
theorem Th24:
  for G being _finite _Graph, S being VNumberingSeq of G, i being
Nat, a,b being set st a in dom (S.i) & (S.i).a < (S.i).b holds not a in dom (S.
  (S.Lifespan() -' (S.i).b))
proof
  let G be _finite _Graph, S be VNumberingSeq of G, i be Nat, a,b being set
  such that
A1: a in dom (S.i) and
A2: (S.i).a < (S.i).b;
  set GN = S.Lifespan();
  set CSI = S.i;
  set VLI = CSI;
  set k = GN -' VLI.a;
  VLI.a <= GN by Th15;
  then
A3: k = GN - VLI.a by XREAL_1:233;
  set CSK = S.k;
  set j = GN -' VLI.b;
  set CSJ = S.j;
  set VLJ = CSJ;
  set VLK = CSK;
  VLI.b <= GN by Th15;
  then j = GN - VLI.b by XREAL_1:233;
  then j < k by A2,A3,XREAL_1:15;
  then VLJ c= VLK by Th17;
  then
A4: dom VLJ c= dom VLK by RELAT_1:11;
  assume
A5: a in dom CSJ;
  1 <= VLI.a by A1,Th15;
  then
A6: k < GN by A3,XREAL_1:44;
  S.PickedAt(k) = a by A1,Th20;
  hence contradiction by A6,A5,A4,Def9;
end;
