set G = the finite Tree-like _Graph;
set F = NAT --> the finite Tree-like _Graph;
A1: dom (NAT --> the finite Tree-like _Graph) = NAT by FUNCOP_1:13;
reconsider F = NAT --> the finite Tree-like _Graph as ManySortedSet of NAT ;

now

let x be Nat; :: thesis: F . x is _Graph
x in NAT by ORDINAL1:def 12;
then F . x in rng F by A1, FUNCT_1:3;
then F . x in { the finite Tree-like _Graph} by FUNCOP_1:8;
hence F . x is _Graph by TARSKI:def 1; :: thesis: verum

end;

then reconsider F = F as GraphSeq by GLIB_000:def 53;
F . (1 + 1) in rng F by A1, FUNCT_1:3;
then F . (1 + 1) in { the finite Tree-like _Graph} by FUNCOP_1:8;
then A2: F . (1 + 1) = the finite Tree-like _Graph by TARSKI:def 1;
take F ; :: thesis: ( F is halting & F is finite & F is Tree-like )
F . 1 in rng F by A1, FUNCT_1:3;
then F . 1 in { the finite Tree-like _Graph} by FUNCOP_1:8;
then F . 1 = the finite Tree-like _Graph by TARSKI:def 1;
hence F is halting by A2, GLIB_000:def 54; :: thesis: ( F is finite & F is Tree-like )

now

let x be Nat; :: thesis: ( F . x is finite & F . x is Tree-like )
x in NAT by ORDINAL1:def 12;
then F . x in rng F by A1, FUNCT_1:3;
then F . x in { the finite Tree-like _Graph} by FUNCOP_1:8;
hence ( F . x is finite & F . x is Tree-like ) by TARSKI:def 1; :: thesis: verum

end;

hence ( F is finite & F is Tree-like ) by Def14, GLIB_000:def 57; :: thesis: verum