let s be State of ; :: thesis: for I being Program of holds
( Initialized I is_halting_on s iff I is_halting_on Initialize s )
let I be Program of ; :: thesis: ( Initialized I is_halting_on s iff I is_halting_on Initialize s )
X: s +* ((Initialized I) +* (Start-At (insloc 0 ))) = (Initialize s) +* (I +* (Start-At (insloc 0 ))) by Th16;

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 ProgramPart (s +* ((Initialized I) +* (Start-At (insloc 0 )))) halts_on s +* ((Initialized I) +* (Start-At (insloc 0 ))) by SCMFSA7B:def 8;
then ProgramPart ((Initialize s) +* (I +* (Start-At (insloc 0 )))) halts_on (Initialize s) +* (I +* (Start-At (insloc 0 ))) by Th16, X;
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 ProgramPart ((Initialize s) +* (I +* (Start-At (insloc 0 )))) halts_on (Initialize s) +* (I +* (Start-At (insloc 0 ))) by SCMFSA7B:def 8;
then ProgramPart (s +* ((Initialized I) +* (Start-At (insloc 0 )))) halts_on s +* ((Initialized I) +* (Start-At (insloc 0 ))) by Th16, X;
hence Initialized I is_halting_on s by SCMFSA7B:def 8; :: thesis: verum