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

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

assume Initialized I is_halting_on s ; :: thesis: I is_halting_on Initialize s
then s +* ((Initialized I) +* (Start-At (insloc 0 ))) is halting by SCMFSA7B:def 8;
then (Initialize s) +* (I +* (Start-At (insloc 0 ))) is halting by Th16;
hence I is_halting_on Initialize s by SCMFSA7B:def 8; :: thesis: verum

end;

assume I is_halting_on Initialize s ; :: thesis: Initialized I is_halting_on s
then (Initialize s) +* (I +* (Start-At (insloc 0 ))) is halting by SCMFSA7B:def 8;
then s +* ((Initialized I) +* (Start-At (insloc 0 ))) is halting by Th16;
hence Initialized I is_halting_on s by SCMFSA7B:def 8; :: thesis: verum