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 Th7:
  for f being FinSequence of X, f1 being FinSequence of the
  carrier of PGraph(X) st len f>=1 & f=f1 holds f1 is_oriented_vertex_seq_of
  PairF(f)
proof
  let f be FinSequence of X, f1 be FinSequence of the carrier of PGraph(X);
  assume that
A1: len f>=1 and
A2: f=f1;
A3: for n st 1<=n & n<=len PairF(f) holds ((PairF(f)).n) orientedly_joins f1
  /.n, f1/.(n+1)
  proof
A4: len f-'1<len f-'1+1 by NAT_1:13;
    let n;
    assume that
A5: 1<=n and
A6: n<=len PairF(f);
A7: 1<n+1 by A5,NAT_1:13;
A8: len PairF(f)=len f-'1 by Def2;
    then 1<=len f-'1 by A5,A6,XXREAL_0:2;
    then len f-'1=len f-1 by NAT_D:39;
    then
A9: n<len f by A6,A8,A4,XXREAL_0:2;
    then n+1<=len f by NAT_1:13;
    then
A10: n+1 in dom f by A7,FINSEQ_3:25;
    then
A11: f.(n+1) in rng f by FUNCT_1:def 3;
A12: n in dom f by A5,A9,FINSEQ_3:25;
    then
A13: f.n in rng f by FUNCT_1:def 3;
    then
A14: (pr1(X,X)).(f.n,f.(n+1))=f.n by A11,FUNCT_3:def 4;
A15: (pr2(X,X)).(f.n,f.(n+1))=f.(n+1) by A13,A11,FUNCT_3:def 5;
    f1/.(n+1)=f1.(n+1) by A2,A10,PARTFUN1:def 6;
    then
A16: (the Target of PGraph(X)).((PairF(f)).n) = f1/.(n+1) by A2,A5,A9,A15,Def2;
    f1/.n=f1.n by A2,A12,PARTFUN1:def 6;
    then (the Source of PGraph(X)).((PairF(f)).n) = f1/.n by A2,A5,A9,A14,Def2;
    hence thesis by A16,GRAPH_4:def 1;
  end;
A17:for n being Nat
  st 1<=n & n<=len PairF(f) holds ((PairF(f)).n) orientedly_joins f1
  /.n, f1/.(n+1) by A3;
  len f-'1=len f-1 by A1,XREAL_1:233;
  then len f-1+1=len PairF(f)+1 by Def2;
  hence thesis by A2,A17,GRAPH_4:def 5;
end;
