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 +* (Start-At (0,SCM+FSA)) holds
for J being Program of SCM+FSA
for k being Element of NAT st k <= LifeSpan ((P +* I),(s +* (Start-At (0,SCM+FSA)))) holds
NPP (Comput ((P +* I),(s +* (Start-At (0,SCM+FSA))),k)) = NPP (Comput ((P +* (I ';' J)),(s +* (Start-At (0,SCM+FSA))),k))
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 +* (Start-At (0,SCM+FSA)) holds
for J being Program of SCM+FSA
for k being Element of NAT st k <= LifeSpan ((P +* I),(s +* (Start-At (0,SCM+FSA)))) holds
NPP (Comput ((P +* I),(s +* (Start-At (0,SCM+FSA))),k)) = NPP (Comput ((P +* (I ';' J)),(s +* (Start-At (0,SCM+FSA))),k))
let I be paraclosed Program of SCM+FSA; :: thesis: ( P +* I halts_on s +* (Start-At (0,SCM+FSA)) implies for J being Program of SCM+FSA
for k being Element of NAT st k <= LifeSpan ((P +* I),(s +* (Start-At (0,SCM+FSA)))) holds
NPP (Comput ((P +* I),(s +* (Start-At (0,SCM+FSA))),k)) = NPP (Comput ((P +* (I ';' J)),(s +* (Start-At (0,SCM+FSA))),k)) )
assume A1: P +* I halts_on s +* (Start-At (0,SCM+FSA)) ; :: thesis: for J being Program of SCM+FSA
for k being Element of NAT st k <= LifeSpan ((P +* I),(s +* (Start-At (0,SCM+FSA)))) holds
NPP (Comput ((P +* I),(s +* (Start-At (0,SCM+FSA))),k)) = NPP (Comput ((P +* (I ';' J)),(s +* (Start-At (0,SCM+FSA))),k))
set s1 = s +* (Start-At (0,SCM+FSA));
let J be Program of SCM+FSA; :: thesis: for k being Element of NAT st k <= LifeSpan ((P +* I),(s +* (Start-At (0,SCM+FSA)))) holds
NPP (Comput ((P +* I),(s +* (Start-At (0,SCM+FSA))),k)) = NPP (Comput ((P +* (I ';' J)),(s +* (Start-At (0,SCM+FSA))),k))
B2: Start-At (0,SCM+FSA) c= s +* (Start-At (0,SCM+FSA)) 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 +* (Start-At (0,SCM+FSA));
defpred S₁[ Nat] means ( $1 <= LifeSpan ((P +* I),(s +* (Start-At (0,SCM+FSA)))) implies NPP (Comput ((P +* I),(s +* (Start-At (0,SCM+FSA))),$1)) = NPP (Comput ((P +* (I ';' J)),(s +* (Start-At (0,SCM+FSA))),$1)) );
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 +* (Start-At (0,SCM+FSA)))) implies NPP (Comput ((P +* I),(s +* (Start-At (0,SCM+FSA))),m)) = NPP (Comput ((P +* (I ';' J)),(s +* (Start-At (0,SCM+FSA))),m)) ) ; :: thesis: S₁[m + 1]
A8: Comput ((P +* I),(s +* (Start-At (0,SCM+FSA))),(m + 1)) = Following ((P +* I),(Comput ((P +* I),(s +* (Start-At (0,SCM+FSA))),m))) by EXTPRO_1:4
.= Exec ((CurInstr ((P +* I),(Comput ((P +* I),(s +* (Start-At (0,SCM+FSA))),m)))),(Comput ((P +* I),(s +* (Start-At (0,SCM+FSA))),m))) ;
A9: Comput ((P +* (I ';' J)),(s +* (Start-At (0,SCM+FSA))),(m + 1)) = Following ((P +* (I ';' J)),(Comput ((P +* (I ';' J)),(s +* (Start-At (0,SCM+FSA))),m))) by EXTPRO_1:4
.= Exec ((CurInstr ((P +* (I ';' J)),(Comput ((P +* (I ';' J)),(s +* (Start-At (0,SCM+FSA))),m)))),(Comput ((P +* (I ';' J)),(s +* (Start-At (0,SCM+FSA))),m))) ;
A10: IC (Comput ((P +* I),(s +* (Start-At (0,SCM+FSA))),m)) in dom I by Def2, A3, A4, B2;
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 +* (Start-At (0,SCM+FSA))),m))) = (P +* I) . (IC (Comput ((P +* I),(s +* (Start-At (0,SCM+FSA))),m))) by PARTFUN1:def 8
.= I . (IC (Comput ((P +* I),(s +* (Start-At (0,SCM+FSA))),m))) by A10, A11, GRFUNC_1:8 ;
assume A13: m + 1 <= LifeSpan ((P +* I),(s +* (Start-At (0,SCM+FSA)))) ; :: thesis: NPP (Comput ((P +* I),(s +* (Start-At (0,SCM+FSA))),(m + 1))) = NPP (Comput ((P +* (I ';' J)),(s +* (Start-At (0,SCM+FSA))),(m + 1)))
then A14: IC (Comput ((P +* I),(s +* (Start-At (0,SCM+FSA))),m)) = IC (Comput ((P +* (I ';' J)),(s +* (Start-At (0,SCM+FSA))),m)) by A7, COMPOS_1:230, 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 +* (Start-At (0,SCM+FSA)))) by A13, NAT_1:13;
then I . (IC (Comput ((P +* I),(s +* (Start-At (0,SCM+FSA))),m))) <> halt SCM+FSA by A1, A12, EXTPRO_1:def 14;
then CurInstr ((P +* I),(Comput ((P +* I),(s +* (Start-At (0,SCM+FSA))),m))) = (I ';' J) . (IC (Comput ((P +* I),(s +* (Start-At (0,SCM+FSA))),m))) by A10, A12, SCMFSA6A:54
.= (P +* (I ';' J)) . (IC (Comput ((P +* (I ';' J)),(s +* (Start-At (0,SCM+FSA))),m))) by A14, A10, A6, A15, GRFUNC_1:8
.= CurInstr ((P +* (I ';' J)),(Comput ((P +* (I ';' J)),(s +* (Start-At (0,SCM+FSA))),m))) by A16, PARTFUN1:def 8 ;
hence NPP (Comput ((P +* I),(s +* (Start-At (0,SCM+FSA))),(m + 1))) = NPP (Comput ((P +* (I ';' J)),(s +* (Start-At (0,SCM+FSA))),(m + 1))) by A8, A9, A7, A13, AMISTD_2:def 20, NAT_1:13; :: thesis: verum

end;

A18: ( Comput ((P +* I),(s +* (Start-At (0,SCM+FSA))),0) = s +* (Start-At (0,SCM+FSA)) & Comput ((P +* (I ';' J)),(s +* (Start-At (0,SCM+FSA))),0) = s +* (Start-At (0,SCM+FSA)) ) by EXTPRO_1:3;
A21: S₁[ 0 ] by A18;
thus for k being Element of NAT holds S₁[k] from NAT_1:sch 1(A21, A5); :: thesis: verum