let s be State of SCM+FSA; :: thesis: for I being Program of SCM+FSA holds
( Initialized I is_closed_on s iff I is_closed_on Initialized s )
let I be Program of SCM+FSA; :: thesis: ( Initialized I is_closed_on s iff I is_closed_on Initialized s )

hereby :: thesis: ( I is_closed_on Initialized s implies Initialized I is_closed_on s )

assume A1: Initialized I is_closed_on s ; :: thesis: I is_closed_on Initialized s

now

let k be Element of NAT ; :: thesis: IC (Comput ((ProgramPart ((Initialized s) +* (I +* (Start-At (0,SCM+FSA))))),((Initialized s) +* (I +* (Start-At (0,SCM+FSA)))),k)) in dom I
X: s +* ((Initialized I) +* (Start-At (0,SCM+FSA))) = (Initialized s) +* (I +* (Start-At (0,SCM+FSA))) by Th16;
IC (Comput ((ProgramPart (s +* ((Initialized I) +* (Start-At (0,SCM+FSA))))),(s +* ((Initialized I) +* (Start-At (0,SCM+FSA)))),k)) in dom (Initialized I) by A1, SCMFSA7B:def 7;
then IC (Comput ((ProgramPart ((Initialized s) +* (I +* (Start-At (0,SCM+FSA))))),((Initialized s) +* (I +* (Start-At (0,SCM+FSA)))),k)) in dom (Initialized I) by X;
hence IC (Comput ((ProgramPart ((Initialized s) +* (I +* (Start-At (0,SCM+FSA))))),((Initialized s) +* (I +* (Start-At (0,SCM+FSA)))),k)) in dom I by Th20; :: thesis: verum

end;

hence I is_closed_on Initialized s by SCMFSA7B:def 7; :: thesis: verum

end;

assume A2: I is_closed_on Initialized s ; :: thesis: Initialized I is_closed_on s

now

let k be Element of NAT ; :: thesis: IC (Comput ((ProgramPart (s +* ((Initialized I) +* (Start-At (0,SCM+FSA))))),(s +* ((Initialized I) +* (Start-At (0,SCM+FSA)))),k)) in dom (Initialized I)
X: s +* ((Initialized I) +* (Start-At (0,SCM+FSA))) = (Initialized s) +* (I +* (Start-At (0,SCM+FSA))) by Th16;
IC (Comput ((ProgramPart ((Initialized s) +* (I +* (Start-At (0,SCM+FSA))))),((Initialized s) +* (I +* (Start-At (0,SCM+FSA)))),k)) in dom I by A2, SCMFSA7B:def 7;
then IC (Comput ((ProgramPart ((Initialized s) +* (I +* (Start-At (0,SCM+FSA))))),((Initialized s) +* (I +* (Start-At (0,SCM+FSA)))),k)) in dom (Initialized I) by Th20;
hence IC (Comput ((ProgramPart (s +* ((Initialized I) +* (Start-At (0,SCM+FSA))))),(s +* ((Initialized I) +* (Start-At (0,SCM+FSA)))),k)) in dom (Initialized I) by X; :: thesis: verum

end;

hence Initialized I is_closed_on s by SCMFSA7B:def 7; :: thesis: verum