let s1, s2 be State of SCM+FSA; :: thesis: for I being parahalting Program of SCM+FSA st I +* (Start-At (0,SCM+FSA)) c= s1 & I +* (Start-At (0,SCM+FSA)) 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 (Comput ((ProgramPart s1),s1,k))),(Comput ((ProgramPart s1),s1,k))) = CurInstr ((ProgramPart (Comput ((ProgramPart s2),s2,k))),(Comput ((ProgramPart s2),s2,k))) )
let I be parahalting Program of SCM+FSA; :: thesis: ( I +* (Start-At (0,SCM+FSA)) c= s1 & I +* (Start-At (0,SCM+FSA)) 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 (Comput ((ProgramPart s1),s1,k))),(Comput ((ProgramPart s1),s1,k))) = CurInstr ((ProgramPart (Comput ((ProgramPart s2),s2,k))),(Comput ((ProgramPart s2),s2,k))) ) )
assume that
A1: I +* (Start-At (0,SCM+FSA)) c= s1 and
A2: I +* (Start-At (0,SCM+FSA)) 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 (Comput ((ProgramPart s1),s1,k))),(Comput ((ProgramPart s1),s1,k))) = CurInstr ((ProgramPart (Comput ((ProgramPart s2),s2,k))),(Comput ((ProgramPart s2),s2,k))) )
A4: I c= s2 by A2, Th5;
A5: I c= s1 by A1, Th5;

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 (Comput ((ProgramPart s2),s2,k))),(Comput ((ProgramPart s2),s2,k))) = CurInstr ((ProgramPart (Comput ((ProgramPart s1),s1,k))),(Comput ((ProgramPart s1),s1,k))) )
A6: IC (Comput ((ProgramPart s1),s1,k)) in dom I by A1, Def2;
A7: IC (Comput ((ProgramPart s2),s2,k)) in dom I by A2, Def2;
for m being Element of NAT st m < k holds
IC (Comput ((ProgramPart s2),s2,m)) in dom I by A2, Def2;
hence Comput ((ProgramPart s1),s1,k), Comput ((ProgramPart s2),s2,k) equal_outside NAT by A3, A5, A4, Th21; :: thesis: CurInstr ((ProgramPart (Comput ((ProgramPart s2),s2,k))),(Comput ((ProgramPart s2),s2,k))) = CurInstr ((ProgramPart (Comput ((ProgramPart s1),s1,k))),(Comput ((ProgramPart s1),s1,k)))
then A8: IC (Comput ((ProgramPart s1),s1,k)) = IC (Comput ((ProgramPart s2),s2,k)) by COMPOS_1:24;
Y: (ProgramPart (Comput ((ProgramPart s2),s2,k))) /. (IC (Comput ((ProgramPart s2),s2,k))) = (Comput ((ProgramPart s2),s2,k)) . (IC (Comput ((ProgramPart s2),s2,k))) by COMPOS_1:38;
Z: (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;
thus CurInstr ((ProgramPart (Comput ((ProgramPart s2),s2,k))),(Comput ((ProgramPart s2),s2,k))) = s2 . (IC (Comput ((ProgramPart s2),s2,k))) by Y, AMI_1:54
.= 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 (Comput ((ProgramPart s1),s1,k))),(Comput ((ProgramPart s1),s1,k))) by Z, AMI_1:54 ; :: thesis: verum

end;