reserve x, x1, x2, y, X, D for set,
  i, j, k, l, m, n, N for Nat,
  p, q for XFinSequence of NAT,
  q9 for XFinSequence,
  pd, qd for XFinSequence of D;
reserve pN, qN for Element of NAT^omega;
reserve seq1,seq2,seq3,seq4 for Real_Sequence,
  r,s,e for Real,
  Fr,Fr1, Fr2 for XFinSequence of REAL;

theorem
  for r ex Catal be Real_Sequence st (for n holds Catal.n = Catalan(n+1)
* r|^n) & (|.r.| < 1/4 implies Catal is absolutely_summable & Sum Catal = 1 + r
  * (Sum Catal) |^ 2 & Sum Catal = 2 / (1 + sqrt(1-4*r)) & (r<>0 implies Sum
  Catal = (1 - sqrt(1-4*r)) / (2*r)) )
proof
  defpred E[object,object] means
   for r st $1=r ex Catal be Real_Sequence st (for n
  holds Catal.n = Catalan(n+1) * r|^n) & (|.r.| < 1/4 implies Catal is
  absolutely_summable & Sum Catal = 1 + r * (Sum Catal) |^ 2 & $2=Sum Catal);
A1: for x being object st x in REAL ex y being object st y in REAL & E[x,y]
  proof
    let x be object;
A2: |.1 .|=1 by ABSVALUE:def 1;
    assume x in REAL;
    then reconsider r=x as Real;
    set a=4*|.r.|;
    deffunc C(Nat)= In(Catalan($1+1) * r|^$1,REAL);
    consider Cat be Real_Sequence such that
A3: for n be Element of NAT holds Cat.n = C(n) from FUNCT_2:sch 4;
    set G=a GeoSeq;
    defpred P[Nat] means abs(Cat).$1 <= G.$1;
A4: for n st P[n] holds P[n+1]
    proof
A5:   |.r.|>=0 by COMPLEX1:46;
      let n;
      assume P[n];
      then
A6:   a*abs(Cat).n <= a*G.n by A5,XREAL_1:64;
      set n1=n+1;
A7:   |.r|^n1.| >=0 & r|^n1= r*r|^n by COMPLEX1:46,NEWTON:6;
      Catalan(n1+1)>=0 by CATALAN1:17;
      then
A8:   |.Catalan(n1+1).|=Catalan(n1+1) by ABSVALUE:def 1;
      Catalan(n1)>=0 by CATALAN1:17;
      then |.Catalan(n1).|=Catalan(n1) by ABSVALUE:def 1;
      then |.Catalan(n1+1).| < 4*|.Catalan(n1).| by A8,CATALAN1:21;
      then
A9:   |.r|^n1.|*|.Catalan(n1+1).|<=(4*|.Catalan(n1).|)*|.r*r|^n.| by A7,
XREAL_1:64;
      |.r*r|^n.| =|.r.|*|.r|^n.| by COMPLEX1:65;
      then
      |.C(n1).| <=a*(|.Catalan(n1).|*|.r|^n.|) by A9,COMPLEX1:65;
      then |.Cat.n1.|<=a*(|.Catalan(n1).|*|.r|^n.|) by A3;
      then |.Cat.n1.|<=a*|.C(n).| & n in NAT
        by COMPLEX1:65,ORDINAL1:def 12;
      then
A10:  |.Cat.n1.|<=a*|.Cat.n.| by A3;
      |.Cat.n.|=abs(Cat).n by SEQ_1:12;
      then abs(Cat).n1<=a*abs(Cat).n by A10,SEQ_1:12;
      then abs(Cat).n1<=a*G.n by A6,XXREAL_0:2;
      hence thesis by PREPOWER:3;
    end;
    Cat.0=C(0) by A3;
    then
A11: abs(Cat).0=|.r|^0.| by CATALAN1:11,SEQ_1:12;
    r|^0=1 & a|^0= 1 by NEWTON:4;
    then
A12: P[0] by A11,A2,PREPOWER:def 1;
 for n holds P[n] from NAT_1:sch 2(A12,A4);
then A13: for n be Nat holds P[n];
A14: now
      let n be Nat;
      abs(Cat).n=|.Cat.n.| by SEQ_1:12;
      hence abs(Cat).n>=0 by COMPLEX1:46;
    end;
    take Sum Cat;
    thus Sum Cat in REAL by XREAL_0:def 1;
    let s such that
A15: x=s;
    for y being object st y in NAT holds (Cat^\1).y=(Cat (##) (r(#)Cat)).y
    proof
      let y be object;
      assume y in NAT;
      then reconsider n=y as Nat;
      set n1=n+1;
      consider Fr1 such that
A16:  dom Fr1 = n1 and
A17:  for i st i in n+1 holds Fr1.i = Cat.i * (r(#)Cat).(n-'i) and
A18:  Sum Fr1 = (Cat (##) (r(#)Cat)).n by Def4;
      consider Catal be XFinSequence of NAT such that
A19:  Sum Catal = Catalan (n1+1) and
A20:  dom Catal = n1 and
A21:  for j st j<n1 holds Catal.j=Catalan(j+1)*Catalan(n1-'j) by Th39;
      reconsider CatalR=Catal as XFinSequence of REAL;
      defpred Q[set,set] means for k st k=$1 holds $2 = r |^ n1 * Catal.k;
A22:  for k st k in Segm n1 ex x be Element of REAL st Q[k,x]
      proof
        let k such that
        k in Segm n1;
         reconsider rr = r |^ n1 * Catal.k as Element of REAL
    by XREAL_0:def 1;
        take rr;
        thus thesis;
      end;
      consider Fr2 such that
A23:  dom Fr2 = Segm n1 and
A24:  for k be Nat st k in Segm n1 holds Q[k,Fr2.k] from STIRL2_1:sch 5(A22);
A25:  now
        let k be Nat such that
A26:    k in dom Fr2;
        k < len Fr2 by A26,AFINSQ_1:86;
        then
A27:    k<n+1 by A23;
        then
A28:    n1-'k=n1-k by XREAL_1:233;
A29:    n=k+(n-k);
        k<=n by A27,NAT_1:13;
        then
A30:    n-'k=n-k by XREAL_1:233;
        then Fr1.k = Cat.k * (r(#)Cat).(n-k) by A17,A23,A26
          .= C(k) * (r(#)Cat).(n-k) by A26,A3
          .=Catalan(k+1) * r|^k * (r(#)Cat).(n-k)
          .=Catalan(k+1) * r|^k * (r * Cat.(n-k)) by A30,SEQ_1:9
          .=Catalan(k+1) * r|^k*(r*C(n-'k)) by A3,A30
          .=Catalan(k+1) * Catalan(n1-'k) * r *(r|^k*r|^(n-'k)) by A30,A28
          .=Catalan(k+1) * Catalan(n1-'k) * r *(r|^n) by A30,A29,NEWTON:8
          .=Catal.k * r * r|^n by A21,A27
          .=Catal.k * (r * r|^n)
          .=Catal.k * r|^n1 by NEWTON:6
          .=Fr2.k by A23,A24,A26;
        hence Fr1.k=Fr2.k;
      end;
      for k st k in len Fr2 holds Fr2.k= r|^n1 * CatalR.k by A23,A24;
      then Sum Fr2 = r|^n1 * (Sum CatalR) by A20,A23,Th44
        .=C(n1) by A19
        .=Cat.n1 by A3
        .=(Cat^\1).n by NAT_1:def 3;
      hence thesis by A16,A18,A23,A25,AFINSQ_1:8;
    end;
    then
A31: Cat^\1= Cat(##)(r(#)Cat) by FUNCT_2:12;
    |.r.|>=0 by COMPLEX1:46;
    then
A32: |.a.|=a by ABSVALUE:def 1;
    take Cat;
   hereby let  n;
  n in NAT by ORDINAL1:def 12;
    hence Cat.n = C(n) by A3
      .= Catalan(n+1) * s|^n by A15;
   end;
A33: r|^0=1 by NEWTON:4;
    Cat.0 = C(0) by A3
       .=Catalan((0 qua Nat)+1)*r|^0;
    then
A34: Partial_Sums(Cat).0=1 by A33,CATALAN1:11,SERIES_1:def 1;
    assume |.s.| <1/4;
    then a<4*(1/4) by A15,XREAL_1:68;
    then G is summable by A32,SERIES_1:24;
    then abs(Cat) is summable by A14,A13,SERIES_1:20;
    hence
A35: Cat is absolutely_summable by SERIES_1:def 4;
    then Cat is summable;
    then r(#)Cat is summable & Sum(r(#)Cat)=r*Sum Cat by SERIES_1:10;
    then Sum(Cat^\((0 qua Nat)+1))=Sum Cat *(r*Sum Cat) by A35,A31,Th53;
    then Sum Cat= 1 + r*(Sum Cat*Sum Cat) by A35,A34,SERIES_1:15;
    hence thesis by A15,WSIERP_1:1;
  end;
  consider SumC be Function of REAL,REAL such that
A36: for x being object st x in REAL holds E[x,SumC.x] from FUNCT_2:sch 1(A1);
A37: for r,s st 0<s & s <= r & r < 1/4 holds SumC.s <= SumC.r
  proof
    let r,s such that
A38: 0<s and
A39: s <= r and
A40: r < 1/4;
    r in REAL by XREAL_0:def 1;
    then consider Cr be Real_Sequence such that
A41: for n holds Cr.n = Catalan(n+1) * r|^n and
A42: |.r.| < 1/4 implies Cr is absolutely_summable & Sum Cr = 1 + r *
    (Sum Cr) |^ 2 & SumC.r=Sum Cr by A36;
    s in REAL by XREAL_0:def 1;
    then consider Cs be Real_Sequence such that
A43: for n holds Cs.n = Catalan(n+1) * s|^n and
A44: |.s.| < 1/4 implies Cs is absolutely_summable & Sum Cs = 1 + s *
    (Sum Cs) |^ 2 & SumC.s=Sum Cs by A36;
A45: now
      let n be Nat;
      s|^n <= r|^n & Catalan(n+1)>=0 by A38,A39,CATALAN1:17,PREPOWER:9;
      then
A46:  Catalan(n+1) * s|^n <= Catalan(n+1)*r|^n by XREAL_1:64;
      Catalan(n+1)*r|^n=Cr.n by A41;
      hence Cs.n <= Cr.n by A43,A46;
    end;
A47: s < 1/4 by A39,A40,XXREAL_0:2;
    thus thesis by A38,A39,A40,A47,A44,A42,A45,ABSVALUE:def 1,TIETZE:5;
  end;
  set R={r where r is Real:0<r & r<1/4 &
         SumC.r=(1+sqrt(1-4*r))/(2*r)};
A48: for r st r<>0 & |.r.| < 1/4 holds SumC.r=(1-sqrt(1-4*r))/(2*r) or SumC.
  r=(1+sqrt(1-4*r))/(2*r)
  proof
    let r such that
A49: r<>0 and
A50: |.r.| < 1/4;
    r<=1/4 by A50,ABSVALUE:5;
    then 4*r <= (1/4)*4 by XREAL_1:64;
    then
A51: 4*r-4*r<=1-4*r by XREAL_1:9;
    r in REAL by XREAL_0:def 1;
    then consider Catal be Real_Sequence such that
    for n holds Catal.n = Catalan(n+1) * r|^n and
A52: |.r.| < 1/4 implies Catal is absolutely_summable & Sum Catal = 1
    + r * (Sum Catal) |^ 2 & SumC.r=Sum Catal by A36;
    set S=Sum Catal;
    S=1+r*S^2 by A50,A52,WSIERP_1:1;
    then
A53: r*S^2+(-1)*S+1=0;
    delta(r,-1,1)= 1-4*r & -(-1)=1;
    hence thesis by A49,A50,A52,A53,A51,FIB_NUM:6;
  end;
A54: for r,s st 0<r & r<s & s<1/4 holds (1+sqrt(1-4*r))/(2*r) > (1+sqrt(1-4*
  s))/(2*s)
  proof
    let r,s such that
A55: 0<r and
A56: r<s and
A57: s<1/4;
    4*s<4*(1/4) by A57,XREAL_1:68;
    then
A58: 4*s-4*s < 1-4*s by XREAL_1:9;
    then
A59: sqrt(1-4*s)>0 by SQUARE_1:25;
    4*r < 4*s by A56,XREAL_1:68;
    then 1-4*r >= 1-4*s by XREAL_1:10;
    then sqrt(1-4*r)>= sqrt(1-4*s) by A58,SQUARE_1:26;
    then 1+sqrt(1-4*r)>=1+sqrt(1-4*s) by XREAL_1:7;
    then
A60: (1+sqrt(1-4*r))/(2*r)>=(1+sqrt(1-4*s))/(2*r) by A55,XREAL_1:72;
    2*r>2*0 & 2*r<2*s by A55,A56,XREAL_1:68;
    then (1+sqrt(1-4*s))/(2*r)>(1+sqrt(1-4*s))/(2*s) by A59,XREAL_1:76;
    hence thesis by A60,XXREAL_0:2;
  end;
A61: R={}
  proof
    assume R<>{};
    then consider x being object such that
A62: x in R by XBOOLE_0:def 1;
    consider r be Real such that
    x=r and
A63: 0<r and
A64: r<1/4 and
A65: SumC.r=(1+sqrt(1-4*r))/(2*r) by A62;
    consider s be Real such that
A66: r<s and
A67: s< 1/4 by A64,XREAL_1:5;
A68: |.s.|=s by A63,A66,ABSVALUE:def 1;
    4*s<4*(1/4) by A67,XREAL_1:68;
    then 4*s-4*s < 1-4*s by XREAL_1:9;
    then sqrt(1-4*s)>0 by SQUARE_1:25;
    then 1-sqrt(1-4*s) <= 1-0 by XREAL_1:10;
    then
A69: (1-sqrt(1-4*s))/(2*s)<=1/(2*s) by A63,A66,XREAL_1:72;
A70: 2*r>2*0 by A63,XREAL_1:68;
    R c= {r}
    proof
      assume not R c= {r};
      then R\{r}<>{} by XBOOLE_1:37;
      then consider y being object such that
A71:  y in R\{r} by XBOOLE_0:def 1;
      y in R by A71;
      then consider s be Real such that
A72:  y=s and
A73:  0<s and
A74:  s<1/4 and
A75:  SumC.s=(1+sqrt(1-4*s))/(2*s);
A76:  r<>s by A71,A72,ZFMISC_1:56;
      now
        per cases by A76,XXREAL_0:1;
        suppose
A77:      r>s;
          then SumC.s > SumC.r by A54,A64,A65,A73,A75;
          hence contradiction by A37,A64,A73,A77;
        end;
        suppose
A78:      r<s;
          then SumC.r > SumC.s by A54,A63,A65,A74,A75;
          hence contradiction by A37,A63,A74,A78;
        end;
      end;
      hence contradiction;
    end;
    then not s in R by A66,TARSKI:def 1;
    then SumC.s<>(1+sqrt(1-4*s))/(2*s) by A63,A66,A67;
    then
A79: SumC.s=(1-sqrt(1-4*s))/(2*s) by A48,A63,A66,A67,A68;
    4*r<4*(1/4) by A64,XREAL_1:68;
    then 4*r-4*r < 1-4*r by XREAL_1:9;
    then sqrt(1-4*r)>0 by SQUARE_1:25;
    then 1+(0 qua Nat)<1+sqrt(1-4*r) by XREAL_1:8;
    then
A80: 1/(2*r) < SumC.r by A65,A70,XREAL_1:74;
    2*r<2* s by A66,XREAL_1:68;
    then 1/(2*s)<1/(2*r) by A70,XREAL_1:76;
    then SumC.s<1/(2*r) by A79,A69,XXREAL_0:2;
    then SumC.s<SumC.r by A80,XXREAL_0:2;
    hence thesis by A37,A63,A66,A67;
  end;
A81: for r st 0<r & |.r.| <1/4 holds SumC.r=(1-sqrt(1-4*r))/(2*r)
  proof
    let r such that
A82: 0<r and
A83: |.r.|<1/4;
    assume SumC.r<>(1-sqrt(1-4*r))/(2*r);
    then
A84: SumC.r=(1+sqrt(1-4*r))/(2*r) by A48,A82,A83;
    |.r.|=r by A82,ABSVALUE:def 1;
    then r in R by A82,A83,A84;
    hence thesis by A61;
  end;
A85: for r st r<0 & |.r.|<1/4 holds SumC.r=(1-sqrt(1-4*r))/(2*r)
  proof
    let r such that
A86: r<0 and
A87: |.r.|<1/4;
    2*r<2*0 by A86,XREAL_1:68;
    then
A88: |.2*r.|=-(2*r) & 0 qua Nat-2*r > 0 qua Nat-0 by ABSVALUE:def 1;
A89: |.-r.|<1/4 by A87,COMPLEX1:52;
    then 1/4>= -r by ABSVALUE:5;
    then 4*(1/4)>=4*(-r) by XREAL_1:64;
    then 1-4*(-r)>=4*(-r)-4*(-r) by XREAL_1:9;
    then sqrt(1-4*(-r))>=0 by SQUARE_1:def 2;
    then
A90: 1-sqrt(1-4*(-r))<=1-0 by XREAL_1:10;
A91: sqrt(1-4*r)>0 by A86,SQUARE_1:25;
    then 1+sqrt(1-4*r)> 1+(0 qua Nat) by XREAL_1:8;
    then
A92: 1-sqrt(1-4*(-r)) < 1+sqrt(1-4*r) by A90,XXREAL_0:2;
    1+sqrt(1-4*r)=|.1+sqrt(1-4*r).| by A91,ABSVALUE:def 1;
    then
A93: (1-sqrt(1-4*(-r)))/(2*(-r))<|.1+sqrt(1-4*r).|/|.2*r.| by A92,A88,
XREAL_1:74;
    -r in REAL by XREAL_0:def 1;
    then E[-r,SumC.-r] by A36;
    then
for s st -r=s ex Catal be Real_Sequence st (for n
  holds Catal.n = Catalan(n+1) * s|^n) & (|.s.| < 1/4 implies Catal is
  absolutely_summable & Sum Catal = 1 + s * (Sum Catal) |^ 2
      & SumC.-r=Sum Catal);
     then consider CR be Real_Sequence such that
 A94: for n holds CR.n = Catalan(n+1) * (-r)|^n and
 A95: |.-r.| < 1/4 implies CR is absolutely_summable & Sum CR = 1 +
     (-r) * (Sum CR) |^ 2 & SumC.(-r)=Sum CR;
    assume
A96: SumC.r<>(1-sqrt(1-4*r))/(2*r);
    r in REAL by XREAL_0:def 1;
    then consider Cr be Real_Sequence such that
A97: for n holds Cr.n = Catalan(n+1) * (r|^n) and
A98: |.r.| < 1/4 implies Cr is absolutely_summable & Sum Cr = 1 + r *
    (Sum Cr) |^ 2 & SumC.r=Sum Cr by A36;
    now
      let x be object;
      assume x in NAT;
      then reconsider n=x as Element of NAT;
      (-r)|^n=((-1) *r)|^n .=(-1)|^n* r|^n by NEWTON:7;
      then
A99: |.(-r)|^n.|=|.(-1)|^n.|*|.r|^n.| by COMPLEX1:65
        .=1*|.r|^n.| by SERIES_2:1;
      Catalan(n+1)>=0 by CATALAN1:17;
      then
A100: |.Catalan(n+1).|=Catalan(n+1) by ABSVALUE:def 1;
      (-r)|^ n>=0 by A86,POWER:3;
      then |.(-r)|^ n.|=(-r)|^ n by ABSVALUE:def 1;
      then CR.n=|.r|^n.|*|.Catalan(n+1).| by A94,A99,A100
        .=|.r|^n*Catalan(n+1).| by COMPLEX1:65
        .=|.Cr.n.| by A97
        .=abs(Cr).n by SEQ_1:12;
      hence abs(Cr).x=CR.x;
    end;
    then
A101: abs(Cr)=CR by FUNCT_2:12;
    0 qua Nat-r > 0 qua Nat-0 by A86;
    then
A102: Sum CR=(1-sqrt(1-4*(-r)))/(2*(-r)) by A81,A95,A89;
    |.Sum Cr.|<=Sum abs Cr by A87,A98,TIETZE:6;
    then |.(1+sqrt(1-4*r))/(2*r).|<= Sum CR by A48,A86,A87,A98,A101,A96;
    hence thesis by A102,A93,COMPLEX1:67;
  end;
  let r;
  r in REAL by XREAL_0:def 1;
  then consider Cat be Real_Sequence such that
A103: for n holds Cat.n = Catalan(n+1) * r|^n and
A104: |.r.| < 1/4 implies Cat is absolutely_summable & Sum Cat = 1 + r *
  (Sum Cat) |^ 2 & SumC.r=Sum Cat by A36;
  set s=sqrt(1-4*r);
  take Cat;
  thus for n holds Cat.n = Catalan(n+1) * r|^n by A103;
  assume
A105: |.r.|<1/4;
  hence Cat is absolutely_summable & Sum Cat = 1 + r*(Sum Cat)|^ 2 by A104;
A106: r<>0 implies Sum Cat = (1 - sqrt(1-4*r)) / (2*r)
  proof
    assume r<>0;
    then r>0 or r<0;
    hence thesis by A81,A85,A104,A105;
  end;
  now
    per cases;
    suppose
      r=0;
      hence 2 / (1 + s)=Sum Cat by A104,A105;
    end;
    suppose r<>0;
      then
A108: 2*r<>0;
      r<=1/4 by A105,ABSVALUE:5;
      then 4*r <=4*(1/4) by XREAL_1:64;
      then
A109: 1-4*r >= 4*r-4*r by XREAL_1:9;
      then s >=0 by SQUARE_1:def 2;
      then (1+s)/(1+s)=1 by XCMPLX_1:60;
      then (1-s) / (2*r)=((1-s) / (2*r)) * ((1+s)/(1+s))
        .=((1-s)*(1+s)) / ((2*r)*(1+s)) by XCMPLX_1:76
        .=(1^2-s^2)/ ((2*r)*(1+s))
        .=(1-(1-4*r))/ ((2*r)*(1+s)) by A109,SQUARE_1:def 2
        .=((2*r)*2)/((2*r)*(1+s))
        .=((2*r)/(2*r))*(2/(1+s)) by XCMPLX_1:76
        .=(1)*(2/(1+s)) by A108,XCMPLX_1:60;
      hence Sum Cat=2/(1+s) by A106;
    end;
  end;
  hence thesis by A106;
end;
