reserve n,m,i,j,k for Nat,
  x,y,e,X,V,U for set,
  W,f,g for Function;
reserve p,q for FinSequence;
reserve G for Graph,
  pe,qe for FinSequence of the carrier' of G;

theorem Th13:
  len pe = 1 implies pe is Simple oriented Chain of G
proof
  set p=pe;
  set v1=(the Source of G).(p.1), v2=(the Target of G).(p.1);
A1: now
    let n;
    assume 1 <= n & n <= len p;
    then n in dom p by FINSEQ_3:25;
    hence p.n in the carrier' of G by FINSEQ_2:11;
  end;
  assume
A2: len p = 1;
  then 1 in dom p by FINSEQ_3:25;
  then reconsider v1, v2 as Element of G by FINSEQ_2:11,FUNCT_2:5;
  set vs=<*v1,v2*>;
A3: len vs = len p +1 by A2,FINSEQ_1:44;
A4: now
    let n;
    assume 1 <= n & n <= len p;
    then
A5: n=1 by A2,XXREAL_0:1;
    take v1,v2;
    thus v1= vs.n & v2 = vs.(n+1) by A5;
    thus p.n joins v1, v2 by A5,GRAPH_1:def 12;
  end;
A6: len vs = 2 by FINSEQ_1:44;
A7: now
    let n,m;
    assume that
A8: 1<=n and
A9: n<m and
A10: m<=len vs and
    vs.n=vs.m;
    n < 1+1 by A6,A9,A10,XXREAL_0:2;
    then n <= 1 by NAT_1:13;
    hence n=1 by A8,XXREAL_0:1;
    1 < m by A8,A9,XXREAL_0:2;
    then 1 + 1 < m + 1 by XREAL_1:8;
    then 2 <= m by NAT_1:13;
    hence m=len vs by A6,A10,XXREAL_0:1;
  end;
A11: now
    let n;
    assume that
A12: 1 <= n and
A13: n <= len vs;
    per cases;
    suppose
      n < 1+1;
      then n <= 1 by NAT_1:13;
      then n=1 by A12,XXREAL_0:1;
      then vs.n=v1;
      hence vs.n in the carrier of G;
    end;
    suppose
      n >= 2;
      then n=2 by A6,A13,XXREAL_0:1;
      then vs.n=v2;
      hence vs.n in the carrier of G;
    end;
  end;
A14: for n st 1 <= n & n < len p & (the Source of G).(p.(n+1)) <> (the
  Target of G).(p.n) holds contradiction by A2;
  now
    let n;
    assume 1<=n & n<=len p;
    then
A15: n=1 by A2,XXREAL_0:1;
    vs/.1= v1 & vs/.(1+1)= v2 by FINSEQ_4:17;
    hence p.n orientedly_joins vs/.n, vs/.(n+1) by A15,GRAPH_4:def 1;
  end;
  then vs is_oriented_vertex_seq_of p by A3,GRAPH_4:def 5;
  hence thesis by A3,A7,A1,A11,A4,A14,GRAPH_1:def 14,def 15,GRAPH_4:def 7;
end;
