let s be State of SCM+FSA; :: thesis: for P being the Instructions of SCM+FSA -valued ManySortedSet of NAT
for I being Program of SCM+FSA st I is_closed_on Initialized s,P & I is_halting_on Initialized s,P holds
for k being Element of NAT st k <= LifeSpan ((P +* I),(s +* (Initialize ((intloc 0) .--> 1)))) holds
( NPP (Comput ((P +* I),(s +* (Initialize ((intloc 0) .--> 1))),k)) = NPP (Comput ((P +* (Directed I)),(s +* (Initialize ((intloc 0) .--> 1))),k)) & CurInstr ((P +* (Directed I)),(Comput ((P +* (Directed I)),(s +* (Initialize ((intloc 0) .--> 1))),k))) <> halt SCM+FSA )
let P be the Instructions of SCM+FSA -valued ManySortedSet of NAT ; :: thesis: for I being Program of SCM+FSA st I is_closed_on Initialized s,P & I is_halting_on Initialized s,P holds
for k being Element of NAT st k <= LifeSpan ((P +* I),(s +* (Initialize ((intloc 0) .--> 1)))) holds
( NPP (Comput ((P +* I),(s +* (Initialize ((intloc 0) .--> 1))),k)) = NPP (Comput ((P +* (Directed I)),(s +* (Initialize ((intloc 0) .--> 1))),k)) & CurInstr ((P +* (Directed I)),(Comput ((P +* (Directed I)),(s +* (Initialize ((intloc 0) .--> 1))),k))) <> halt SCM+FSA )
let I be Program of SCM+FSA; :: thesis: ( I is_closed_on Initialized s,P & I is_halting_on Initialized s,P implies for k being Element of NAT st k <= LifeSpan ((P +* I),(s +* (Initialize ((intloc 0) .--> 1)))) holds
( NPP (Comput ((P +* I),(s +* (Initialize ((intloc 0) .--> 1))),k)) = NPP (Comput ((P +* (Directed I)),(s +* (Initialize ((intloc 0) .--> 1))),k)) & CurInstr ((P +* (Directed I)),(Comput ((P +* (Directed I)),(s +* (Initialize ((intloc 0) .--> 1))),k))) <> halt SCM+FSA ) )
set s1 = s +* (Initialize ((intloc 0) .--> 1));
set s2 = s +* (Initialize ((intloc 0) .--> 1));
A2: dom (P +* (Directed I)) = NAT by PARTFUN1:def 4;
A3: dom (P +* I) = NAT by PARTFUN1:def 4;
A4: Directed I c= P +* (Directed I) by FUNCT_4:26;
defpred S₁[ Nat] means ( $1 <= LifeSpan ((P +* I),(s +* (Initialize ((intloc 0) .--> 1)))) implies ( NPP (Comput ((P +* I),(s +* (Initialize ((intloc 0) .--> 1))),$1)) = NPP (Comput ((P +* (Directed I)),(s +* (Initialize ((intloc 0) .--> 1))),$1)) & CurInstr ((P +* (Directed I)),(Comput ((P +* (Directed I)),(s +* (Initialize ((intloc 0) .--> 1))),$1))) <> halt SCM+FSA ) );
A5: s +* (Initialize ((intloc 0) .--> 1)) = Initialize (Initialized s) by Th13;
assume A6: I is_closed_on Initialized s,P ; :: thesis: ( not I is_halting_on Initialized s,P or for k being Element of NAT st k <= LifeSpan ((P +* I),(s +* (Initialize ((intloc 0) .--> 1)))) holds
( NPP (Comput ((P +* I),(s +* (Initialize ((intloc 0) .--> 1))),k)) = NPP (Comput ((P +* (Directed I)),(s +* (Initialize ((intloc 0) .--> 1))),k)) & CurInstr ((P +* (Directed I)),(Comput ((P +* (Directed I)),(s +* (Initialize ((intloc 0) .--> 1))),k))) <> halt SCM+FSA ) )

A7: now

let k be Element of NAT ; :: thesis: ( NPP (Comput ((P +* I),(s +* (Initialize ((intloc 0) .--> 1))),k)) = NPP (Comput ((P +* (Directed I)),(s +* (Initialize ((intloc 0) .--> 1))),k)) implies not CurInstr ((P +* (Directed I)),(Comput ((P +* (Directed I)),(s +* (Initialize ((intloc 0) .--> 1))),k))) = halt SCM+FSA )
dom (Directed I) = dom I by FUNCT_4:105;
then A8: IC (Comput ((P +* I),(s +* (Initialize ((intloc 0) .--> 1))),k)) in dom (Directed I) by A6, A5, SCMFSA7B:def 7;
A9: (P +* (Directed I)) /. (IC (Comput ((P +* (Directed I)),(s +* (Initialize ((intloc 0) .--> 1))),k))) = (P +* (Directed I)) . (IC (Comput ((P +* (Directed I)),(s +* (Initialize ((intloc 0) .--> 1))),k))) by A2, PARTFUN1:def 8;
assume NPP (Comput ((P +* I),(s +* (Initialize ((intloc 0) .--> 1))),k)) = NPP (Comput ((P +* (Directed I)),(s +* (Initialize ((intloc 0) .--> 1))),k)) ; :: thesis: not CurInstr ((P +* (Directed I)),(Comput ((P +* (Directed I)),(s +* (Initialize ((intloc 0) .--> 1))),k))) = halt SCM+FSA
then CurInstr ((P +* (Directed I)),(Comput ((P +* (Directed I)),(s +* (Initialize ((intloc 0) .--> 1))),k))) = (P +* (Directed I)) . (IC (Comput ((P +* I),(s +* (Initialize ((intloc 0) .--> 1))),k))) by A9, COMPOS_1:230
.= (Directed I) . (IC (Comput ((P +* I),(s +* (Initialize ((intloc 0) .--> 1))),k))) by A8, GRFUNC_1:8, A4 ;
then A10: CurInstr ((P +* (Directed I)),(Comput ((P +* (Directed I)),(s +* (Initialize ((intloc 0) .--> 1))),k))) in rng (Directed I) by A8, FUNCT_1:def 5;
assume CurInstr ((P +* (Directed I)),(Comput ((P +* (Directed I)),(s +* (Initialize ((intloc 0) .--> 1))),k))) = halt SCM+FSA ; :: thesis: contradiction
hence contradiction by A10, SCMFSA6A:18; :: thesis: verum

end;

assume A11: I is_halting_on Initialized s,P ; :: thesis: for k being Element of NAT st k <= LifeSpan ((P +* I),(s +* (Initialize ((intloc 0) .--> 1)))) holds
( NPP (Comput ((P +* I),(s +* (Initialize ((intloc 0) .--> 1))),k)) = NPP (Comput ((P +* (Directed I)),(s +* (Initialize ((intloc 0) .--> 1))),k)) & CurInstr ((P +* (Directed I)),(Comput ((P +* (Directed I)),(s +* (Initialize ((intloc 0) .--> 1))),k))) <> halt SCM+FSA )

now

A12: P +* I halts_on s +* (Initialize ((intloc 0) .--> 1)) by A11, A5, SCMFSA7B:def 8;
A13: dom I c= dom (Directed I) by FUNCT_4:105;
let k be Element of NAT ; :: thesis: ( ( k <= LifeSpan ((P +* I),(s +* (Initialize ((intloc 0) .--> 1)))) implies NPP (Comput ((P +* I),(s +* (Initialize ((intloc 0) .--> 1))),k)) = NPP (Comput ((P +* (Directed I)),(s +* (Initialize ((intloc 0) .--> 1))),k)) ) & k + 1 <= LifeSpan ((P +* I),(s +* (Initialize ((intloc 0) .--> 1)))) implies ( NPP (Comput ((P +* I),(s +* (Initialize ((intloc 0) .--> 1))),(k + 1))) = NPP (Comput ((P +* (Directed I)),(s +* (Initialize ((intloc 0) .--> 1))),(k + 1))) & CurInstr ((P +* (Directed I)),(Comput ((P +* (Directed I)),(s +* (Initialize ((intloc 0) .--> 1))),(k + 1)))) <> halt SCM+FSA ) )
assume A14: ( k <= LifeSpan ((P +* I),(s +* (Initialize ((intloc 0) .--> 1)))) implies NPP (Comput ((P +* I),(s +* (Initialize ((intloc 0) .--> 1))),k)) = NPP (Comput ((P +* (Directed I)),(s +* (Initialize ((intloc 0) .--> 1))),k)) ) ; :: thesis: ( k + 1 <= LifeSpan ((P +* I),(s +* (Initialize ((intloc 0) .--> 1)))) implies ( NPP (Comput ((P +* I),(s +* (Initialize ((intloc 0) .--> 1))),(k + 1))) = NPP (Comput ((P +* (Directed I)),(s +* (Initialize ((intloc 0) .--> 1))),(k + 1))) & CurInstr ((P +* (Directed I)),(Comput ((P +* (Directed I)),(s +* (Initialize ((intloc 0) .--> 1))),(k + 1)))) <> halt SCM+FSA ) )
A15: Comput ((P +* (Directed I)),(s +* (Initialize ((intloc 0) .--> 1))),(k + 1)) = Following ((P +* (Directed I)),(Comput ((P +* (Directed I)),(s +* (Initialize ((intloc 0) .--> 1))),k))) by EXTPRO_1:4
.= Exec ((CurInstr ((P +* (Directed I)),(Comput ((P +* (Directed I)),(s +* (Initialize ((intloc 0) .--> 1))),k)))),(Comput ((P +* (Directed I)),(s +* (Initialize ((intloc 0) .--> 1))),k))) ;
A16: IC (Comput ((P +* I),(s +* (Initialize ((intloc 0) .--> 1))),k)) in dom I by A6, A5, SCMFSA7B:def 7;
A17: I c= P +* I by FUNCT_4:26;
A18: CurInstr ((P +* I),(Comput ((P +* I),(s +* (Initialize ((intloc 0) .--> 1))),k))) = (P +* I) . (IC (Comput ((P +* I),(s +* (Initialize ((intloc 0) .--> 1))),k))) by A3, PARTFUN1:def 8
.= I . (IC (Comput ((P +* I),(s +* (Initialize ((intloc 0) .--> 1))),k))) by A16, A17, GRFUNC_1:8 ;
A19: k + 0 < k + 1 by XREAL_1:8;
assume A20: k + 1 <= LifeSpan ((P +* I),(s +* (Initialize ((intloc 0) .--> 1)))) ; :: thesis: ( NPP (Comput ((P +* I),(s +* (Initialize ((intloc 0) .--> 1))),(k + 1))) = NPP (Comput ((P +* (Directed I)),(s +* (Initialize ((intloc 0) .--> 1))),(k + 1))) & CurInstr ((P +* (Directed I)),(Comput ((P +* (Directed I)),(s +* (Initialize ((intloc 0) .--> 1))),(k + 1)))) <> halt SCM+FSA )
then k < LifeSpan ((P +* I),(s +* (Initialize ((intloc 0) .--> 1)))) by A19, XXREAL_0:2;
then I . (IC (Comput ((P +* I),(s +* (Initialize ((intloc 0) .--> 1))),k))) <> halt SCM+FSA by A18, A12, EXTPRO_1:def 14;
then A21: CurInstr ((P +* I),(Comput ((P +* I),(s +* (Initialize ((intloc 0) .--> 1))),k))) = (Directed I) . (IC (Comput ((P +* I),(s +* (Initialize ((intloc 0) .--> 1))),k))) by A18, FUNCT_4:111
.= (P +* (Directed I)) . (IC (Comput ((P +* I),(s +* (Initialize ((intloc 0) .--> 1))),k))) by A4, A16, A13, GRFUNC_1:8
.= (P +* (Directed I)) . (IC (Comput ((P +* (Directed I)),(s +* (Initialize ((intloc 0) .--> 1))),k))) by A14, A20, A19, COMPOS_1:230, XXREAL_0:2
.= CurInstr ((P +* (Directed I)),(Comput ((P +* (Directed I)),(s +* (Initialize ((intloc 0) .--> 1))),k))) by A2, PARTFUN1:def 8 ;
Comput ((P +* I),(s +* (Initialize ((intloc 0) .--> 1))),(k + 1)) = Following ((P +* I),(Comput ((P +* I),(s +* (Initialize ((intloc 0) .--> 1))),k))) by EXTPRO_1:4
.= Exec ((CurInstr ((P +* I),(Comput ((P +* I),(s +* (Initialize ((intloc 0) .--> 1))),k)))),(Comput ((P +* I),(s +* (Initialize ((intloc 0) .--> 1))),k))) ;
hence NPP (Comput ((P +* I),(s +* (Initialize ((intloc 0) .--> 1))),(k + 1))) = NPP (Comput ((P +* (Directed I)),(s +* (Initialize ((intloc 0) .--> 1))),(k + 1))) by A14, A20, A19, A21, A15, AMISTD_2:def 20, XXREAL_0:2; :: thesis: CurInstr ((P +* (Directed I)),(Comput ((P +* (Directed I)),(s +* (Initialize ((intloc 0) .--> 1))),(k + 1)))) <> halt SCM+FSA
hence CurInstr ((P +* (Directed I)),(Comput ((P +* (Directed I)),(s +* (Initialize ((intloc 0) .--> 1))),(k + 1)))) <> halt SCM+FSA by A7; :: thesis: verum

end;

then A22: for k being Element of NAT st S₁[k] holds
S₁[k + 1] ;

now

assume 0 <= LifeSpan ((P +* I),(s +* (Initialize ((intloc 0) .--> 1)))) ; :: thesis: ( NPP (Comput ((P +* I),(s +* (Initialize ((intloc 0) .--> 1))),0)) = NPP (Comput ((P +* (Directed I)),(s +* (Initialize ((intloc 0) .--> 1))),0)) & CurInstr ((P +* (Directed I)),(Comput ((P +* (Directed I)),(s +* (Initialize ((intloc 0) .--> 1))),0))) <> halt SCM+FSA )
( Comput ((P +* I),(s +* (Initialize ((intloc 0) .--> 1))),0) = s +* (Initialize ((intloc 0) .--> 1)) & Comput ((P +* (Directed I)),(s +* (Initialize ((intloc 0) .--> 1))),0) = s +* (Initialize ((intloc 0) .--> 1)) ) by EXTPRO_1:3;
hence NPP (Comput ((P +* I),(s +* (Initialize ((intloc 0) .--> 1))),0)) = NPP (Comput ((P +* (Directed I)),(s +* (Initialize ((intloc 0) .--> 1))),0)) ; :: thesis: CurInstr ((P +* (Directed I)),(Comput ((P +* (Directed I)),(s +* (Initialize ((intloc 0) .--> 1))),0))) <> halt SCM+FSA
hence CurInstr ((P +* (Directed I)),(Comput ((P +* (Directed I)),(s +* (Initialize ((intloc 0) .--> 1))),0))) <> halt SCM+FSA by A7; :: thesis: verum

end;

then A24: S₁[ 0 ] ;
thus for k being Element of NAT holds S₁[k] from NAT_1:sch 1(A24, A22); :: thesis: verum