
theorem Th51:
  for G being _finite _Graph, i,j being Nat, a,b being Vertex of G
  st a in dom ((LexBFS:CSeq(G)).i)`1 & b in dom ((LexBFS:CSeq(G)).i)`1 & ((
  LexBFS:CSeq(G)).i)`1.a < ((LexBFS:CSeq(G)).i)`1.b & j = G.order() -' ((
  LexBFS:CSeq(G)).i)`1.b holds (((LexBFS:CSeq(G)).j)`2.a,1)-bag <= ((((
  LexBFS:CSeq(G)).j)`2).b,1)-bag, InvLexOrder NAT
proof
  let G be _finite _Graph;
  let i,j be Nat, a,b be Vertex of G such that
A1: a in dom ((LexBFS:CSeq(G)).i)`1 and
A2: b in dom ((LexBFS:CSeq(G)).i)`1 and
A3: ((LexBFS:CSeq(G)).i)`1.a < ((LexBFS:CSeq(G)).i)`1.b and
A4: j = G.order() -' ((LexBFS:CSeq(G)).i)`1.b;
  set VL = (LexBFS:CSeq(G))``1;
  set CSJ = (LexBFS:CSeq(G)).j;
  set VLI = VL.i, VLJ = VL.j;
A5: ((LexBFS:CSeq(G)).i)`1.b = ((LexBFS:CSeq(G))``1.i).b by Def15;
A6: a in the_Vertices_of G;
A7: ((LexBFS:CSeq(G)).i)`1 = ((LexBFS:CSeq(G))``1.i) by Def15;
A8: (LexBFS:CSeq(G)).Lifespan() = VL.Lifespan() by Th39;
A9: G.order() = (LexBFS:CSeq(G)).Lifespan() by Th37;
  then VLI.b <= G.order() by A8,Th15;
  then
A10: G.order() -' VLI.b = G.order() - VLI.b by XREAL_1:233;
  then
A11: G.order() -' j = G.order() - (G.order() - VLI.b) by A4,A5,NAT_D:35
,XREAL_1:233;
A12: now
    assume a in dom (CSJ`1);
    then
A13: a in dom VLJ by Def15;
    then VLI.b < VLJ.a by A9,A8,A11,Th22;
    hence contradiction by A1,A3,A7,A13,Th19;
  end;
  VL.PickedAt(j) = b by A2,A4,A7,A9,A8,Th20;
  then LexBFS:PickUnnumbered(CSJ) = b by A3,A4,A5,A10,Th41,XREAL_1:44;
  hence thesis by A6,A12,Th29;
end;
