set G = the _finite Tree-like _Graph;
set F = NAT --> the _finite Tree-like _Graph;
A1: dom (NAT --> the _finite Tree-like _Graph) = NAT ;
reconsider F = NAT --> the _finite Tree-like _Graph as ManySortedSet of NAT ;
reconsider F = F as GraphSeq ;
A2: F . (1 + 1) = the _finite Tree-like _Graph ;
take F ; :: thesis: ( F is halting & F is _finite & F is Tree-like )
F . 1 = the _finite Tree-like _Graph ;
hence F is halting by A2, GLIB_000:def 55; :: thesis: ( F is _finite & F is Tree-like )
now :: thesis: for x being Nat holds
( F . x is _finite & F . x is Tree-like )
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 GLIB_000:def 74; :: thesis: verum