reserve G for Graph,
  v, v1, v2 for Vertex of G,
  c for Chain of G,
  p, p1, p2 for Path of G,
  vs, vs1, vs2 for FinSequence of the carrier of G,
  e, X for set,
  n, m for Nat;

theorem Th9:
  m+1 in dom p implies len (((m+1,len p)-cut p)^(1,m)-cut p) = len
p & rng (((m+1,len p)-cut p)^(1,m)-cut p) = rng p & (((m+1,len p)-cut p)^(1,m)
  -cut p).1 = p.(m+1)
proof
  set r2 = (m+1,len p)-cut p;
  set r1 = (1,m)-cut p;
  set r = r2^r1;
  set n = m+1;
  assume
A1: m+1 in dom p;
  then
A2: m+1 <= len p by FINSEQ_3:25;
  then
A3: n < len p +1 by NAT_1:13;
  m <= m+1 by NAT_1:11;
  then
A4: p = r1 ^ r2 by A2,FINSEQ_6:135,XXREAL_0:2;
  thus len r = len r1 + len r2 by FINSEQ_1:22
    .= len p by A4,FINSEQ_1:22;
  thus rng r = rng r1 \/ rng r2 by FINSEQ_1:31
    .= rng p by A4,FINSEQ_1:31;
A5: 1 <= m+1 by A1,FINSEQ_3:25;
  then len r2 +n = len p +1 by A2,FINSEQ_6:def 4;
  then 1+n <= len r2 +n by A3,NAT_1:13;
  then
A6: 1 <= len r2 by XREAL_1:6;
  then 1 in dom r2 by FINSEQ_3:25;
  hence r.1 = r2.(0+1) by FINSEQ_1:def 7
    .= p.(n+0) by A5,A2,A6,FINSEQ_6:def 4
    .= p.n;
end;
