let s be State of SCM+FSA; :: thesis: for P being the Instructions of SCM+FSA -valued ManySortedSet of NAT
for I being paraclosed Program of SCM+FSA st P +* I halts_on s +* (Initialize I) holds
for J being Program of SCM+FSA
for k being Element of NAT st k <= LifeSpan ((P +* I),(s +* (Initialize I))) holds
Comput ((P +* I),(s +* (Initialize I)),k), Comput ((P +* (I ';' J)),(s +* (Initialize (I ';' J))),k) equal_outside NAT
let P be the Instructions of SCM+FSA -valued ManySortedSet of NAT ; :: thesis: for I being paraclosed Program of SCM+FSA st P +* I halts_on s +* (Initialize I) holds
for J being Program of SCM+FSA
for k being Element of NAT st k <= LifeSpan ((P +* I),(s +* (Initialize I))) holds
Comput ((P +* I),(s +* (Initialize I)),k), Comput ((P +* (I ';' J)),(s +* (Initialize (I ';' J))),k) equal_outside NAT
let I be paraclosed Program of SCM+FSA; :: thesis: ( P +* I halts_on s +* (Initialize I) implies for J being Program of SCM+FSA
for k being Element of NAT st k <= LifeSpan ((P +* I),(s +* (Initialize I))) holds
Comput ((P +* I),(s +* (Initialize I)),k), Comput ((P +* (I ';' J)),(s +* (Initialize (I ';' J))),k) equal_outside NAT )
assume A1: P +* I halts_on s +* (Initialize I) ; :: thesis: for J being Program of SCM+FSA
for k being Element of NAT st k <= LifeSpan ((P +* I),(s +* (Initialize I))) holds
Comput ((P +* I),(s +* (Initialize I)),k), Comput ((P +* (I ';' J)),(s +* (Initialize (I ';' J))),k) equal_outside NAT
set s1 = s +* (Initialize I);
let J be Program of SCM+FSA; :: thesis: for k being Element of NAT st k <= LifeSpan ((P +* I),(s +* (Initialize I))) holds
Comput ((P +* I),(s +* (Initialize I)),k), Comput ((P +* (I ';' J)),(s +* (Initialize (I ';' J))),k) equal_outside NAT
A2: Initialize I c= s +* (Initialize I) by FUNCT_4:26;
ProgramPart (Initialize I) = I by COMPOS_1:144;
then A3: ProgramPart (Initialize I) c= P +* I by FUNCT_4:26;
A4: ProgramPart (Initialize I) = I by COMPOS_1:144;
set s2 = s +* (Initialize (I ';' J));
defpred S₁[ Nat] means ( $1 <= LifeSpan ((P +* I),(s +* (Initialize I))) implies Comput ((P +* I),(s +* (Initialize I)),$1), Comput ((P +* (I ';' J)),(s +* (Initialize (I ';' J))),$1) equal_outside NAT );
A5: for m being Element of NAT st S₁[m] holds
S₁[m + 1]

proof

dom (I ';' J) = (dom (Directed I)) \/ (dom (Reloc (J,(card I)))) by FUNCT_4:def 1
.= (dom I) \/ (dom (Reloc (J,(card I)))) by FUNCT_4:105 ;
then A6: dom I c= dom (I ';' J) by XBOOLE_1:7;
let m be Element of NAT ; :: thesis: ( S₁[m] implies S₁[m + 1] )
assume A7: ( m <= LifeSpan ((P +* I),(s +* (Initialize I))) implies Comput ((P +* I),(s +* (Initialize I)),m), Comput ((P +* (I ';' J)),(s +* (Initialize (I ';' J))),m) equal_outside NAT ) ; :: thesis: S₁[m + 1]
A8: Comput ((P +* I),(s +* (Initialize I)),(m + 1)) = Following ((P +* I),(Comput ((P +* I),(s +* (Initialize I)),m))) by EXTPRO_1:4
.= Exec ((CurInstr ((P +* I),(Comput ((P +* I),(s +* (Initialize I)),m)))),(Comput ((P +* I),(s +* (Initialize I)),m))) ;
A9: Comput ((P +* (I ';' J)),(s +* (Initialize (I ';' J))),(m + 1)) = Following ((P +* (I ';' J)),(Comput ((P +* (I ';' J)),(s +* (Initialize (I ';' J))),m))) by EXTPRO_1:4
.= Exec ((CurInstr ((P +* (I ';' J)),(Comput ((P +* (I ';' J)),(s +* (Initialize (I ';' J))),m)))),(Comput ((P +* (I ';' J)),(s +* (Initialize (I ';' J))),m))) ;
A10: IC (Comput ((P +* I),(s +* (Initialize I)),m)) in dom I by Def2, A2, A3, A4;
A11: I c= P +* I by FUNCT_4:26;
dom (P +* I) = NAT by PARTFUN1:def 4;
then A12: CurInstr ((P +* I),(Comput ((P +* I),(s +* (Initialize I)),m))) = (P +* I) . (IC (Comput ((P +* I),(s +* (Initialize I)),m))) by PARTFUN1:def 8
.= I . (IC (Comput ((P +* I),(s +* (Initialize I)),m))) by A10, GRFUNC_1:8, A11 ;
assume A13: m + 1 <= LifeSpan ((P +* I),(s +* (Initialize I))) ; :: thesis: Comput ((P +* I),(s +* (Initialize I)),(m + 1)), Comput ((P +* (I ';' J)),(s +* (Initialize (I ';' J))),(m + 1)) equal_outside NAT
then A14: IC (Comput ((P +* I),(s +* (Initialize I)),m)) = IC (Comput ((P +* (I ';' J)),(s +* (Initialize (I ';' J))),m)) by A7, COMPOS_1:24, NAT_1:13;
A15: I ';' J c= P +* (I ';' J) by FUNCT_4:26;
A16: dom (P +* (I ';' J)) = NAT by PARTFUN1:def 4;
m < LifeSpan ((P +* I),(s +* (Initialize I))) by A13, NAT_1:13;
then I . (IC (Comput ((P +* I),(s +* (Initialize I)),m))) <> halt SCM+FSA by A1, A12, EXTPRO_1:def 14;
then A17: CurInstr ((P +* I),(Comput ((P +* I),(s +* (Initialize I)),m))) = (I ';' J) . (IC (Comput ((P +* I),(s +* (Initialize I)),m))) by A10, A12, SCMFSA6A:54
.= (P +* (I ';' J)) . (IC (Comput ((P +* (I ';' J)),(s +* (Initialize (I ';' J))),m))) by A14, A10, A6, GRFUNC_1:8, A15
.= CurInstr ((P +* (I ';' J)),(Comput ((P +* (I ';' J)),(s +* (Initialize (I ';' J))),m))) by A16, PARTFUN1:def 8 ;
Comput ((P +* I),(s +* (Initialize I)),m), Comput ((P +* (I ';' J)),(s +* (Initialize (I ';' J))),m) equal_outside NAT by A7, A13, NAT_1:13;
hence Comput ((P +* I),(s +* (Initialize I)),(m + 1)), Comput ((P +* (I ';' J)),(s +* (Initialize (I ';' J))),(m + 1)) equal_outside NAT by A8, A9, AMISTD_2:def 20, A17; :: thesis: verum

end;

A18: ( Comput ((P +* I),(s +* (Initialize I)),0) = s +* (Initialize I) & Comput ((P +* (I ';' J)),(s +* (Initialize (I ';' J))),0) = s +* (Initialize (I ';' J)) ) by EXTPRO_1:3;
A19: ( (Initialize s) +* I, Initialize s equal_outside NAT & Initialize s,(Initialize s) +* (I ';' J) equal_outside NAT ) by FUNCT_7:28, FUNCT_7:132;
A20: s +* (Initialize (I ';' J)) = Initialize (s +* (I ';' J)) by FUNCT_4:15
.= (Initialize s) +* (I ';' J) by COMPOS_1:83 ;
s +* (Initialize I) = Initialize (s +* I) by FUNCT_4:15
.= (Initialize s) +* I by COMPOS_1:83 ;
then A21: S₁[ 0 ] by A20, A19, A18, FUNCT_7:29;
thus for k being Element of NAT holds S₁[k] from NAT_1:sch 1(A21, A5); :: thesis: verum