
theorem Th12:
  for G being _Graph, W being Walk of G for m, n, i being odd Nat
st m <= n & n <= len W & i <= len W.cut(m,n) ex j being odd Nat st W.cut(m,n).i
  = W.j & j = m+i-1 & j <= len W
proof
  let G be _Graph, W be Walk of G;
  let m, n, i being odd Nat such that
A1: m <= n and
A2: n <= len W and
A3: i <= len W.cut(m,n);
  set j = m+i-1;
  m >= 1 & i >= 1 by ABIAN:12;
  then m+i >= 1+1 by XREAL_1:7;
  then m+i-1 >= 1+1-1 by XREAL_1:9;
  then j is odd Element of NAT by INT_1:3;
  then reconsider j as odd Nat;
  take j;
  reconsider m9= m, n9 = n as odd Element of NAT by ORDINAL1:def 12;
  i >= 1 by ABIAN:12;
  then i-1 >= 1-1 by XREAL_1:9;
  then reconsider i1 = i-1 as Element of NAT by INT_1:3;
  i < len W.cut(m,n) +1 by A3,NAT_1:13;
  then
A4: i1 < len W.cut(m,n) +1 -1 by XREAL_1:9;
  thus W.cut(m,n).i = W.cut(m9,n9).(i1+1) .= W.(m+i1) by A1,A2,A4,GLIB_001:36
    .= W.j;
  thus j = m+i-1;
  m+i <= len W.cut(m,n)+m by A3,XREAL_1:7;
  then m9+i <= n9+1 by A1,A2,GLIB_001:36;
  then m+i-1 <= n+1-1 by XREAL_1:9;
  hence thesis by A2,XXREAL_0:2;
end;
