
theorem Th23:
  for G being _finite _Graph, S being VNumberingSeq of G, i being
Nat, a,b being set st a in dom (S.i) & b in dom (S.i) & (S.i).a < (S.i).b holds
  b in dom (S.(S.Lifespan() -' (S.i).a))
proof
  let G be _finite _Graph, S be VNumberingSeq of G;
  let i be Nat, a,b be set such that
A1: a in dom (S.i) and
A2: b in dom (S.i) and
A3: (S.i).a < (S.i).b;
  set GN = S.Lifespan();
  set CSI = S.i;
  set VLI = CSI;
  set j = S.Lifespan() -' VLI.a;
  set CSJ = S.j;
  set VLJ = CSJ;
  VLI.a <= GN by Th15;
  then
A4: GN -' VLI.a = GN - VLI.a by XREAL_1:233;
  then GN -' j = GN - (GN - VLI.a) by NAT_D:35,XREAL_1:233;
  then consider w being Vertex of G such that
A5: w in dom CSJ and
A6: VLJ.w = VLI.b by A3,Th15,Th16;
  now
    assume j >= i;
    then VLI c= VLJ by Th17;
    then
A7: dom VLI c= dom VLJ by RELAT_1:11;
    0 < VLI.a by A1,Th15;
    then
A8: j < GN by A4,XREAL_1:44;
    a = S.PickedAt(j) by A1,Th20;
    hence contradiction by A1,A7,A8,Def9;
  end;
  then
A9: VLJ c= VLI by Th17;
A10: [w,VLI.b] in VLJ by A5,A6,FUNCT_1:1;
  then
A11: VLI.w = VLI.b by A9,FUNCT_1:1;
A12: VLI is one-to-one by Th18;
  w in dom VLI by A9,A10,FUNCT_1:1;
  hence thesis by A2,A5,A11,A12,FUNCT_1:def 4;
end;
