let s1, s2 be State of SCM+FSA; :: thesis: for I being InitHalting Program of SCM+FSA st Initialized I c= s1 & Initialized I c= s2 & s1,s2 equal_outside NAT holds
for k being Element of NAT holds
( Comput ((ProgramPart s1),s1,k), Comput ((ProgramPart s2),s2,k) equal_outside NAT & CurInstr ((ProgramPart s1),(Comput ((ProgramPart s1),s1,k))) = CurInstr ((ProgramPart s2),(Comput ((ProgramPart s2),s2,k))) )
let I be InitHalting Program of SCM+FSA; :: thesis: ( Initialized I c= s1 & Initialized I c= s2 & s1,s2 equal_outside NAT implies for k being Element of NAT holds
( Comput ((ProgramPart s1),s1,k), Comput ((ProgramPart s2),s2,k) equal_outside NAT & CurInstr ((ProgramPart s1),(Comput ((ProgramPart s1),s1,k))) = CurInstr ((ProgramPart s2),(Comput ((ProgramPart s2),s2,k))) ) )
assume that
A1: Initialized I c= s1 and
A2: Initialized I c= s2 and
A3: s1,s2 equal_outside NAT ; :: thesis: for k being Element of NAT holds
( Comput ((ProgramPart s1),s1,k), Comput ((ProgramPart s2),s2,k) equal_outside NAT & CurInstr ((ProgramPart s1),(Comput ((ProgramPart s1),s1,k))) = CurInstr ((ProgramPart s2),(Comput ((ProgramPart s2),s2,k))) )
A4: I c= s2 by A2, SCM_HALT:13;
A5: I c= s1 by A1, SCM_HALT:13;

hereby :: thesis: verum

let k be Element of NAT ; :: thesis: ( Comput ((ProgramPart s1),s1,k), Comput ((ProgramPart s2),s2,k) equal_outside NAT & CurInstr ((ProgramPart s2),(Comput ((ProgramPart s2),s2,k))) = CurInstr ((ProgramPart s1),(Comput ((ProgramPart s1),s1,k))) )
A6: IC (Comput ((ProgramPart s1),s1,k)) in dom I by A1, SCM_HALT:def 1;
A7: IC (Comput ((ProgramPart s2),s2,k)) in dom I by A2, SCM_HALT:def 1;
for m being Element of NAT st m < k holds
IC (Comput ((ProgramPart s2),s2,m)) in dom I by A2, SCM_HALT:def 1;
hence Comput ((ProgramPart s1),s1,k), Comput ((ProgramPart s2),s2,k) equal_outside NAT by A3, A5, A4, SCMFSA6B:21; :: thesis: CurInstr ((ProgramPart s2),(Comput ((ProgramPart s2),s2,k))) = CurInstr ((ProgramPart s1),(Comput ((ProgramPart s1),s1,k)))
then A8: IC (Comput ((ProgramPart s1),s1,k)) = IC (Comput ((ProgramPart s2),s2,k)) by COMPOS_1:24;
thus CurInstr ((ProgramPart s2),(Comput ((ProgramPart s2),s2,k))) = s2 . (IC (Comput ((ProgramPart s2),s2,k))) by COMPOS_1:38
.= I . (IC (Comput ((ProgramPart s2),s2,k))) by A4, A7, GRFUNC_1:8
.= s1 . (IC (Comput ((ProgramPart s1),s1,k))) by A5, A8, A6, GRFUNC_1:8
.= CurInstr ((ProgramPart s1),(Comput ((ProgramPart s1),s1,k))) by COMPOS_1:38 ; :: thesis: verum

end;