 reserve x,y,z for object, X for set,
         i,k,n,m for Nat,
         R for Relation,
         P for finite Relation,
         p,q for FinSequence;
reserve FSG for Friendship_Graph;

theorem
  FSG is non empty implies FSG is with_universal_friend
proof
  set F=field FSG;
  defpred Q1[FinSequence] means for i be Nat st 1<= i & i < len $1
    holds [$1.i,$1.(i+1)] in FSG;
  deffunc W(object) = Im(FSG,$1);
  assume A1:FSG is non empty without_universal_friend;
  then consider x be object such that
    A2:x in F by XBOOLE_0:def 1;
  reconsider F as non empty finite set by A2;
  set m = card Im(FSG,x);
  reconsider m1=m-1 as Nat by Th10,A1,A2,NAT_1:20;
  m1 +1 >2 by Th10,A1,A2;
  then consider p be Element of NAT such that
        A3: p is prime
    and A4: p divides m1 by NAT_1:13,INT_2:31;
  A5:p>1 by A3,INT_2:def 4;
  reconsider p as non zero Element of NAT by A3,INT_2:def 4;
  A6: 2 divides m by Th8;
  A7: p >2
  proof
    A8: m1+1=m;
    assume A9:p <= 2;
    p >= 1+1 by A5,NAT_1:13;
    then 2 divides m1 by A9,XXREAL_0:1,A4;
    then 2 divides 1 by A8,INT_2:1,A6;
    hence contradiction by INT_2:13;
  end;
  then reconsider p2 =p-2 as Nat by NAT_1:21;
  reconsider p1 =p-1 as Nat by NAT_1:20, A3,INT_2:def 4;
  set T1=p1-tuples_on F;
  set XFSG1 = {f where f is Element of T1:Q1[f] & f.1=f.p1};
  set XFSG2 = {f where f is Element of T1:Q1[f] & f.1<>f.p1};
  defpred C[object,object] means
for f be FinSequence st f=$1 holds $2=f|p1;
A10:for x being object st x in F holds W(x) in bool F
  proof
    let x be object;
    assume x in F;
    W(x) c= F
    proof
      let y be object;
      assume y in W(x);
      then [x,y] in FSG by RELAT_1:169;
      hence thesis by RELAT_1:15;
    end;
    hence thesis;
  end;
  consider IM be Function of F,bool F such that
A11:for x being object st x in F holds IM.x=W(x) from FUNCT_2:sch 2(A10);
  A12:for x st x in F holds card (IM.x)=m
  proof
    let y;
    assume A13:y in F;
    hence card (IM.y) = card W(y) by A11
         .= m by A2,Th11,A1,A13;
  end;
  defpred Q[FinSequence] means [$1.p,$1.1] in FSG & Q1[$1];
  set T = p-tuples_on F;
  set XT={f where f is Element of T:Q[f]};
  set XFSG = {f where f is Element of (p2+1)-tuples_on F: f.1 in F &
     for i be Nat st i in Seg p2 holds f.(i+1) in IM.(f.i)};
  F c= F;
  then A14:card XFSG = card F * (m|^p2) by A12,Th6;
  then reconsider XFSG as finite set;
  A15:for f be Element of p1-tuples_on F holds f in XFSG iff Q1[f]
  proof
    let f be Element of T1;
    A16: len f = p2+1 by CARD_1:def 7;
    hereby
      assume f in XFSG;
      then A17: ex g be Element of T1 st f=g & g.1 in F &
        for i be Nat st i in Seg p2 holds g.(i+1) in IM.(g.i);
      thus Q1[f]
      proof
        let i such that
              A18: 1<= i
          and A19: i < len f;
         i in dom f by A18,A19,FINSEQ_3:25;
        then f.i in rng f by FUNCT_1:def 3;
        then A20: W(f.i)=IM.(f.i) by A11;
        i <= p2 by A19,A16,NAT_1:13;
        then i in Seg p2 by A18;
        hence thesis by A17,A20,RELAT_1:169;
      end;
    end;
    assume A21:Q1[f];
    A22: for i be Nat st i in Seg p2 holds f.(i+1) in IM.(f.i)
    proof
      let i be Nat;
      assume A23: i in Seg p2;
      then A24: 1 <= i by FINSEQ_1:1;
      i <= p2 by A23,FINSEQ_1:1;
      then A25: i < p2+1 by NAT_1:13;
      then i in dom f by A24,A16,FINSEQ_3:25;
      then A26: f.i in rng f by FUNCT_1:def 3;
      [f.i,f.(i+1)] in FSG by A24,A25,A21,A16;
      then f.(i+1) in W(f.i) by RELAT_1:169;
      hence thesis by A26,A11;
    end;
    1<= len f by A16,NAT_1:11;
    then 1 in dom f by FINSEQ_3:25;
    then f.1 in rng f by FUNCT_1:def 3;
    hence thesis by A22;
  end;
  A27:XFSG1 c= XFSG
  proof
    let x be object;
    assume x in XFSG1;
    then ex f be Element of T1 st x= f & Q1[f] & f.1=f.p1;
    hence thesis by A15;
  end;
  A28:XFSG2 c= XFSG
  proof
    let x be object;
    assume x in XFSG2;
    then ex f be Element of T1 st x= f & Q1[f] & f.1<>f.p1;
    hence thesis by A15;
  end;
  then reconsider XFSG1,XFSG2 as finite set by A27;
  A29:p>1 by A3,INT_2:def 4;
  then not p divides (m1+1) by A4,NEWTON:39;
  then A30: not p divides (m|^p2) by A3,NAT_3:5;
  A31:XFSG1 misses XFSG2
  proof
    assume XFSG1 meets XFSG2;
    then consider x being object such that
    A32: x in XFSG1 & x in XFSG2 by XBOOLE_0:3;
      ( ex f be Element of T1 st x= f & Q1[f] & f.1=f.p1)&
      ex f be Element of T1 st x= f & Q1[f] & f.1<>f.p1 by A32;
    hence contradiction;
  end;
  A33:for x being object st x in XT ex y being object st y in XFSG & C[x,y]
  proof
    let x be object;
    assume x in XT;
    then consider f be Element of T such that
          A34: x=f
      and A35: Q[f];
    set g = f|p1;
    A36:len f = p1+1 by CARD_1:def 7;
    then p1 < len f by NAT_1:13;
    then A37:len g = p1 by FINSEQ_1:59;
    then reconsider g as Element of T1 by FINSEQ_2:92;
    take g;
    Q1[g]
    proof
      let i be Nat;
      assume that
            A38: 1 <= i
        and A39: i < len g;
      i in dom g by A38,A39, FINSEQ_3:25;
      then A40: f.i=g.i by FUNCT_1:47;
      i < len f by A39, A36,NAT_1:13,A37;
      then A41: [f.i,f.(i+1)] in FSG by A38,A35;
      1 < i+1 & i+1<= len g by A38,NAT_1:13, A39;
      then i+1 in dom g by FINSEQ_3:25;
      hence thesis by A40,FUNCT_1:47,A41;
    end;
    hence thesis by A15,A34;
  end;
  consider FSGR be Function of XT,XFSG such that
A42: for x being object st x in XT holds C[x,FSGR.x] from FUNCT_2:sch 1(A33);
  reconsider pr=FSGR~ as Relation;
  m>2 by Th10,A1,A2;
  then m|^p2>0 by NEWTON:83;
  then A43: XFSG is non empty by XREAL_1:129,A14;
  then A44:dom FSGR=XT by FUNCT_2:def 1;
  A45: XFSG c= XFSG1\/XFSG2
  proof
    let x be object;
    assume A46:x in XFSG;
    then consider f be Element of (p2+1)-tuples_on F such that
          A47: x=f
      and f.1 in F &
          for i be Nat st i in Seg p2 holds f.(i+1) in IM.(f.i);
    A48: f.1=f.p1 or f.1<>f.p1;
    Q1[f] by A15,A46,A47;
    then f in XFSG1 or f in XFSG2 by A48;
    hence thesis by XBOOLE_0:def 3,A47;
  end;
  A49:p=p1+1;
  then A50: p1>=1 by A29,NAT_1:13;
  A51:for f be Element of T1 st Q1[f] & [f.p1,y] in FSG & [y,f.1] in FSG
    holds f^<*y*> in XT
   proof
     let f be Element of T1 such that
           A52: Q1[f]
       and A53: [f.p1,y] in FSG
       and A54: [y,f.1] in FSG;
     set fy = f^<*y*>;
     y in F by A53,RELAT_1:15;
     then <*y*> is FinSequence of F by FINSEQ_1:74;
     then A55: fy is FinSequence of F by FINSEQ_1:75;
     A56: len fy = p1+1 by CARD_1:def 7;
     A57:len f = p1 by CARD_1:def 7;
     then A58:fy.p=y by FINSEQ_1:42;
     reconsider fy as Element of T by A55,A56,FINSEQ_2:92;
     A59: Q1[fy]
     proof
       let i such that
             A60: 1 <= i
         and A61: i < len fy;
       A62:i <= p1 by A61,A56,NAT_1:13;
       then A63:i in dom f by A60,A57,FINSEQ_3:25;
       then A64:f.i=fy.i by FINSEQ_1:def 7;
       per cases by A62,XXREAL_0:1;
         suppose i=p1;
           hence thesis by A58, A63,FINSEQ_1:def 7,A53;
         end;
         suppose A65: i<p1;
           then i+1<= p1 by NAT_1:13;
           then A66: i+1 in dom f by NAT_1:11,A57,FINSEQ_3:25;
           [f.i,f.(i+1)] in FSG by A65,A57,A52,A60;
           hence thesis by A66,FINSEQ_1:def 7,A64;
         end;
       end;
       1 in dom f by A50,A57,FINSEQ_3:25;
       then [fy.p,fy.1] in FSG by FINSEQ_1:def 7,A58,A54;
       hence thesis by A59;
     end;
     A67:for f be Element of T1 st Q1[f] & f.1 = f.p1
         for y st y in Im(FSG,f.1) holds f^<*y*> in XT
     proof
       let f be Element of T1 such that
         A68: Q1[f] & f.1 = f.p1;
       let y such that
         A69: y in Im(FSG,f.1);
       A70:[f.1,y] in FSG by RELAT_1:169,A69;
       then [y,f.1] in FSG by Lm3;
       hence thesis by A68,A70,A51;
     end;
     A71:for x st x in XFSG1 holds card Im(pr,x) = m
     proof
       let x;
       assume x in XFSG1;
       then consider f be Element of T1 such that
             A72: f=x
         and A73: Q1[f] & f.1 = f.p1;
       deffunc D(object) = f^<*$1*>;
       A74:len f = p1 by CARD_1:def 7;
       A75:for y being object st y in Im(FSG,f.1) holds D(y) in Im(pr,f)
       proof
         let y be object;
         assume A76:y in Im(FSG,f.1);
         then A77:D(y) in XT by A67,A73;
         then consider fy be Element of T such that
             A78: D(y) = fy
         and Q[fy];
         reconsider yy=<*y*> as FinSequence of F by A78,FINSEQ_1:36;
         FSGR.D(y) = (f^yy) |p1 by A76,A67,A73,A42
           .= f by A74,FINSEQ_5:23;
         then [fy,f] in FSGR by A77,A78, A44,FUNCT_1:def 2;
         then [f,fy] in pr by RELAT_1:def 7;
         hence thesis by A78,RELAT_1:169;
       end;
       consider d be Function of Im(FSG,f.1),Im(pr,f) such that
A79: for y being object st y in Im(FSG,f.1) holds d.y = D(y)
         from FUNCT_2:sch 2(A75);
       A80:d is one-to-one
       proof
         let x1,x2 be object;
         assume that
               A81: x1 in dom d
           and A82: x2 in dom d & d.x1=d.x2;
         d.x1=D(x1) by A81,A79;
         then D(x1)=D(x2) by A82,A79;
         hence thesis by FINSEQ_2:17;
       end;
       A83:1 in dom f by A74,A50,FINSEQ_3:25;
       then f.1 in rng f by FUNCT_1:def 3;
       then A84:card Im(FSG,f.1)=m by A1,Th11,A2;
       then Im(FSG,f.1) is non empty by Th10,A1,A2;
       then ex xx be object st xx in Im(FSG,f.1);
       then Im(pr,f) is non empty by A75;
       then A85:dom d = Im(FSG,f.1) by FUNCT_2:def 1;
       Im(pr,f) c=rng d
       proof
         let y be object;
         assume y in Im(pr,f);
         then [f,y] in pr by RELAT_1:169;
         then A86: [y,f] in FSGR by RELAT_1:def 7;
         then A87:y in dom FSGR by XTUPLE_0:def 12;
         then consider g be Element of T such that
               A88: y=g
           and A89: Q[g] by A44;
         A90: len g = p1+1 by CARD_1:def 7;
         f =FSGR.y by A87,A86,FUNCT_1:def 2;
         then g|p1 = f by A87,A42,A88;
         then A91:g = f ^ <*g.p*> by A90,FINSEQ_3:55;
         then g.1 = f.1 by A83,FINSEQ_1:def 7;
         then A92: [f.1,g.p] in FSG by A89,Lm3;
         then g.p in dom d by RELAT_1:169,A85;
         then d.(g.p) in rng d by FUNCT_1:def 3;
         hence thesis by A92,RELAT_1:169,A91,A79,A88;
       end;
       then Im(pr,f) =rng d;
       hence thesis by A85,A80,WELLORD2:def 4,A84,CARD_1:5,A72;
     end;
     XFSG1\/XFSG2 c= XFSG by A27,XBOOLE_1:8,A28;
     then A93:XFSG = XFSG1\/XFSG2 by A45;
     then card XFSG = card XFSG1+card XFSG2 by A31,CARD_2:40;
     then A94:(1+ m*m1) * (m|^p2) = card XFSG1+card XFSG2 by A14, A1,A2,Th12;
     A95:for f be Element of T1 st Q1[f] & f.1 <> f.p1 ex y st y in F &
        f^<*y*> in XT
     proof
       let f be Element of T1 such that
             A96: Q1[f]
         and A97: f.1 <> f.p1;
        A98: len f =p1 by CARD_1:def 7;
        then p1 in dom f by A50,FINSEQ_3:25;
        then A99: f.p1 in rng f by FUNCT_1:def 3;
        1 in dom f by A98,A50,FINSEQ_3:25;
        then f.1 in rng f by FUNCT_1:def 3;
        then consider w be object such that
           A100: {w} = Im(FSG,f.p1)/\Coim(FSG,f.1) by A99, A97,Def3;
        take w;
        A101: w in {w} by TARSKI:def 1;
        Coim(FSG,f.1) =Im(FSG,f.1) by Th2;
        then w in Im(FSG,f.1) by A101,A100,XBOOLE_0:def 4;
        then [f.1,w] in FSG by RELAT_1:169;
        then A102: [w,f.1] in FSG by Lm3;
        w in Im(FSG,f.p1) by A101,A100,XBOOLE_0:def 4;
        then [f.p1,w] in FSG by RELAT_1:169;
        hence thesis by A102,A51,A96,RELAT_1:15;
      end;
      A103:for x st x in XFSG2 holds card Im(pr | XFSG2,x) = 1
      proof
        let x such that
          A104:x in XFSG2;
        consider f be Element of T1 such that
              A105: x = f
          and A106: Q1[f]
          and A107: f.1<>f.p1 by A104;
        consider y such that
              y in F
          and A109: f^<*y*> in XT by A95,A106,A107;
        A111:len f = p1 by CARD_1:def 7;
        A112: Im(pr | XFSG2,f) c= {f^<*y*>}
        proof
          let z be object;
          consider w be Element of T such that
                A113: w = f^<*y*>
            and A114: Q[w] by A109;
          A115: w.p = y by A113,A111, FINSEQ_1:42;
          A116: p1 in dom f by A50,A111,FINSEQ_3:25;
          then A117: w.p1=f.p1 by A113,FINSEQ_1:def 7;
          A118: 1 in dom f by A50,A111,FINSEQ_3:25;
          then A119: w.1=f.1 by A113,FINSEQ_1:def 7;
          assume z in Im(pr | XFSG2,f);
          then [f,z] in pr | XFSG2 by RELAT_1:169;
          then [f,z] in pr by RELAT_1:def 11;
          then A120: [z,f] in FSGR by RELAT_1:def 7;
          then A121:z in dom FSGR by XTUPLE_0:def 12;
          then z in XT;
          then consider g be Element of T such that
               A122: g=z
            and A123: Q[g];
          A124: p1 < p by A49,NAT_1:13;
          f = FSGR.z by A121,FUNCT_1:def 2, A120;
          then A125: f = g|p1 by A121,A42,A122;
          then A126: g.1=f.1 by A118,FUNCT_1:47;
          len w = p by CARD_1:def 7;
          then A127: [w.p1,w.p] in FSG by A49,A29,NAT_1:13,A124,A114;
          A128: len g = p by CARD_1:def 7;
          then A129: g = f^<*g.p*> by A125,FINSEQ_3:55;
          A130: g.p1=f.p1 by A116,A125,FUNCT_1:47;
          [g.p1,g.p] in FSG by A49,A29,NAT_1:13,A124,A123,A128;
          then g.p = w.p by A119,A126,A117,A130,A123,A114,A107,A127,Lm5;
          hence thesis by A129,A115,A122,TARSKI:def 1;
        end;
        FSGR.(f^<*y*>) = (f^<*y*>) | p1 by A42,A109
        .= f by FINSEQ_5:23,A111;
        then [f^<*y*>,f] in FSGR by FUNCT_1:def 2,A44,A109;
        then [f,f^<*y*>] in pr by RELAT_1:def 7;
        then [f,f^<*y*>] in pr|XFSG2 by A105,A104,RELAT_1:def 11;
        then f^<*y*> in Im(pr | XFSG2,f) by RELAT_1:169;
        then Im(pr | XFSG2,f) = {f^<*y*>} by A112,ZFMISC_1:33;
        hence thesis by A105,CARD_1:30;
      end;
      dom (pr|XFSG2) \ XFSG2 = {} by RELAT_1:58,XBOOLE_1:37;
      then card ((pr| XFSG2) | (dom (pr|XFSG2) \ XFSG2 ))=0;
      then A131: card (pr|XFSG2) = 0 +` (1 *` card XFSG2) by A103,SIMPLEX1:1
      .= 1*` card XFSG2 by CARD_2:18
      .= card XFSG2 by CARD_2:21;
      XFSG c= rng FSGR
      proof
        let xp be object;
        assume A132: xp in XFSG;
        per cases by A132, A45,XBOOLE_0:def 3;
        suppose xp in XFSG1;
          then consider f be Element of T1 such that
                A133: xp=f
            and A134: Q1[f] & f.1 = f.p1;
          len f = p1 by CARD_1:def 7;
          then 1 in dom f by A50,FINSEQ_3:25;
          then f.1 in rng f by FUNCT_1:def 3;
          then card Im(FSG,f.1)=m by A1,Th11,A2;
          then Im(FSG,f.1) is non empty by Th10,A1,A2;
          then consider y be object such that
          A135:y in Im(FSG,f.1);
          set fy=y;
          set fy=f^<*y*>;
          f^<*y*> in XT by A67,A134,A135;
          then A136: FSGR.fy in rng FSGR by A44,FUNCT_1:def 3;
          [f.1,y] in FSG by A135,RELAT_1:169;
          then y in F by RELAT_1:15;
          then <*y*> is FinSequence of F by FINSEQ_1:74;
          then A138: len fy = p1+1 & fy is FinSequence of F
            by CARD_1:def 7,FINSEQ_1:75;
          A139: len f = p1 by CARD_1:def 7;
          FSGR.fy = fy|p1 by A42, A67,A134,A135;
          hence thesis by A139,FINSEQ_5:23,A133,A136;
          reconsider fy as Element of T by A138,FINSEQ_2:92;
        end;
        suppose xp in XFSG2;
          then consider f be Element of T1 such that
                A140: xp=f
            and A141: Q1[f] & f.1 <> f.p1;
          consider y such that
                A142: y in F
            and A143: f^<*y*> in XT by A141,A95;
          set fy=f^<*y*>;
          A144: FSGR.fy = fy|p1 & FSGR.fy in rng FSGR
            by A42,A143,A44,FUNCT_1:def 3;
          <*y*> is FinSequence of F by A142,FINSEQ_1:74;
          then len fy = p1+1 & fy is FinSequence of F
            by CARD_1:def 7,FINSEQ_1:75;
          then reconsider fy as Element of T by FINSEQ_2:92;
          len f = p1 by CARD_1:def 7;
         hence thesis by FINSEQ_5:23,A140,A144;
       end;
     end;
     then XFSG = rng FSGR;
     then dom pr = XFSG by RELAT_1:20;
     then dom pr \ XFSG1=XFSG2 by XBOOLE_1:88,A93,A31;
     then
     A146:card pr = card XFSG2 +` (m *` card XFSG1) by A131,A71,SIMPLEX1:1
     .= card XFSG2 +` (m * card XFSG1) by Lm1
     .= card XFSG2 + m*card XFSG1 by Lm1;
     A147:for f1,f2 be FinSequence of F st f1^f2 is p -element & Q[f1^f2]
       holds Q[f2^f1]
     proof
       let f1,f2 be FinSequence of F;
       assume that
       A148: f1^f2 is p -element
         and A149: Q[f1^f2];
       set f12=f1^f2,f21=f2^f1,L1=len f1,L2=len f2;
       A150: len f12 = p by CARD_1:def 7,A148;
       A151: len f21 = L2+L1 by FINSEQ_1:22;
       A152: len f12 = L1+L2 by FINSEQ_1:22;
       per cases;
         suppose f1={} or f2={};
           then f12=f2 & f21=f2 or f12=f1 & f21=f1 by FINSEQ_1:34;
           hence thesis by A149;
         end;
         suppose A153: f1<>{} & f2 <>{};
           then L2 >= 1 by FINSEQ_1:20;
           then A154: 1 in dom f2 by FINSEQ_3:25;
           then A155: f12.(L1+1) = f2.1 by FINSEQ_1:def 7;
           A156: L1+1 <= p by A153,FINSEQ_1:20,A150,A152,XREAL_1:6;
           A157: L1 >= 1 by A153,FINSEQ_1:20;
           then A158: 1 in dom f1 by FINSEQ_3:25;
           A159: L1 in dom f1 by A157,FINSEQ_3:25;
           then A160: f21.p = f1.L1 by A150,A152,FINSEQ_1:def 7;
           f12.L1 = f1.L1 by A159,FINSEQ_1:def 7;
           then [f1.L1,f2.1] in FSG by A156,NAT_1:13,A149,A157,A155, A150;
           hence [f21.p,f21.1] in FSG by A160, A154,FINSEQ_1:def 7;
           let i;
           assume that
                 A161: 1<= i
             and A162: i < len f21;
           A163:i in dom f21 by A161,A162,FINSEQ_3:25;
           per cases by A163,FINSEQ_1:25;
           suppose A164:i in dom f2;
             then A165: i <= L2 by FINSEQ_3:25;
             A166: f21.i = f2.i by A164,FINSEQ_1:def 7;
             per cases;
             suppose A167: i = L2;
               A168: f1.1 = f12.1 by A158,FINSEQ_1:def 7;
               f2.i = f12.p by A167,A150,A152,A164,FINSEQ_1:def 7;
               hence thesis by A168, A167,A158,FINSEQ_1:def 7,A149,A166;
             end;
             suppose A169: i <>L2;
               A170: 1+0 <= L1+i by XREAL_1:7,A157;
               A171: i < L2 by A169,A165,XXREAL_0:1;
               then L1+i < p by A150,A152,XREAL_1:6;
               then A172: [f12.(L1+i),f12.(L1+i+1)] in FSG by A170,A149, A150;
               A173: f12.(L1+i)=f2.i by A164, FINSEQ_1:def 7;
               i+1<=L2 by A171, NAT_1:13;
               then A174: i+1 in dom f2 by NAT_1:11,FINSEQ_3:25;
               then f12.(L1+(i+1)) = f2.(i+1) by FINSEQ_1:def 7;
               hence thesis by A174, FINSEQ_1:def 7, A172,A173, A166;
             end;
           end;
           suppose ex n be Nat st n in dom f1 & i=L2 + n;
             then consider n be Nat such that
             A175: n in dom f1
             and
             A176: i=L2 + n;
             A177: f21.i = f1.n & f12.n = f1.n by A175,A176,FINSEQ_1:def 7;
             A178: 1<= n by A175,FINSEQ_3:25;
             A179: n < L1 by A176,A162,A151,XREAL_1:6;
             then A180: n+0 < L1+L2 by XREAL_1:8;
             n+1 <= L1 by A179,NAT_1:13;
             then A181: n+1 in dom f1 by NAT_1:11,FINSEQ_3:25;
             i+1 = L2+(n+1) by A176;
             then A182: f21.(i+1) = f1.(n+1) by A181,FINSEQ_1:def 7;
             f12.(n+1) = f1.(n+1) by A181,FINSEQ_1:def 7;
             hence thesis by A178,A180,A149,A152,A177,A182;
           end;
         end;
       end;
   A183:for f1 be Element of T st Q[f1] for i be Nat st i < p &
      f1 = (f1/^i)^(f1|i) holds i = 0
   proof
     let f1 be Element of T such that
         A184:Q[f1];
     A185: len f1 = p by CARD_1:def 7;
     then 2 in dom f1 by A7,FINSEQ_3: 25;
     then A186: f1.2 in rng f1 by FUNCT_1:def 3;
     let i be Nat such that
         A187: i < p & f1 = (f1/^i)^(f1|i) & i <> 0;
     rng f1 c= {f1.1} by A187,A185,A3,Th7;
     then A188: f1.2 =f1.1 by A186,TARSKI:def 1;
     [f1.1,f1.(1+1)] in FSG by A3,INT_2:def 4,A184,A185;
     hence contradiction by A188,Lm2;
   end;
   consider C be Cardinal such that
       A189:p*`C = card XT from Sch(A147,A183);
   A190:card pr = card FSGR by Th1
       .= card dom FSGR by CARD_1:62
       .= card XT by A43,FUNCT_2:def 1;
   then C is finite by A146,A189;
   then reconsider C as Nat;
   p*C = card XFSG2 + m*card XFSG1 by A189,Lm1,A146,A190;
   then A191:p divides (1+ m*m1) * (m|^p2) + (m-1) *card XFSG1
     by A94,INT_1:def 3;
   not p divides m*m1 +1 by A4,INT_2:2,NEWTON:39,A29;
   then A192: not p divides (1+ m*m1) * (m|^p2) by A30,A3,INT_5:7;
   p divides m1 *card XFSG1 by A4,INT_2:2;
   hence contradiction by A192,INT_2:1,A191;
 end;
