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 Th5:
  for n being Nat,f being FinSequence of X st 1 <= n & n
  <= len PairF(f) holds (PairF(f)).n in the carrier' of PGraph(X)
proof
  let n be Nat,f be FinSequence of X;
  assume that
A1: 1 <= n and
A2: n <= len PairF(f);
A3: len f-'1<len f-'1+1 by NAT_1:13;
A4: len PairF(f)=len f-'1 by Def2;
  then 1<=len f-'1 by A1,A2,XXREAL_0:2;
  then len f-'1=len f-1 by NAT_D:39;
  then
A5: n<len f by A2,A4,A3,XXREAL_0:2;
  then
A6: n+1<=len f by NAT_1:13;
  1<n+1 by A1,NAT_1:13;
  then n+1 in dom f by A6,FINSEQ_3:25;
  then
A7: f.(n+1) in rng f by FUNCT_1:def 3;
  n in dom f by A1,A5,FINSEQ_3:25;
  then
A8: f.n in rng f by FUNCT_1:def 3;
  (PairF(f)).n =[f.n,f.(n+1)] by A1,A5,Def2;
  hence thesis by A8,A7,ZFMISC_1:def 2;
end;
