
        then
A46:      (A*Fy).i = A*((n choose (2*i-'1)) * a |^ (n1-'2*i) * A |^ i1)
            by VALUED_1:6
          .= (n choose (2*i-'1)) * a |^ (n1-'2*i) * A |^ i by A45;
        2*i <= n1 by A44,XREAL_1:83;
        then
A47:      n1-'2*i=n1-2*i by XREAL_1:233;
A48:      (a*Fx0).i = a * (Fx0.i) by VALUED_1:6;
A49:      (a*Fx0).i  = (n choose (2*i)) * a |^ (n1-'2*i) * A |^ i
        proof
          per cases;
          suppose
A50:        i <= k;
            then
A51:        2*i < n by A19,NAT_1:13,XREAL_1:66;
            then i <= n / 2 by XREAL_1:77;
            then
A52:        Fx.i = (n choose (2*i)) * a |^ (n-'2*i) * A |^ i by A11,A44;
A53:        i in dom Fx by A44,A50,A10,A20,FINSEQ_3:25;
A54:        n-'2*i = n- 2*i by A51,XREAL_1:233;
            n1-'2*i = 1+ (n-'2*i) by A54,A47;
            then
A55:        a |^ (n1-'2*i) = a * a |^ (n-'2*i) by NEWTON:6;
            (a*Fx0).i = a*((n choose (2*i)) * a |^ (n-'2*i) * A |^ i)
                by A48,A52,A53,FINSEQ_1:def 7
              .= (n choose (2*i)) * a |^ (n1-'2*i) * A |^ i by A55;
            hence thesis;
          end;
          suppose i > k;
            then i >= k1 by NAT_1:13;
            then
A56:          2*i >= 2*k1 by XREAL_1:66;
            then
A57:          2*i >= n1 by A19;
            2*i <= n1 by A44,XREAL_1:83;
            then A58:2*i = 2*k1 by A56,A19,XXREAL_0:1;
            2*i> n by A57,NAT_1:13;
            then n choose (2*i)=0 by NEWTON:def 3;
            hence thesis by A58,A10,A20,A48;
          end;
        end;
A59:      2*i-'1+1 =2*i by XREAL_1:235,NAT_1:14,A44;
        thus FX.i = (AFx0).i + (AFy).i by RVSUM_1:11
          .= ((n choose (2*i-'1))+(n choose (2*i)))* a |^ (n1-'2*i) * A |^ i
            by A46,A49
          .= (n1 choose (2*i))* a |^ (n1-'2*i) * A |^ i by A59,NEWTON:22;
      end;
    end;
    suppose n is even;
      then consider k such that
A60:    n = 2*k by ABIAN:def 2;set k1=k+1;
A61:    [\ n1/2 /] = k  & [\ n/2 /] = k by A60,Lm1;
      then
A62:    len Fxa =k1 & len Fy0 = k1 & len (a*Fy0) = len Fy0
        by FINSEQ_2:16,FINSEQ_5:8,A8,A10,RVSUM_1:117;
      rng Fxa c= REAL & rng (a*Fy0) c= REAL;
      then Fxa is FinSequence of REAL & a*Fy0 is FinSequence of REAL
        by FINSEQ_1:def 4;
      then reconsider FxA=Fxa,aFy0=a*Fy0 as Element of k1-tuples_on REAL
        by A62,FINSEQ_2:92;
      rng (Fxa + a*Fy0) c= NAT by VALUED_0:def 6;
      then reconsider FY = Fxa + a*Fy0 as FinSequence of NAT by FINSEQ_1:def 4;
A63:    len (a*Fx) = len Fx = k & len (A*Fy) = k by A61,A8,A10,RVSUM_1:117;
      reconsider AFx=a*Fx,AFy=A*Fy as Element of k-tuples_on REAL
        by A63,FINSEQ_2:92;
      rng (a*Fx + A*Fy) c= NAT by VALUED_0:def 6;
      then reconsider FX = a*Fx + A*Fy as FinSequence of NAT by FINSEQ_1:def 4;
      take FY,FX;
      thus Sum FY = Sum (FxA) + Sum aFy0 by RVSUM_1:89
        .= Py(a,n1) by A17,A14,RVSUM_1:87;
      len (FxA+aFy0) = k1 by CARD_1:def 7;
      hence len FY = [\(n1+1)/2/] by A60;
      thus for i be Nat st 1 <= i <= (n1+1)/2 holds
        FY.i = (n1 choose (2*i-'1)) * a |^ ((n1+1)-'2*i) * A |^ (i-'1)
      proof
        let i be Nat such that
A64:      1 <= i <= (n1+1)/2;
        set i1=i-'1;
A65:      i1 +1 = i by A64,XREAL_1:235;
A66:      2*i <= n1+1 by A64,XREAL_1:83;
        then
A67:      2*i1+2 <= n+2 by A65;
        then
A68:      2*i1 <= n by XREAL_1:8;
A69:      n-'2*i1 = n-2*i1 & n1+1-'2*i = n1+1-2*i by A67,A66,XREAL_1:8,233;
        2*1 <=2*i by A64,XREAL_1:66;
        then
A70:      2*i-'1 = 2*i-1 by XREAL_1:233,XXREAL_0:2;
A71:      FxA.i = (n choose (2*i1)) * a |^ (n1+1-'2*i) * A |^ i1
        proof
          per cases;
          suppose
A72:          i > 1;
            then
A73:          i1 >=1 by A65,NAT_1:13;
            i1 <= n/2 by A68,XREAL_1:77;
            then
A74:          Fx.i1 = (n choose (2*i1)) * a |^ (n-'2*i1) * A |^ i1
              by A11,A72,A65,NAT_1:13;
            i1 <= k by A68,A60,XREAL_1:68;
            then i1 in dom Fx by A73,A61,A10,FINSEQ_3:25;
            hence thesis by A69,A74,A65,FINSEQ_3:103;
          end;
          suppose i <=1;
            then
A75:          i=1 by A64,XXREAL_0:1;
A76:          n choose (2*i1) = 1 by A75,A65,NEWTON:19;
            n1+1-'2*i = n+2-2*1 by A75,NAT_1:11,XREAL_1:233;
            hence thesis by A76,NEWTON:4,A75,A65;
          end;
        end;
A77:    (a*Fy0).i  = (n choose (2*i1+1)) * a |^ (n1+1-'2*i) * A |^ i1
        proof
          per cases;
          suppose
A78:          i <= k;
            then
A79:          2*i <= n1 by NAT_1:13,A60,XREAL_1:66;
            then
A80:          2*i <= n1+1 by NAT_1:13;
            i <= n1 / 2 by A79, XREAL_1:77;
            then
A81:          Fy.i = (n choose (2*i-'1)) * a |^ (n1-'2*i) * A |^ i1 by A9,A64;
            i in dom Fy by A64,A78,A8,A61,FINSEQ_3:25;
            then
A82:          Fy.i = Fy0.i by FINSEQ_1:def 7;
A83:          n1-'2*i = n1- 2*i & n1+1-'2*i = n1+1-2*i by A79,A80,XREAL_1:233;
            n1+1-'2*i = 1+ (n1-'2*i) by A83;
            then
A84:          a |^ (n1+1-'2*i) = a * a |^ (n1-'2*i) by NEWTON:6;
            (a*Fy0).i = a*((n choose (2*i-'1)) * a |^ (n1-'2*i) * A |^ i1)
              by A81,A82,VALUED_1:6
              .= (n choose (2*i-'1)) * a |^ (n1+1-'2*i) * A |^ i1 by A84;
            hence thesis by A70,A65;
          end;
          suppose
A85:          i > k;
            then i1 >=k by A65,NAT_1:13;
            then
A86:          2*i1+1 > n by NAT_1:13,A60,XREAL_1:66;
            i >= k1 by A85,NAT_1:13;
            then 2*i >= 2*k1 by XREAL_1:66;
            then 2*i = 2*k1 by A60,A66,XXREAL_0:1;
            then Fy0.i = 0 by A8,A61;
            then
A87:          (a*Fy0).i=a*0 by VALUED_1:6;
            n choose (2*i1+1)=0 by A86,NEWTON:def 3;
            hence thesis by A87;
          end;
        end;
        thus FY.i = (FxA.i) + (aFy0.i) by RVSUM_1:11
            .= ((n choose (2*i1))+
               (n choose (2*i1+1))) * a |^ (n1+1-'2*i) * A |^ i1 by A71,A77
            .= (n1 choose (2*i-'1)) * a |^ (n1+1-'2*i) * A |^ i1
              by A70,A65,NEWTON:22;
      end;
      Sum FX = Sum AFx + Sum AFy by RVSUM_1:89
        .= Sum (a*Fx) + Sum (A*Fy);
      hence a|^n1 + Sum FX = Px(a,n1) by A12,A13;
      len (AFx + AFy) = k by CARD_1:def 7;
      hence len FX = [\ n1/2 /]  by A60,Lm1;
      thus for i be Nat st 1 <= i <= n1/2 holds
      FX.i = (n1 choose (2*i)) * a |^ (n1-'2*i) * A |^ i
      proof
        let i be Nat such that
A88:      1 <= i <= n1/2;
        set i1=i-'1;
        i1 +1 = i by A88,XREAL_1:235;
        then
A89:      A* A |^ i1 = A |^ i by NEWTON:6;
        Fy.i = (n choose (2*i-'1)) * a |^ (n1-'2*i) * A |^ i1 by A9,A88;
        then
A90:      (A*Fy).i = A*((n choose (2*i-'1)) * a |^ (n1-'2*i) * A |^ i1)
            by VALUED_1:6
          .= (n choose (2*i-'1)) * a |^ (n1-'2*i) * A |^ i by A89;
A91:      2*i <= n1 by A88,XREAL_1:83;
        then
A92:      n1-'2*i=n1-2*i by XREAL_1:233;
A93:      2*i < n1 by A91,A60,XXREAL_0:1;
        then
A94:      2*i <= n by NAT_1:13;
A95:      i <= n/2 by A93,NAT_1:13, XREAL_1:77;
A96:     n-'2*i = n- 2*i by A94,XREAL_1:233;
        n1-'2*i = 1+ (n-'2*i) by A96,A92;
        then
A97:     a |^ (n1-'2*i) = a * a |^ (n-'2*i) by NEWTON:6;
A98:     (a*Fx).i = a * (Fx.i) by VALUED_1:6
          .= a*((n choose (2*i)) * a |^ (n-'2*i) * A |^ i) by A95,A11,A88
          .= (n choose (2*i)) * a |^ (n1-'2*i) * A |^ i by A97;
A99:   2*i-'1+1 =2*i by XREAL_1:235,NAT_1:14,A88;
        thus FX.i = (AFx).i + (AFy).i by RVSUM_1:11
          .= ((n choose (2*i-'1))+(n choose (2*i)))* a |^ (n1-'2*i) * A |^ i
             by A90,A98
          .= (n1 choose (2*i))* a |^ (n1-'2*i) * A |^ i by A99,NEWTON:22;
      end;
    end;
  end;
  for n holds P[n] from NAT_1:sch 2(A1,A7);
  hence thesis;
end;
