consider G being finite loopless trivial non-multi non-Dmulti simple Dsimple _Graph;
set F = NAT --> G;
A1: dom (NAT --> G) = NAT by FUNCOP_1:19;
reconsider F = NAT --> G as ManySortedSet of NAT ;

now

let x be Nat; :: thesis: F . x is _Graph
x in NAT by ORDINAL1:def 13;
then F . x in rng F by A1, FUNCT_1:12;
then F . x in {G} by FUNCOP_1:14;
hence F . x is _Graph by TARSKI:def 1; :: thesis: verum

end;

then reconsider F = F as GraphSeq by Def55;
F . (1 + 1) in rng F by A1, FUNCT_1:12;
then F . (1 + 1) in {G} by FUNCOP_1:14;
then A2: F . (1 + 1) = G by TARSKI:def 1;
take F ; :: thesis: ( F is halting & F is finite & F is loopless & F is trivial & F is non-multi & F is non-Dmulti & F is simple & F is Dsimple )
F . 1 in rng F by A1, FUNCT_1:12;
then F . 1 in {G} by FUNCOP_1:14;
then F . 1 = G by TARSKI:def 1;
hence F is halting by A2, Def56; :: thesis: ( F is finite & F is loopless & F is trivial & F is non-multi & F is non-Dmulti & F is simple & F is Dsimple )

now

let x be Nat; :: thesis: ( F . x is finite & F . x is loopless & F . x is trivial & F . x is non-multi & F . x is non-Dmulti & F . x is simple & F . x is Dsimple )
x in NAT by ORDINAL1:def 13;
then F . x in rng F by A1, FUNCT_1:12;
then F . x in {G} by FUNCOP_1:14;
hence ( F . x is finite & F . x is loopless & F . x is trivial & F . x is non-multi & F . x is non-Dmulti & F . x is simple & F . x is Dsimple ) by TARSKI:def 1; :: thesis: verum

end;

hence ( F is finite & F is loopless & F is trivial & F is non-multi & F is non-Dmulti & F is simple & F is Dsimple ) by Def60, Def61, Def62, Def64, Def65, Def66, Def67; :: thesis: verum