reserve p, q for FinSequence,
  X, Y, x, y, e for set,
  D for non empty set,
  i, j, k, l, m, n, r for Nat;
reserve G for Graph;
reserve v, v1, v2, v3, v4 for Element of G;
reserve vs, vs1, vs2 for FinSequence of the carrier of G,
  c, c1, c2 for Chain of G;

theorem Th33:
  ex vs st vs is_vertex_seq_of c
proof
  consider p such that
A1: len p = len c + 1 and
A2: for n st 1<=n & n<=len p holds p.n in the carrier of G and
A3: for n st 1<=n & n<=len c ex v1,v2 st v1=p.n & v2=p.(n+1) & c.n joins
  v1,v2 by GRAPH_1:def 14;
  rng p c= the carrier of G
  proof
    let y be object;
    assume y in rng p;
    then consider x being object such that
A4: x in dom p and
A5: y = p.x by FUNCT_1:def 3;
    reconsider n=x as Element of NAT by A4;
A6: n<=len p by A4,FINSEQ_3:25;
    1<=n by A4,FINSEQ_3:25;
    hence thesis by A2,A5,A6;
  end;
  then reconsider p as FinSequence of the carrier of G by FINSEQ_1:def 4;
  take p;
  thus len p = len c + 1 by A1;
  let n be Nat;
  assume that
A7: 1<=n and
A8: n<=len c;
A9: n<=len p by A1,A8,NAT_1:12;
  1<=n+1 by NAT_1:12;
  then
A10: p/.(n+1)=p.(n+1) by A1,A8,FINSEQ_4:15,XREAL_1:7;
  ex v1,v2 st v1=p.n & v2=p.(n+1) & c.n joins v1,v2 by A3,A7,A8;
  hence thesis by A7,A9,A10,FINSEQ_4:15;
end;
