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 Th5:
  (m,n)-cut p is Path of G
proof
  per cases;
  suppose
    not 1<=m or not n<=len p or m > n;
    then (m,n)-cut p = {} by FINSEQ_6:def 4;
    hence thesis by GRAPH_1:14;
  end;
  suppose
    1<=m & m<=n & n<=len p;
    then reconsider q = (m,n)-cut p as Chain of G by GRAPH_2:41;
    assume not (m,n)-cut p is Path of G;
    then consider a, b being Nat such that
A1: 1 <= a and
A2: a < b and
A3: b <= len q and
A4: q.a = q.b by GRAPH_1:def 16;
    1 <= b by A1,A2,XXREAL_0:2;
    then b in dom q by A3,FINSEQ_3:25;
    then consider kb being Nat such that
A5: kb in dom p & p.kb = q.b and
A6: kb+1 = m+b and
    m <= kb and
    kb <= n by Th2;
    a <= len q by A2,A3,XXREAL_0:2;
    then a in dom q by A1,FINSEQ_3:25;
    then consider ka being Nat such that
A7: ka in dom p & p.ka = q.a and
A8: ka+1 = m+a and
    m <= ka and
    ka <= n by Th2;
    ka = kb by A4,A7,A5,FUNCT_1:def 4;
    hence contradiction by A2,A8,A6;
  end;
end;
