
theorem Th40: :: TopPath04
  for G being _Graph,
      S being non empty Subset of the_Vertices_of G
  for H being inducedSubgraph of G,S for W being Walk of H st W is minlength
  for m,n being odd Nat st m+2 < n & n <= len W holds
  not ex e being object st e Joins W.m,W.n,G
proof
  let G be _Graph, S be non empty Subset of the_Vertices_of G;
  let GA be inducedSubgraph of G,S;
  let P be Walk of GA;
A1: S = the_Vertices_of GA by GLIB_000:def 37;
  assume
A2: P is minlength;
  now
    let m,n be odd Nat such that
A3: m+2 < n and
A4: n <= len P;
    m + 0 <= m + 2 by XREAL_1:7;
    then
A5: m <= n by A3,XXREAL_0:2;
    n in NAT by ORDINAL1:def 12;
    then
A6: P.n in the_Vertices_of GA by A4,GLIB_001:7;
    m in NAT by ORDINAL1:def 12;
    then
A7: P.m in the_Vertices_of GA by A4,A5,GLIB_001:7,XXREAL_0:2;
    let e be object;
    assume e Joins P.m,P.n,G;
    hence contradiction by A2,A1,A3,A4,A7,A6,Th19,Th39;
  end;
  hence thesis;
end;
