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

theorem
  for c, c1 being directed Chain of G st 1<=m & m<=n & n<=len c & c1 = (
  m,n)-cut c holds vertex-seq c1 = (m,n+1)-cut vertex-seq c
proof
  let c, c1 be directed Chain of G;
  assume that
A1: 1<=m and
A2: m<=n and
A3: n<=len c and
A4: c1 = (m,n)-cut c;
  set mn1c=(m,n+1)-cut vertex-seq c;
A5: c is non empty by A1,A2,A3;
  then
A6: vertex-seq c is_vertex_seq_of c by GRAPH_2:def 10;
  then
A7: mn1c is_vertex_seq_of c1 by A1,A2,A3,A4,GRAPH_2:42;
  set vsc = vertex-seq c;
A8: m<=n+1 by A2,NAT_1:12;
A9: len vsc = len c +1 by A6;
  then
A10: n+1<=len vsc by A3,XREAL_1:6;
A11: c1 is non empty by A1,A2,A3,A4,A5,Th11;
  then 0<len c1;
  then
A12: c1.(0+1) = c.(m+0) by A1,A2,A3,A4,FINSEQ_6:def 4;
A13: m<=n+1 by A2,NAT_1:12;
  vsc is non empty by A9;
  then mn1c is non empty by A1,A8,A10,Th11;
  then 0<len mn1c;
  then
A14: vsc.(m+0) = mn1c.(0+1) by A1,A10,A13,FINSEQ_6:def 4;
  m<=len c by A2,A3,XXREAL_0:2;
  then mn1c.1 = (the Source of G).(c1.1) by A1,A5,A12,A14,Th10;
  hence thesis by A7,A11,GRAPH_2:def 10;
end;
