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;
reserve sc for simple Chain of G;

theorem
  sc|(Seg n) is simple Chain of G
proof
  reconsider q9 = sc|Seg n as Chain of G by GRAPH_1:4;
  consider vs such that
A1: vs is_vertex_seq_of sc and
A2: for n, m st 1<=n & n < m & m<=len vs & vs.n = vs.m holds n=1 & m=
  len vs by Def9;
  reconsider p9 = vs|Seg(n+1) as FinSequence of the carrier of G by FINSEQ_1:18
;
  now
    take p9;
    thus p9 is_vertex_seq_of q9 by A1,Th40;
    let k, m;
    assume that
A3: 1<=k and
A4: k < m and
A5: m<=len p9 and
A6: p9.k = p9.m;
    k<=len p9 by A4,A5,XXREAL_0:2;
    then
A7: p9.k = vs.k by A3,FINSEQ_6:128;
    1<=m by A3,A4,XXREAL_0:2;
    then
A8: p9.m = vs.m by A5,FINSEQ_6:128;
A9: len p9<=len vs by FINSEQ_6:128;
    then
A10: m<=len vs by A5,XXREAL_0:2;
    hence k=1 by A2,A3,A4,A6,A7,A8;
    len p9 = len vs or len p9 < len vs by A9,XXREAL_0:1;
    hence m = len p9 by A2,A3,A4,A5,A6,A7,A8,A10;
  end;
  hence thesis by Def9;
end;
