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 Th4:
  for f being FinSequence of X holds PairF(f) is FinSequence of the
  carrier' of PGraph(X)
proof
  let f be FinSequence of X;
  rng PairF(f) c= [:X,X:]
  proof
    let y be object;
A1: len f-'1<len f-'1+1 by NAT_1:13;
    assume y in rng PairF(f);
    then consider x being object such that
A2: x in dom PairF(f) and
A3: y=(PairF(f)).x by FUNCT_1:def 3;
    reconsider n=x as Element of NAT by A2;
A4: x in Seg len PairF(f) by A2,FINSEQ_1:def 3;
    then
A5: 1<=n by FINSEQ_1:1;
A6: len PairF(f)=len f-'1 by Def2;
A7: n<=len PairF(f) by A4,FINSEQ_1:1;
    then 1<=len f-'1 by A5,A6,XXREAL_0:2;
    then len f-'1=len f-1 by NAT_D:39;
    then
A8: n<len f by A7,A6,A1,XXREAL_0:2;
    then
A9: n+1<=len f by NAT_1:13;
    1<n+1 by A5,NAT_1:13;
    then n+1 in dom f by A9,FINSEQ_3:25;
    then
A10: f.(n+1) in rng f by FUNCT_1:def 3;
    n in dom f by A5,A8,FINSEQ_3:25;
    then
A11: f.n in rng f by FUNCT_1:def 3;
    (PairF(f)).n =[f.n,f.(n+1)] by A5,A8,Def2;
    hence thesis by A3,A11,A10,ZFMISC_1:def 2;
  end;
  hence thesis by FINSEQ_1:def 4;
end;
