reserve x,y for set;
reserve G for Graph;
reserve vs,vs9 for FinSequence of the carrier of G;
reserve IT for oriented Chain of G;
reserve N for Nat;
reserve n,m,k,i,j for Nat;
reserve r,r1,r2 for Real;
reserve X for non empty set;

theorem
  for f being FinSequence of X holds f is one-to-one & len f>1 implies
  PairF(f) is Simple & f.1<>f.len f
proof
  let f be FinSequence of X;
  assume that
A1: f is one-to-one and
A2: len f>1;
A3: 1 in dom f & len f in dom f by A2,FINSEQ_3:25;
  reconsider f1=f as FinSequence of the carrier of PGraph(X);
A4: for i,j being Nat
     st 1<=i & i<j & j<=len f1 & f1.i=f1.j holds i=1 & j=len f1
  proof
    let i,j be Nat;
    assume that
A5: 1<=i and
A6: i<j and
A7: j<=len f1 and
A8: f1.i=f1.j;
    1<j by A5,A6,XXREAL_0:2;
    then j in Seg len f by A7,FINSEQ_1:1;
    then
A9: j in dom f by FINSEQ_1:def 3;
    i<len f by A6,A7,XXREAL_0:2;
    then i in Seg len f by A5,FINSEQ_1:1;
    then i in dom f by FINSEQ_1:def 3;
    hence thesis by A1,A6,A8,A9;
  end;
  f1 is_oriented_vertex_seq_of PairF(f) by A2,Th7;
  hence thesis by A1,A2,A3,A4,GRAPH_4:def 7;
end;
