let s be State of SCM+FSA ; :: thesis: for I being Program of SCM+FSA st I is_closed_on s holds
0 in dom I
let I be Program of SCM+FSA ; :: thesis: ( I is_closed_on s implies 0 in dom I )
reconsider n = IC (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),0 ) as Element of NAT ;
assume A1: I is_closed_on s ; :: thesis: 0 in dom I
then A2: n in dom I by SCMFSA7B:def 7;

per cases ( n = 0 or 0 < n ) ;

suppose n = 0 ; :: thesis: 0 in dom I

hence 0 in dom I by A1, SCMFSA7B:def 7; :: thesis: verum

end;

suppose 0 < n ; :: thesis: 0 in dom I

hence 0 in dom I by A2, SCMNORM:def 1; :: thesis: verum

end;

end;