let s be State of SCM+FSA; :: thesis: for I being Program of SCM+FSA holds
( I is_closed_onInit s iff I is_closed_on Initialized s )
let I be Program of SCM+FSA; :: thesis: ( I is_closed_onInit s iff I is_closed_on Initialized s )
set s1 = s +* (Initialized I);
set s2 = (Initialized s) +* (I +* (Start-At (0,SCM+FSA)));
A1: s +* (Initialized I) = (Initialized s) +* (I +* (Start-At (0,SCM+FSA))) by SCMFSA8A:13;
( I is_closed_onInit s iff for k being Element of NAT holds IC (Comput ((ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),k)) in dom I ) by Def4;
hence ( I is_closed_onInit s iff I is_closed_on Initialized s ) by A1, SCMFSA7B:def 7; :: thesis: verum