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

theorem
  for c being directed Chain of G st 1<=m & m<=n & n<=len c holds (m,n)
  -cut c is directed Chain of G
proof
  let c be directed Chain of G;
  assume that
A1: 1<=m and
A2: m<=n and
A3: n<=len c;
  reconsider mnc = (m,n)-cut c as Chain of G by A1,A2,A3,GRAPH_2:41;
A4: len mnc +m = n+1 by A1,A2,A3,FINSEQ_6:def 4;
  now
A5: len mnc +m<=len c +1 by A3,A4,XREAL_1:6;
    let k;
    assume that
A6: 1 <= k and
A7: k < len mnc;
    0+1<=k by A6;
    then consider i being Nat such that
    0<=i and
A8: i<len mnc and
A9: k=i+1 by A7,FINSEQ_6:127;
A10: 1<=m+i by A1,NAT_1:12;
    m+(i+1)<len mnc +m by A7,A9,XREAL_1:6;
    then m+i+1 < len c + 1 by A5,XXREAL_0:2;
    then
A11: m+i<len c by XREAL_1:6;
A12: mnc.(k+1) = c.(m+k) by A1,A2,A3,A7,FINSEQ_6:def 4;
    mnc.(i+1) = c.(m+i) & m+k = (m+i)+1 by A1,A2,A3,A8,A9,FINSEQ_6:def 4;
    hence (the Source of G).(mnc.(k+1)) = (the Target of G).(mnc.k) by A12,A10
,A11,GRAPH_1:def 15;
  end;
  hence thesis by GRAPH_1:def 15;
end;
