let p be the Instructions of SCM+FSA -valued ManySortedSet of NAT ; :: thesis: for s being State of SCM+FSA
for I being InitHalting Program of SCM+FSA st Initialized I c= s & I c= p holds
for k being Element of NAT st k <= LifeSpan (p,s) holds
CurInstr ((p +* (loop I)),(Comput ((p +* (loop I)),(s +* (loop I)),k))) <> halt SCM+FSA
let s be State of SCM+FSA; :: thesis: for I being InitHalting Program of SCM+FSA st Initialized I c= s & I c= p holds
for k being Element of NAT st k <= LifeSpan (p,s) holds
CurInstr ((p +* (loop I)),(Comput ((p +* (loop I)),(s +* (loop I)),k))) <> halt SCM+FSA
let I be InitHalting Program of SCM+FSA; :: thesis: ( Initialized I c= s & I c= p implies for k being Element of NAT st k <= LifeSpan (p,s) holds
CurInstr ((p +* (loop I)),(Comput ((p +* (loop I)),(s +* (loop I)),k))) <> halt SCM+FSA )
set s2 = s +* (loop I);
set p2 = p +* (loop I);
A1: ProgramPart (Initialized I) = I by SCMFSA6A:33;
assume A2: Initialized I c= s ; :: thesis: ( not I c= p or for k being Element of NAT st k <= LifeSpan (p,s) holds
CurInstr ((p +* (loop I)),(Comput ((p +* (loop I)),(s +* (loop I)),k))) <> halt SCM+FSA )
assume A3: I c= p ; :: thesis: for k being Element of NAT st k <= LifeSpan (p,s) holds
CurInstr ((p +* (loop I)),(Comput ((p +* (loop I)),(s +* (loop I)),k))) <> halt SCM+FSA
then A4: p halts_on s by EXTPRO_1:def 10, A2, A1;

hereby :: thesis: verum

let k be Element of NAT ; :: thesis: ( k <= LifeSpan (p,s) implies not CurInstr ((p +* (loop I)),(Comput ((p +* (loop I)),(s +* (loop I)),k))) = halt SCM+FSA )
set lk = IC (Comput (p,s,k));
A5: ( IC (Comput (p,s,k)) in dom I & dom I = dom (loop I) ) by A2, Def1, FUNCT_4:105, A3;
then A6: (loop I) . (IC (Comput (p,s,k))) in rng (loop I) by FUNCT_1:def 5;
assume k <= LifeSpan (p,s) ; :: thesis: not CurInstr ((p +* (loop I)),(Comput ((p +* (loop I)),(s +* (loop I)),k))) = halt SCM+FSA
then IC (Comput (p,s,k)) = IC (Comput ((p +* (loop I)),(s +* (loop I)),k)) by A2, A4, Th65, COMPOS_1:24, A3;
then A7: CurInstr ((p +* (loop I)),(Comput ((p +* (loop I)),(s +* (loop I)),k))) = (p +* (loop I)) . (IC (Comput (p,s,k))) by PBOOLE:158
.= (loop I) . (IC (Comput (p,s,k))) by A5, FUNCT_4:14 ;
assume CurInstr ((p +* (loop I)),(Comput ((p +* (loop I)),(s +* (loop I)),k))) = halt SCM+FSA ; :: thesis: contradiction
hence contradiction by A7, A6, SCMFSA8C:107; :: thesis: verum

end;