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 Initialize s )
let I be Program of SCM+FSA ; :: thesis: ( Initialized I is_closed_on s iff I is_closed_on Initialize s )

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

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

now

let k be Element of NAT ; :: thesis: IC (Computation ((Initialize s) +* (I +* (Start-At (insloc 0 )))),k) in dom I
IC (Computation (s +* ((Initialized I) +* (Start-At (insloc 0 )))),k) in dom (Initialized I) by A1, SCMFSA7B:def 7;
then IC (Computation ((Initialize s) +* (I +* (Start-At (insloc 0 )))),k) in dom (Initialized I) by Th16;
hence IC (Computation ((Initialize s) +* (I +* (Start-At (insloc 0 )))),k) in dom I by Th20; :: thesis: verum

end;

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

end;

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

now

let k be Element of NAT ; :: thesis: IC (Computation (s +* ((Initialized I) +* (Start-At (insloc 0 )))),k) in dom (Initialized I)
IC (Computation ((Initialize s) +* (I +* (Start-At (insloc 0 )))),k) in dom I by A2, SCMFSA7B:def 7;
then IC (Computation ((Initialize s) +* (I +* (Start-At (insloc 0 )))),k) in dom (Initialized I) by Th20;
hence IC (Computation (s +* ((Initialized I) +* (Start-At (insloc 0 )))),k) in dom (Initialized I) by Th16; :: thesis: verum

end;

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