reserve G for Graph,
  k, m, n for Nat;
reserve G for non void Graph;

theorem Th14:
  for c1, c2 being directed non empty Chain of G holds (vertex-seq
c1).(len c1 + 1) = (vertex-seq c2).1 iff c1^c2 is directed non empty Chain of G
proof
  let c1, c2 be directed non empty Chain of G;
  set vsc1 = vertex-seq c1, vsc2 = vertex-seq c2;
A1: len (c1^c2) = len c1 + len c2 by FINSEQ_1:22;
A2: vsc1 is_vertex_seq_of c1 by GRAPH_2:def 10;
  then
A3: vsc2 is_vertex_seq_of c2 & len vsc1 = len c1 +1 by GRAPH_2:def 10;
  hereby
    assume
A4: (vertex-seq c1).(len c1 + 1) = (vertex-seq c2).1;
    then reconsider c1c2 = c1^c2 as Chain of G by A2,A3,GRAPH_2:43;
    c1c2 is directed
    proof
      let n;
      assume that
A5:   1 <= n and
A6:   n < len c1c2;
      per cases by XXREAL_0:1;
      suppose
A7:     n<len c1;
        then 1<=n+1 & n+1<=len c1 by NAT_1:11,13;
        then n+1 in dom c1 by FINSEQ_3:25;
        then
A8:     c1c2.(n+1) = c1.(n+1) by FINSEQ_1:def 7;
        n in dom c1 by A5,A7,FINSEQ_3:25;
        then c1c2.n = c1.n by FINSEQ_1:def 7;
        hence thesis by A5,A7,A8,GRAPH_1:def 15;
      end;
      suppose
A9:     n=len c1;
        then n in dom c1 by FINSEQ_5:6;
        then
A10:    c1c2.n = c1.n by FINSEQ_1:def 7;
        1 in dom c2 by FINSEQ_5:6;
        then
A11:    c1c2.(n+1) = c2.1 by A9,FINSEQ_1:def 7;
        vsc1.(len c1 +1) =(the Target of G).(c1.(len c1)) by Th13;
        hence thesis by A4,A9,A11,A10,GRAPH_2:def 10;
      end;
      suppose
A12:    n>len c1;
        then reconsider k = n-len c1 as Element of NAT by INT_1:5;
A13:    n = len c1 + k;
A14:    n+1 = len c1 +(k+1);
A15:    k<len c2 by A1,A6,XREAL_1:19;
        then 1<=k+1 & k+1<=len c2 by NAT_1:11,13;
        then k+1 in dom c2 by FINSEQ_3:25;
        then
A16:    c1c2.(n+1) = c2.(k+1) by A14,FINSEQ_1:def 7;
        n>=len c1 +1 by A12,NAT_1:13;
        then
A17:    1<=k by XREAL_1:19;
        then k in dom c2 by A15,FINSEQ_3:25;
        then c1c2.n = c2.k by A13,FINSEQ_1:def 7;
        hence thesis by A17,A15,A16,GRAPH_1:def 15;
      end;
    end;
    hence c1^c2 is directed non empty Chain of G;
  end;
  set n = len c1;
  assume c1^c2 is directed non empty Chain of G;
  then reconsider c1c2 = c1^c2 as directed non empty Chain of G;
A18: n<len c1c2 by A1,XREAL_1:29;
A19: n in dom c1 by FINSEQ_5:6;
  then
A20: c1c2.n = c1.n by FINSEQ_1:def 7;
  1<=n by A19,FINSEQ_3:25;
  then
A21: (the Source of G).(c1c2.(n+1)) = (the Target of G).(c1c2.n) by A18,
GRAPH_1:def 15;
  1 in dom c2 by FINSEQ_5:6;
  then
A22: c1c2.(n+1) = c2.1 by FINSEQ_1:def 7;
  vsc1.(len c1 +1) =(the Target of G).(c1.(len c1)) by Th13;
  hence thesis by A21,A22,A20,GRAPH_2:def 10;
end;
