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

theorem Th15:
  for c, c1, c2 being directed non empty Chain of G st c =c1^c2
  holds (vertex-seq c).1 = (vertex-seq c1).1 & (vertex-seq c).(len c +1) = (
  vertex-seq c2).(len c2 +1)
proof
  let c, c1, c2 be directed non empty Chain of G;
  1 in dom c by FINSEQ_5:6;
  then 1<=len c by FINSEQ_3:25;
  then
A1: (vertex-seq c).1 = (the Source of G).(c.1) & (vertex-seq c).(len c +1)
  = ( the Target of G).(c.len c) by Th10;
A2: 1 in dom c1 by FINSEQ_5:6;
  then 1<=len c1 by FINSEQ_3:25;
  then
A3: (vertex-seq c1).1 = (the Source of G).(c1.1) by Th10;
  1 in dom c2 by FINSEQ_5:6;
  then 1<=len c2 by FINSEQ_3:25;
  then
A4: (vertex-seq c2).(len c2 +1) = (the Target of G).(c2.len c2) by Th10;
  assume
A5: c =c1^c2;
  then len c2 in dom c2 & len c = len c1 + len c2 by FINSEQ_1:22,FINSEQ_5:6;
  hence thesis by A5,A2,A3,A1,A4,FINSEQ_1:def 7;
end;
