let I be Program of SCM+FSA ; :: thesis: ( I is paraclosed & I is good implies I is keeping_0 )
assume A1: ( I is paraclosed & I is good ) ; :: thesis: I is keeping_0
then A2: I does_not_destroy intloc 0 by Def5;

now

let s be State of SCM+FSA ; :: thesis: ( I +* (Start-At (insloc 0 )) c= s implies for k being Element of NAT holds (Computation s,k) . (intloc 0 ) = s . (intloc 0 ) )
assume A3: I +* (Start-At (insloc 0 )) c= s ; :: thesis: for k being Element of NAT holds (Computation s,k) . (intloc 0 ) = s . (intloc 0 )
let k be Element of NAT ; :: thesis: (Computation s,k) . (intloc 0 ) = s . (intloc 0 )
A4: I is_closed_on s by A1, Th24;
s +* (I +* (Start-At (insloc 0 ))) = s by A3, FUNCT_4:79;
hence (Computation s,k) . (intloc 0 ) = s . (intloc 0 ) by A2, A4, Th27; :: thesis: verum

end;

hence I is keeping_0 by SCMFSA6B:def 4; :: thesis: verum