let s be State of SCM+FSA ; :: thesis: for I being Program of SCM+FSA holds
( I is_halting_onInit s iff I is_halting_on Initialized s )
let I be Program of SCM+FSA ; :: thesis: ( I is_halting_onInit s iff I is_halting_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_halting_onInit s iff ProgramPart (s +* (Initialized I)) halts_on s +* (Initialized I) ) by Def5;
hence ( I is_halting_onInit s iff I is_halting_on Initialized s ) by A1, SCMFSA7B:def 8; :: thesis: verum