let G be _finite _Graph; :: thesis: (LexBFS:CSeq G) .Lifespan() = G .order()
set CS = LexBFS:CSeq G;
A1: for n being Nat st (LexBFS:CSeq G) . n = (LexBFS:CSeq G) . (n + 1) holds
G .order() <= n

proof

let n be Nat; :: thesis: ( (LexBFS:CSeq G) . n = (LexBFS:CSeq G) . (n + 1) implies G .order() <= n )
assume A2: (LexBFS:CSeq G) . n = (LexBFS:CSeq G) . (n + 1) ; :: thesis: G .order() <= n
set w = LexBFS:PickUnnumbered ((LexBFS:CSeq G) . n);
set VN1 = ((LexBFS:CSeq G) . (n + 1)) `1 ;
set VLN = ((LexBFS:CSeq G) . n) `1 ;
set j = card (dom (((LexBFS:CSeq G) . n) `1));
set wf = (LexBFS:PickUnnumbered ((LexBFS:CSeq G) . n)) .--> ((G .order()) -' (card (dom (((LexBFS:CSeq G) . n) `1))));
assume A3: n < G .order() ; :: thesis: contradiction
then dom (((LexBFS:CSeq G) . n) `1) <> the_Vertices_of G by Th36;
then A4: not LexBFS:PickUnnumbered ((LexBFS:CSeq G) . n) in dom (((LexBFS:CSeq G) . n) `1) by Th30;
card (dom (((LexBFS:CSeq G) . n) `1)) < G .order() by A3, Th32;
then A5: ((LexBFS:CSeq G) . (n + 1)) `1 = (((LexBFS:CSeq G) . n) `1) +* ((LexBFS:PickUnnumbered ((LexBFS:CSeq G) . n)) .--> ((G .order()) -' (card (dom (((LexBFS:CSeq G) . n) `1))))) by Th31;
dom ((LexBFS:PickUnnumbered ((LexBFS:CSeq G) . n)) .--> ((G .order()) -' (card (dom (((LexBFS:CSeq G) . n) `1))))) = {(LexBFS:PickUnnumbered ((LexBFS:CSeq G) . n))} ;
then A6: dom (((LexBFS:CSeq G) . (n + 1)) `1) = (dom (((LexBFS:CSeq G) . n) `1)) \/ {(LexBFS:PickUnnumbered ((LexBFS:CSeq G) . n))} by A5, FUNCT_4:def 1;
LexBFS:PickUnnumbered ((LexBFS:CSeq G) . n) in {(LexBFS:PickUnnumbered ((LexBFS:CSeq G) . n))} by TARSKI:def 1;
hence contradiction by A2, A4, A6, XBOOLE_0:def 3; :: thesis: verum

end;

G .order() <= (G .order()) + 1 by NAT_1:13;
then (LexBFS:CSeq G) . (G .order()) = (LexBFS:CSeq G) . ((G .order()) + 1) by Th33;
hence (LexBFS:CSeq G) .Lifespan() = G .order() by A1, GLIB_000:def 56; :: thesis: verum