let s be State of SCM+FSA; :: thesis: for P being Instruction-Sequence of SCM+FSA
for I being really-closed Program of SCM+FSA st I is_halting_on s,P holds
for k being Nat st k <= LifeSpan ((P +* I),(Initialize s)) holds
( Comput ((P +* I),(Initialize s),k) = Comput ((P +* (Directed I)),(Initialize s),k) & CurInstr ((P +* (Directed I)),(Comput ((P +* (Directed I)),(Initialize s),k))) <> halt SCM+FSA )
let P be Instruction-Sequence of SCM+FSA; :: thesis: for I being really-closed Program of SCM+FSA st I is_halting_on s,P holds
for k being Nat st k <= LifeSpan ((P +* I),(Initialize s)) holds
( Comput ((P +* I),(Initialize s),k) = Comput ((P +* (Directed I)),(Initialize s),k) & CurInstr ((P +* (Directed I)),(Comput ((P +* (Directed I)),(Initialize s),k))) <> halt SCM+FSA )
let I be really-closed Program of SCM+FSA; :: thesis: ( I is_halting_on s,P implies for k being Nat st k <= LifeSpan ((P +* I),(Initialize s)) holds
( Comput ((P +* I),(Initialize s),k) = Comput ((P +* (Directed I)),(Initialize s),k) & CurInstr ((P +* (Directed I)),(Comput ((P +* (Directed I)),(Initialize s),k))) <> halt SCM+FSA ) )
assume A1: I is_halting_on s,P ; :: thesis: for k being Nat st k <= LifeSpan ((P +* I),(Initialize s)) holds
( Comput ((P +* I),(Initialize s),k) = Comput ((P +* (Directed I)),(Initialize s),k) & CurInstr ((P +* (Directed I)),(Comput ((P +* (Directed I)),(Initialize s),k))) <> halt SCM+FSA )
A2: dom (P +* (Directed I)) = NAT by PARTFUN1:def 2;
A3: dom (P +* I) = NAT by PARTFUN1:def 2;
set s2 = Initialize s;
set s1 = Initialize s;
defpred S₁[ Nat] means ( $1 <= LifeSpan ((P +* I),(Initialize s)) implies ( Comput ((P +* I),(Initialize s),$1) = Comput ((P +* (Directed I)),(Initialize s),$1) & CurInstr ((P +* (Directed I)),(Comput ((P +* (Directed I)),(Initialize s),$1))) <> halt SCM+FSA ) );
IC (Initialize s) = 0 by MEMSTR_0:47;
then A4: IC (Initialize s) in dom I by AFINSQ_1:65;
A5: I c= P +* I by FUNCT_4:25;

A6: now :: thesis: for k being Element of NAT st Comput ((P +* I),(Initialize s),k) = Comput ((P +* (Directed I)),(Initialize s),k) holds
not CurInstr ((P +* (Directed I)),(Comput ((P +* (Directed I)),(Initialize s),k))) = halt SCM+FSA

let k be Element of NAT ; :: thesis: ( Comput ((P +* I),(Initialize s),k) = Comput ((P +* (Directed I)),(Initialize s),k) implies not CurInstr ((P +* (Directed I)),(Comput ((P +* (Directed I)),(Initialize s),k))) = halt SCM+FSA )
dom (Directed I) = dom I by FUNCT_4:99;
then A7: IC (Comput ((P +* I),(Initialize s),k)) in dom (Directed I) by AMISTD_1:21, A4, A5;
A8: (P +* (Directed I)) /. (IC (Comput ((P +* (Directed I)),(Initialize s),k))) = (P +* (Directed I)) . (IC (Comput ((P +* (Directed I)),(Initialize s),k))) by A2, PARTFUN1:def 6;
A9: Directed I c= P +* (Directed I) by FUNCT_4:25;
assume Comput ((P +* I),(Initialize s),k) = Comput ((P +* (Directed I)),(Initialize s),k) ; :: thesis: not CurInstr ((P +* (Directed I)),(Comput ((P +* (Directed I)),(Initialize s),k))) = halt SCM+FSA
then CurInstr ((P +* (Directed I)),(Comput ((P +* (Directed I)),(Initialize s),k))) = (P +* (Directed I)) . (IC (Comput ((P +* I),(Initialize s),k))) by A8
.= (Directed I) . (IC (Comput ((P +* I),(Initialize s),k))) by A7, A9, GRFUNC_1:2 ;
then A10: CurInstr ((P +* (Directed I)),(Comput ((P +* (Directed I)),(Initialize s),k))) in rng (Directed I) by A7, FUNCT_1:def 3;
assume CurInstr ((P +* (Directed I)),(Comput ((P +* (Directed I)),(Initialize s),k))) = halt SCM+FSA ; :: thesis: contradiction
hence contradiction by A10, SCMFSA6A:1; :: thesis: verum

end;

now :: thesis: for k being Nat st ( k <= LifeSpan ((P +* I),(Initialize s)) implies Comput ((P +* I),(Initialize s),k) = Comput ((P +* (Directed I)),(Initialize s),k) ) & k + 1 <= LifeSpan ((P +* I),(Initialize s)) holds
( Comput ((P +* I),(Initialize s),(k + 1)) = Comput ((P +* (Directed I)),(Initialize s),(k + 1)) & CurInstr ((P +* (Directed I)),(Comput ((P +* (Directed I)),(Initialize s),(k + 1)))) <> halt SCM+FSA )

A11: P +* I halts_on Initialize s by A1, SCMFSA7B:def 7;
A12: dom I c= dom (Directed I) by FUNCT_4:99;
let k be Nat; :: thesis: ( ( k <= LifeSpan ((P +* I),(Initialize s)) implies Comput ((P +* I),(Initialize s),k) = Comput ((P +* (Directed I)),(Initialize s),k) ) & k + 1 <= LifeSpan ((P +* I),(Initialize s)) implies ( Comput ((P +* I),(Initialize s),(k + 1)) = Comput ((P +* (Directed I)),(Initialize s),(k + 1)) & CurInstr ((P +* (Directed I)),(Comput ((P +* (Directed I)),(Initialize s),(k + 1)))) <> halt SCM+FSA ) )
assume A13: ( k <= LifeSpan ((P +* I),(Initialize s)) implies Comput ((P +* I),(Initialize s),k) = Comput ((P +* (Directed I)),(Initialize s),k) ) ; :: thesis: ( k + 1 <= LifeSpan ((P +* I),(Initialize s)) implies ( Comput ((P +* I),(Initialize s),(k + 1)) = Comput ((P +* (Directed I)),(Initialize s),(k + 1)) & CurInstr ((P +* (Directed I)),(Comput ((P +* (Directed I)),(Initialize s),(k + 1)))) <> halt SCM+FSA ) )
A14: Comput ((P +* (Directed I)),(Initialize s),(k + 1)) = Following ((P +* (Directed I)),(Comput ((P +* (Directed I)),(Initialize s),k))) by EXTPRO_1:3
.= Exec ((CurInstr ((P +* (Directed I)),(Comput ((P +* (Directed I)),(Initialize s),k)))),(Comput ((P +* (Directed I)),(Initialize s),k))) ;
A15: IC (Comput ((P +* I),(Initialize s),k)) in dom I by AMISTD_1:21, A4, A5;
A16: I c= P +* I by FUNCT_4:25;
A17: CurInstr ((P +* I),(Comput ((P +* I),(Initialize s),k))) = (P +* I) . (IC (Comput ((P +* I),(Initialize s),k))) by A3, PARTFUN1:def 6
.= I . (IC (Comput ((P +* I),(Initialize s),k))) by A15, A16, GRFUNC_1:2 ;
A18: k + 0 < k + 1 by XREAL_1:6;
A19: (P +* (Directed I)) /. (IC (Comput ((P +* (Directed I)),(Initialize s),k))) = (P +* (Directed I)) . (IC (Comput ((P +* (Directed I)),(Initialize s),k))) by A2, PARTFUN1:def 6;
A20: Directed I c= P +* (Directed I) by FUNCT_4:25;
assume A21: k + 1 <= LifeSpan ((P +* I),(Initialize s)) ; :: thesis: ( Comput ((P +* I),(Initialize s),(k + 1)) = Comput ((P +* (Directed I)),(Initialize s),(k + 1)) & CurInstr ((P +* (Directed I)),(Comput ((P +* (Directed I)),(Initialize s),(k + 1)))) <> halt SCM+FSA )
then k < LifeSpan ((P +* I),(Initialize s)) by A18, XXREAL_0:2;
then I . (IC (Comput ((P +* I),(Initialize s),k))) <> halt SCM+FSA by A17, A11, EXTPRO_1:def 15;
then A22: CurInstr ((P +* I),(Comput ((P +* I),(Initialize s),k))) = (Directed I) . (IC (Comput ((P +* I),(Initialize s),k))) by A17, FUNCT_4:105
.= (P +* (Directed I)) . (IC (Comput ((P +* I),(Initialize s),k))) by A15, A12, A20, GRFUNC_1:2
.= CurInstr ((P +* (Directed I)),(Comput ((P +* (Directed I)),(Initialize s),k))) by A13, A21, A18, A19, XXREAL_0:2 ;
Comput ((P +* I),(Initialize s),(k + 1)) = Following ((P +* I),(Comput ((P +* I),(Initialize s),k))) by EXTPRO_1:3
.= Exec ((CurInstr ((P +* I),(Comput ((P +* I),(Initialize s),k)))),(Comput ((P +* I),(Initialize s),k))) ;
hence Comput ((P +* I),(Initialize s),(k + 1)) = Comput ((P +* (Directed I)),(Initialize s),(k + 1)) by A13, A21, A18, A22, A14, XXREAL_0:2; :: thesis: CurInstr ((P +* (Directed I)),(Comput ((P +* (Directed I)),(Initialize s),(k + 1)))) <> halt SCM+FSA
hence CurInstr ((P +* (Directed I)),(Comput ((P +* (Directed I)),(Initialize s),(k + 1)))) <> halt SCM+FSA by A6; :: thesis: verum

end;

then A23: for k being Nat st S₁[k] holds
S₁[k + 1] ;

now :: thesis: ( 0 <= LifeSpan ((P +* I),(Initialize s)) implies ( Comput ((P +* I),(Initialize s),0) = Comput ((P +* (Directed I)),(Initialize s),0) & CurInstr ((P +* (Directed I)),(Comput ((P +* (Directed I)),(Initialize s),0))) <> halt SCM+FSA ) )

assume 0 <= LifeSpan ((P +* I),(Initialize s)) ; :: thesis: ( Comput ((P +* I),(Initialize s),0) = Comput ((P +* (Directed I)),(Initialize s),0) & CurInstr ((P +* (Directed I)),(Comput ((P +* (Directed I)),(Initialize s),0))) <> halt SCM+FSA )
thus Comput ((P +* I),(Initialize s),0) = Comput ((P +* (Directed I)),(Initialize s),0) ; :: thesis: CurInstr ((P +* (Directed I)),(Comput ((P +* (Directed I)),(Initialize s),0))) <> halt SCM+FSA
hence CurInstr ((P +* (Directed I)),(Comput ((P +* (Directed I)),(Initialize s),0))) <> halt SCM+FSA by A6; :: thesis: verum

end;

then A24: S₁[ 0 ] ;
thus for k being Nat holds S₁[k] from NAT_1:sch 2(A24, A23); :: thesis: verum