let P1, P2 be the Instructions of SCM+FSA -valued ManySortedSet of NAT ; :: thesis: for s1, s2 being State of SCM+FSA
for I being Program of SCM+FSA st I is_closed_on s1,P1 & I is_halting_on s1,P1 & Start-At (0,SCM+FSA) c= s1 & Start-At (0,SCM+FSA) c= s2 & I c= P1 & I c= P2 & ex k being Element of NAT st NPP (Comput (P1,s1,k)) = NPP s2 holds
NPP (Result (P1,s1)) = NPP (Result (P2,s2))
let s1, s2 be State of SCM+FSA; :: thesis: for I being Program of SCM+FSA st I is_closed_on s1,P1 & I is_halting_on s1,P1 & Start-At (0,SCM+FSA) c= s1 & Start-At (0,SCM+FSA) c= s2 & I c= P1 & I c= P2 & ex k being Element of NAT st NPP (Comput (P1,s1,k)) = NPP s2 holds
NPP (Result (P1,s1)) = NPP (Result (P2,s2))
set A = NAT ;
set D = Data-Locations SCM+FSA;
let I be Program of SCM+FSA; :: thesis: ( I is_closed_on s1,P1 & I is_halting_on s1,P1 & Start-At (0,SCM+FSA) c= s1 & Start-At (0,SCM+FSA) c= s2 & I c= P1 & I c= P2 & ex k being Element of NAT st NPP (Comput (P1,s1,k)) = NPP s2 implies NPP (Result (P1,s1)) = NPP (Result (P2,s2)) )
assume A1: I is_closed_on s1,P1 ; :: thesis: ( not I is_halting_on s1,P1 or not Start-At (0,SCM+FSA) c= s1 or not Start-At (0,SCM+FSA) c= s2 or not I c= P1 or not I c= P2 or for k being Element of NAT holds not NPP (Comput (P1,s1,k)) = NPP s2 or NPP (Result (P1,s1)) = NPP (Result (P2,s2)) )
assume A3: I is_halting_on s1,P1 ; :: thesis: ( not Start-At (0,SCM+FSA) c= s1 or not Start-At (0,SCM+FSA) c= s2 or not I c= P1 or not I c= P2 or for k being Element of NAT holds not NPP (Comput (P1,s1,k)) = NPP s2 or NPP (Result (P1,s1)) = NPP (Result (P2,s2)) )
assume A4: Start-At (0,SCM+FSA) c= s1 ; :: thesis: ( not Start-At (0,SCM+FSA) c= s2 or not I c= P1 or not I c= P2 or for k being Element of NAT holds not NPP (Comput (P1,s1,k)) = NPP s2 or NPP (Result (P1,s1)) = NPP (Result (P2,s2)) )
assume Start-At (0,SCM+FSA) c= s2 ; :: thesis: ( not I c= P1 or not I c= P2 or for k being Element of NAT holds not NPP (Comput (P1,s1,k)) = NPP s2 or NPP (Result (P1,s1)) = NPP (Result (P2,s2)) )
then A5: s2 = Initialize s2 by FUNCT_4:103, FUNCT_4:104;
assume I c= P1 ; :: thesis: ( not I c= P2 or for k being Element of NAT holds not NPP (Comput (P1,s1,k)) = NPP s2 or NPP (Result (P1,s1)) = NPP (Result (P2,s2)) )
then A6: P1 = P1 +* I by FUNCT_4:103, FUNCT_4:104;
assume I c= P2 ; :: thesis: ( for k being Element of NAT holds not NPP (Comput (P1,s1,k)) = NPP s2 or NPP (Result (P1,s1)) = NPP (Result (P2,s2)) )
then A7: P2 = P2 +* I by FUNCT_4:103, FUNCT_4:104;
A8: s1 = Initialize s1 by A4, FUNCT_4:103, FUNCT_4:104;
then A9: P1 halts_on s1 by A3, A6, SCMFSA7B:def 8;
then consider n being Element of NAT such that
A10: CurInstr (P1,(Comput (P1,s1,n))) = halt SCM+FSA by EXTPRO_1:30;
given k being Element of NAT such that A11: NPP (Comput (P1,s1,k)) = NPP s2 ; :: thesis: NPP (Result (P1,s1)) = NPP (Result (P2,s2))
set s3 = Comput (P1,s1,k);
set P3 = P1;
A12: IC in dom (Comput (P1,s1,k)) by COMPOS_1:9;
IC (Comput (P1,s1,k)) = IC s2 by A11, SCMFSA8A:6
.= IC (Initialize s2) by A5
.= 0 by FUNCT_4:121 ;
then (IC ) .--> 0 c= Comput (P1,s1,k) by A12, FUNCOP_1:88;
then Start-At (0,SCM+FSA) c= Comput (P1,s1,k) ;
then A14: Comput (P1,s1,k) = Initialize (Comput (P1,s1,k)) by FUNCT_4:103, FUNCT_4:104;

A15: now

let n be Element of NAT ; :: thesis: IC (Comput (P1,(Comput (P1,s1,k)),n)) in dom I
IC (Comput (P1,(Comput (P1,s1,k)),n)) = IC (Comput (P1,s1,(k + n))) by EXTPRO_1:5;
hence IC (Comput (P1,(Comput (P1,s1,k)),n)) in dom I by A1, A8, A6, SCMFSA7B:def 7; :: thesis: verum

end;

A16: Comput (P1,s1,(n + k)) = Comput (P1,(Comput (P1,s1,k)),n) by EXTPRO_1:5;
A17: Comput (P1,s1,(n + k)) = Comput (P1,s1,n) by A10, EXTPRO_1:6, NAT_1:11;
P1 halts_on Comput (P1,s1,k) by A10, A17, A16, EXTPRO_1:30;
then A18: I is_halting_on Comput (P1,s1,k),P1 by A14, A6, SCMFSA7B:def 8;
A19: DataPart (Comput (P1,s1,k)) = DataPart s2 by A11, SCMFSA8A:6;
consider k being Element of NAT such that
A20: CurInstr (P1,(Comput (P1,s1,k))) = halt SCM+FSA by A9, EXTPRO_1:30;
A22: P1 . (IC (Comput (P1,s1,k))) = halt SCM+FSA by A20, PBOOLE:158;
I is_closed_on Comput (P1,s1,k),P1 by A14, A15, A6, SCMFSA7B:def 7;
then NPP (Result (P1,(Comput (P1,s1,k)))) = NPP (Result (P2,s2)) by A5, A19, A14, A18, Th101, A6, A7;
hence NPP (Result (P1,s1)) = NPP (Result (P2,s2)) by A22, EXTPRO_1:9; :: thesis: verum