
theorem Th41:
  for G being _finite _Graph, n being Nat st n < G.order() holds ((
  LexBFS:CSeq(G))``1).PickedAt(n) = LexBFS:PickUnnumbered((LexBFS:CSeq(G)).n)
proof
  let G be _finite _Graph, n be Nat such that
A1: n < G.order();
  set CS = LexBFS:CSeq(G);
  set CSN = CS.n;
  set CS1 = CS.(n+1);
  set VLN = CSN`1;
  set VL1 = CS1`1;
A2: CS.Lifespan() = G.order() by Th37;
  set PU = LexBFS:PickUnnumbered(CSN);
  set f2 = PU .--> (CS.Lifespan()-'n);
A3: dom f2 = {PU};
  n = card dom VLN by A1,Th32;
  then VL1 = VLN +* (PU .--> (CS.Lifespan()-'n)) by A1,A2,Th31;
  then
A4: dom VL1 = dom VLN \/ {PU} by A3,FUNCT_4:def 1;
A5: CSN`1 = CS``1.n by Def15;
  set PA = CS``1.PickedAt(n);
  set f1 = PA .--> (CS.Lifespan()-'n);
A6: dom f1 = {PA};
A7: CS.Lifespan() = CS``1.Lifespan() by Th39;
  CS1`1 = CS``1.(n+1) by Def15;
  then VL1 = VLN +* (PA .--> (CS.Lifespan()-'n)) by A1,A2,A7,A5,Def9;
  then
A8: dom VL1 = dom VLN \/ {PA} by A6,FUNCT_4:def 1;
A9: not PA in dom VLN by A1,A2,A7,A5,Def9;
  now
    assume PA <> PU;
    then not PA in {PU} by TARSKI:def 1;
    then
A10: not PA in dom VL1 by A9,A4,XBOOLE_0:def 3;
    PA in {PA} by TARSKI:def 1;
    hence contradiction by A8,A10,XBOOLE_0:def 3;
  end;
  hence thesis;
end;
