theorem Th52:
  for c being Element of G-CycleSet st v in G-VSet rng c holds {
  c9 where c9 is Element of G-CycleSet : rng c9 = rng c & ex vs st vs
  is_vertex_seq_of c9 & vs.1 = v} is non empty Subset of G-CycleSet
proof
  let c be Element of G-CycleSet;
  set Cset = { c9 where c9 is Element of G-CycleSet : rng c9 = rng c & ex vs
  being FinSequence of the carrier of G st vs is_vertex_seq_of c9 & vs.1 = v};
  reconsider c9 = c as cyclic Path of G by Def8;
  consider vs being FinSequence of the carrier of G such that
A1: vs is_vertex_seq_of c9 by GRAPH_2:33;
A2: len vs = len c +1 by A1;
  assume
A3: v in G-VSet rng c;
  then
A4: ex vv being Vertex of G st vv = v & ex e being Element of the carrier' of
  G st e in rng c & (vv = (the Source of G).e or vv = (the Target of G).e);
  then G-VSet rng c9 = rng vs by A1,GRAPH_2:31,RELAT_1:38;
  then consider n being Nat such that
A5: n in dom vs and
A6: vs.n = v by A3,FINSEQ_2:10;
  reconsider n as Element of NAT by ORDINAL1:def 12;
  dom vs = Seg len vs by FINSEQ_1:def 3;
  then
A7: 1 <= n & n <= len vs by A5,FINSEQ_1:1;
A8: now
    per cases by A7,XXREAL_0:1;
    suppose
      1 = n & n = len vs;
      then 0+1 = len c +1 by A1;
      then c = {};
      hence Cset is non empty by A4;
    end;
    suppose
      1 = n;
      then c in Cset by A1,A6;
      hence Cset is non empty;
    end;
    suppose
      n = len vs;
      then vs.1 = v by A1,A6,MSSCYC_1:6;
      then c in Cset by A1;
      hence Cset is non empty;
    end;
    suppose
A9:   1 < n & n < len vs;
      set vs2 = (n, len vs)-cut vs;
      consider m being Element of NAT such that
A10:  n = 1+m and
A11:  1 <= m by A9,FINSEQ_4:84;
      set vs1 = (1, m+1)-cut vs;
A12:  1 <= m+1+1 by A9,A10,NAT_1:13;
      then
A13:  len vs1 + 1 = m+1+1 by A9,A10,Lm1;
      then
A14:  vs1.1 = vs.(1+0) by A9,A10,A12,Lm1;
      reconsider c1 = (1,m)-cut c9, c2 = (n,len c)-cut c9 as Path of G by Th5;
A15:  n <= len c by A2,A9,NAT_1:13;
      then
A16:  vs2 is_vertex_seq_of c2 by A1,A9,GRAPH_2:42;
A17:  len vs2 + n = len vs +1 by A9,FINSEQ_6:def 4;
A18:  now
        assume len vs2 = 0;
        then len vs +1 < len vs +0 by A9,A17;
        hence contradiction by XREAL_1:6;
      end;
      then
A19:  1+0 <= len vs2 by NAT_1:13;
      then consider lv2 being Nat such that
      0<=lv2 and
A20:  lv2<len vs2 and
A21:  len vs2=lv2+1 by FINSEQ_6:127;
      reconsider vs21 = vs2^'vs1 as FinSequence of the carrier of G;
A22:  vs21.1 = vs2.(0+1) by A19,FINSEQ_6:140
        .= vs.(n+0) by A9,A18,FINSEQ_6:def 4;
A23:  m <= m+1 by NAT_1:11;
      then m <= len c by A10,A15,XXREAL_0:2;
      then
A24:  vs1 is_vertex_seq_of c1 by A1,A11,GRAPH_2:42;
A25:  vs2.len vs2 = vs.(n+lv2) by A9,A20,A21,FINSEQ_6:def 4
        .= vs1.1 by A1,A17,A21,A14,MSSCYC_1:6;
      now
        given y being object such that
A26:    y in rng c1 and
A27:    y in rng c2;
        consider b being Nat such that
A28:    b in dom c2 and
A29:    c2.b = y by A27,FINSEQ_2:10;
A30:    ex l being Nat st l in dom c9 & c.l = c2.b & l+1 = n+b
        & n <= l & l <= len c by A28,Th2;
        consider a being Nat such that
A31:    a in dom c1 and
A32:    c1.a = y by A26,FINSEQ_2:10;
        consider k being Nat such that
A33:    k in dom c9 & c.k = c1.a and
        k+1 = 1+a and
        1 <= k and
A34:    k <= m by A31,Th2;
        k < n by A10,A34,NAT_1:13;
        hence contradiction by A32,A29,A33,A30,FUNCT_1:def 4;
      end;
      then rng c1 misses rng c2 by XBOOLE_0:3;
      then reconsider c219= c2^c1 as Path of G by A24,A16,A25,Th6;
A35:  vs21 is_vertex_seq_of c219 by A24,A16,A25,GRAPH_2:44;
A36:  c = c1^c2 by A10,A15,A23,FINSEQ_6:135,XXREAL_0:2;
      1 < len vs1 by A11,A13,NAT_1:13;
      then vs21.len vs21 = vs1.len vs1 by FINSEQ_6:142
        .= vs.n by A9,A10,FINSEQ_6:138;
      then reconsider c219 as cyclic Path of G by A22,A35,MSSCYC_1:def 2;
      reconsider c21 = c219 as Element of G-CycleSet by Def8;
      rng c21 = rng c2 \/ rng c1 by FINSEQ_1:31
        .= rng c by A36,FINSEQ_1:31;
      then c21 in Cset by A6,A22,A35;
      hence Cset is non empty;
    end;
  end;
  Cset c= G-CycleSet
  proof
    let x be object;
    assume x in Cset;
    then ex c9 being Element of G-CycleSet st c9 = x & rng c9 = rng c & ex vs
    being FinSequence of the carrier of G st vs is_vertex_seq_of c9 & vs.1 = v;
    hence thesis;
  end;
  hence thesis by A8;
end;
