let s1, s2 be State of SCM+FSA ; :: thesis: for I being Program of SCM+FSA st I is_closed_on s1 & I is_halting_on s1 & I +* (Start-At 0 ,SCM+FSA ) c= s1 & I +* (Start-At 0 ,SCM+FSA ) c= s2 & ex k being Element of NAT st Comput (ProgramPart s1),s1,k,s2 equal_outside NAT holds
Result (ProgramPart s1),s1, Result (ProgramPart s2),s2 equal_outside NAT
set A = NAT ;
set D = Int-Locations \/ FinSeq-Locations ;
let I be Program of SCM+FSA ; :: thesis: ( I is_closed_on s1 & I is_halting_on s1 & I +* (Start-At 0 ,SCM+FSA ) c= s1 & I +* (Start-At 0 ,SCM+FSA ) c= s2 & ex k being Element of NAT st Comput (ProgramPart s1),s1,k,s2 equal_outside NAT implies Result (ProgramPart s1),s1, Result (ProgramPart s2),s2 equal_outside NAT )
assume A1: I is_closed_on s1 ; :: thesis: ( not I is_halting_on s1 or not I +* (Start-At 0 ,SCM+FSA ) c= s1 or not I +* (Start-At 0 ,SCM+FSA ) c= s2 or for k being Element of NAT holds not Comput (ProgramPart s1),s1,k,s2 equal_outside NAT or Result (ProgramPart s1),s1, Result (ProgramPart s2),s2 equal_outside NAT )
assume A2: I is_halting_on s1 ; :: thesis: ( not I +* (Start-At 0 ,SCM+FSA ) c= s1 or not I +* (Start-At 0 ,SCM+FSA ) c= s2 or for k being Element of NAT holds not Comput (ProgramPart s1),s1,k,s2 equal_outside NAT or Result (ProgramPart s1),s1, Result (ProgramPart s2),s2 equal_outside NAT )
assume A3: I +* (Start-At 0 ,SCM+FSA ) c= s1 ; :: thesis: ( not I +* (Start-At 0 ,SCM+FSA ) c= s2 or for k being Element of NAT holds not Comput (ProgramPart s1),s1,k,s2 equal_outside NAT or Result (ProgramPart s1),s1, Result (ProgramPart s2),s2 equal_outside NAT )
then A4: s1 = s1 +* (I +* (Start-At 0 ,SCM+FSA )) by FUNCT_4:79;
then A5: ProgramPart s1 halts_on s1 by A2, SCMFSA7B:def 8;
then consider n being Element of NAT such that
A6: CurInstr (ProgramPart s1),(Comput (ProgramPart s1),s1,n) = halt SCM+FSA by AMI_1:146;
assume I +* (Start-At 0 ,SCM+FSA ) c= s2 ; :: thesis: ( for k being Element of NAT holds not Comput (ProgramPart s1),s1,k,s2 equal_outside NAT or Result (ProgramPart s1),s1, Result (ProgramPart s2),s2 equal_outside NAT )
then A7: s2 = s2 +* (I +* (Start-At 0 ,SCM+FSA )) by FUNCT_4:79;
given k being Element of NAT such that A8: Comput (ProgramPart s1),s1,k,s2 equal_outside NAT ; :: thesis: Result (ProgramPart s1),s1, Result (ProgramPart s2),s2 equal_outside NAT
set s3 = Comput (ProgramPart s1),s1,k;
A9: IC SCM+FSA in dom (Comput (ProgramPart s1),s1,k) by COMPOS_1:9;
I c= I +* (Start-At 0 ,SCM+FSA ) by SCMFSA8A:9;
then I c= s1 by A3, XBOOLE_1:1;
then A10: I c= Comput (ProgramPart s1),s1,k by AMI_1:86;
IC (Comput (ProgramPart s1),s1,k) = IC s2 by A8, SCMFSA8A:6
.= IC ((s2 +* I) +* (Start-At 0 ,SCM+FSA )) by A7, FUNCT_4:15
.= 0 by FUNCT_4:121 ;
then (IC SCM+FSA ) .--> 0 c= Comput (ProgramPart s1),s1,k by A9, FUNCOP_1:88;
then I +* (Start-At 0 ,SCM+FSA ) c= Comput (ProgramPart s1),s1,k by A10, FUNCT_4:92;
then A11: Comput (ProgramPart s1),s1,k = (Comput (ProgramPart s1),s1,k) +* (I +* (Start-At 0 ,SCM+FSA )) by FUNCT_4:79;

A12: now

let n be Element of NAT ; :: thesis: IC (Comput (ProgramPart (Comput (ProgramPart s1),s1,k)),(Comput (ProgramPart s1),s1,k),n) in dom I
T: ProgramPart (Comput (ProgramPart s1),s1,k) = ProgramPart s1 by AMI_1:123;
IC (Comput (ProgramPart s1),(Comput (ProgramPart s1),s1,k),n) = IC (Comput (ProgramPart s1),s1,(k + n)) by AMI_1:51;
hence IC (Comput (ProgramPart (Comput (ProgramPart s1),s1,k)),(Comput (ProgramPart s1),s1,k),n) in dom I by A1, A4, T, SCMFSA7B:def 7; :: thesis: verum

end;

T: ProgramPart s1 = ProgramPart (Comput (ProgramPart s1),s1,k) by AMI_1:123;
x: Comput (ProgramPart s1),s1,(n + k) = Comput (ProgramPart s1),(Comput (ProgramPart s1),s1,k),n by AMI_1:51;
n <= n + k by NAT_1:11;
then yy: Comput (ProgramPart s1),s1,(n + k) = Comput (ProgramPart s1),s1,n by A6, AMI_1:52;
CurInstr (ProgramPart (Comput (ProgramPart s1),s1,k)),(Comput (ProgramPart (Comput (ProgramPart s1),s1,k)),(Comput (ProgramPart s1),s1,k),n) = CurInstr (ProgramPart s1),(Comput (ProgramPart s1),s1,(k + n)) by x, T
.= CurInstr (ProgramPart s1),(Comput (ProgramPart s1),s1,n) by yy ;
then ProgramPart (Comput (ProgramPart s1),s1,k) halts_on Comput (ProgramPart s1),s1,k by A6, AMI_1:146;
then A13: I is_halting_on Comput (ProgramPart s1),s1,k by A11, SCMFSA7B:def 8;
A14: DataPart (Comput (ProgramPart s1),s1,k) = DataPart s2 by A8, SCMFSA8A:6;
TX: ProgramPart s1 = ProgramPart (Comput (ProgramPart s1),s1,k) by AMI_1:123;
consider k being Element of NAT such that
A15: CurInstr (ProgramPart s1),(Comput (ProgramPart s1),s1,k) = halt SCM+FSA by A5, AMI_1:146;
TTX: ProgramPart s1 = ProgramPart (Comput (ProgramPart s1),s1,k) by AMI_1:123;
Y: (ProgramPart (Comput (ProgramPart s1),s1,k)) /. (IC (Comput (ProgramPart s1),s1,k)) = (Comput (ProgramPart s1),s1,k) . (IC (Comput (ProgramPart s1),s1,k)) by COMPOS_1:38;
A16: (ProgramPart s1) . (IC (Comput (ProgramPart s1),s1,k)) = s1 . (IC (Comput (ProgramPart s1),s1,k)) by COMPOS_1:2
.= (Comput (ProgramPart s1),s1,k) . (IC (Comput (ProgramPart s1),s1,k)) by AMI_1:54
.= (ProgramPart (Comput (ProgramPart s1),s1,k)) /. (IC (Comput (ProgramPart s1),s1,k)) by Y
.= (ProgramPart s1) /. (IC (Comput (ProgramPart s1),s1,k)) by TTX
.= CurInstr (ProgramPart s1),(Comput (ProgramPart s1),s1,k)
.= halt SCM+FSA by A15 ;
I is_closed_on Comput (ProgramPart s1),s1,k by A11, A12, SCMFSA7B:def 7;
then Result (ProgramPart (Comput (ProgramPart s1),s1,k)),(Comput (ProgramPart s1),s1,k), Result (ProgramPart s2),s2 equal_outside NAT by A7, A14, A11, A13, Th101;
hence Result (ProgramPart s1),s1, Result (ProgramPart s2),s2 equal_outside NAT by A16, TX, AMI_1:57; :: thesis: verum