let G be finite _Graph; :: thesis: (MCS:CSeq G) .Lifespan() = G .order()
set CS = MCS:CSeq G;
G .order() <= (G .order() ) + 1 by NAT_1:13;
then A1: (MCS:CSeq G) . (G .order() ) = (MCS:CSeq G) . ((G .order() ) + 1) by Th83;
for n being Nat st (MCS:CSeq G) . n = (MCS:CSeq G) . (n + 1) holds
G .order() <= n

proof

let n be Nat; :: thesis: ( (MCS:CSeq G) . n = (MCS:CSeq G) . (n + 1) implies G .order() <= n )
assume A2: (MCS:CSeq G) . n = (MCS:CSeq G) . (n + 1) ; :: thesis: G .order() <= n
assume A3: n < G .order() ; :: thesis: contradiction
set VLN = ((MCS:CSeq G) . n) `1 ;
set VN1 = ((MCS:CSeq G) . (n + 1)) `1 ;
set w = MCS:PickUnnumbered ((MCS:CSeq G) . n);
set j = card (dom (((MCS:CSeq G) . n) `1 ));
dom (((MCS:CSeq G) . n) `1 ) <> the_Vertices_of G by A3, Th86;
then A5: not MCS:PickUnnumbered ((MCS:CSeq G) . n) in dom (((MCS:CSeq G) . n) `1 ) by Th73;
card (dom (((MCS:CSeq G) . n) `1 )) < G .order() by A3, Th82;
then A6: ((MCS:CSeq G) . (n + 1)) `1 = (((MCS:CSeq G) . n) `1 ) +* ((MCS:PickUnnumbered ((MCS:CSeq G) . n)) .--> ((G .order() ) -' (card (dom (((MCS:CSeq G) . n) `1 ))))) by Th81;
set wf = (MCS:PickUnnumbered ((MCS:CSeq G) . n)) .--> ((G .order() ) -' (card (dom (((MCS:CSeq G) . n) `1 ))));
( dom ((MCS:PickUnnumbered ((MCS:CSeq G) . n)) .--> ((G .order() ) -' (card (dom (((MCS:CSeq G) . n) `1 ))))) = {(MCS:PickUnnumbered ((MCS:CSeq G) . n))} & rng ((MCS:PickUnnumbered ((MCS:CSeq G) . n)) .--> ((G .order() ) -' (card (dom (((MCS:CSeq G) . n) `1 ))))) = {((G .order() ) -' (card (dom (((MCS:CSeq G) . n) `1 ))))} ) by FUNCOP_1:14, FUNCOP_1:19;
then A7: dom (((MCS:CSeq G) . (n + 1)) `1 ) = (dom (((MCS:CSeq G) . n) `1 )) \/ {(MCS:PickUnnumbered ((MCS:CSeq G) . n))} by A6, FUNCT_4:def 1;
MCS:PickUnnumbered ((MCS:CSeq G) . n) in {(MCS:PickUnnumbered ((MCS:CSeq G) . n))} by TARSKI:def 1;
hence contradiction by A2, A5, A7, XBOOLE_0:def 3; :: thesis: verum

end;

hence (MCS:CSeq G) .Lifespan() = G .order() by A1, GLIB_000:def 57; :: thesis: verum