let s be State of SCM+FSA; :: thesis: for P being the Instructions of SCM+FSA -valued ManySortedSet of NAT
for I being keeping_0 Program of SCM+FSA st not P +* I halts_on s +* (Start-At (0,SCM+FSA)) holds
for J being Program of SCM+FSA
for k being Element of NAT 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 keeping_0 Program of SCM+FSA st not P +* I halts_on s +* (Start-At (0,SCM+FSA)) holds
for J being Program of SCM+FSA
for k being Element of NAT 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 keeping_0 Program of SCM+FSA; :: thesis: ( not P +* I halts_on s +* (Start-At (0,SCM+FSA)) implies for J being Program of SCM+FSA
for k being Element of NAT 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: not P +* I halts_on s +* (Start-At (0,SCM+FSA)) ; :: thesis: for J being Program of SCM+FSA
for k being Element of NAT 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 holds NPP (Comput ((P +* I),(s +* (Start-At (0,SCM+FSA))),k)) = NPP (Comput ((P +* (I ';' J)),(s +* (Start-At (0,SCM+FSA))),k))
A2: Start-At (0,SCM+FSA) c= s +* (Start-At (0,SCM+FSA)) by FUNCT_4:26;
set s2 = s +* (Start-At (0,SCM+FSA));
defpred S₁[ Nat] means NPP (Comput ((P +* I),(s +* (Start-At (0,SCM+FSA))),$1)) = NPP (Comput ((P +* (I ';' J)),(s +* (Start-At (0,SCM+FSA))),$1));
A3: 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 A4: dom I c= dom (I ';' J) by XBOOLE_1:7;
let m be Element of NAT ; :: thesis: ( S₁[m] implies S₁[m + 1] )
A5: 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))) ;
A6: 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))) ;
A7: I c= P +* I by FUNCT_4:26;
then A8: IC (Comput ((P +* I),(s +* (Start-At (0,SCM+FSA))),m)) in dom I by Def2, A2;
assume A9: 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]
then A10: IC (Comput ((P +* I),(s +* (Start-At (0,SCM+FSA))),m)) = IC (Comput ((P +* (I ';' J)),(s +* (Start-At (0,SCM+FSA))),m)) by COMPOS_1:230;
dom (P +* I) = NAT by PARTFUN1:def 4;
then A11: (P +* I) /. (IC (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;
dom (P +* (I ';' J)) = NAT by PARTFUN1:def 4;
then A12: (P +* (I ';' J)) /. (IC (Comput ((P +* (I ';' J)),(s +* (Start-At (0,SCM+FSA))),m))) = (P +* (I ';' J)) . (IC (Comput ((P +* (I ';' J)),(s +* (Start-At (0,SCM+FSA))),m))) by PARTFUN1:def 8;
A13: I ';' J c= P +* (I ';' J) by FUNCT_4:26;
A14: CurInstr ((P +* I),(Comput ((P +* I),(s +* (Start-At (0,SCM+FSA))),m))) = I . (IC (Comput ((P +* I),(s +* (Start-At (0,SCM+FSA))),m))) by A8, A11, A7, GRFUNC_1:8;
then I . (IC (Comput ((P +* I),(s +* (Start-At (0,SCM+FSA))),m))) <> halt SCM+FSA by A1, EXTPRO_1:30;
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 A8, A14, SCMFSA6A:54
.= CurInstr ((P +* (I ';' J)),(Comput ((P +* (I ';' J)),(s +* (Start-At (0,SCM+FSA))),m))) by A10, A8, A4, A12, A13, GRFUNC_1:8 ;
hence S₁[m + 1] by A9, A5, A6, AMISTD_2:def 20; :: thesis: verum

end;

A15: ( 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;
A18: S₁[ 0 ] by A15;
thus for k being Element of NAT holds S₁[k] from NAT_1:sch 1(A18, A3); :: thesis: verum