let I be Program of SCM+FSA ; :: thesis: ( I is paraclosed iff for s being State of SCM+FSA holds I is_closed_on s )

hereby :: thesis: ( ( for s being State of SCM+FSA holds I is_closed_on s ) implies I is paraclosed )

assume A1: I is paraclosed ; :: thesis: for s being State of SCM+FSA holds I is_closed_on s
let s be State of SCM+FSA ; :: thesis: I is_closed_on s
I +* (Start-At (insloc 0 )) c= s +* (I +* (Start-At (insloc 0 ))) by FUNCT_4:26;
then for k being Element of NAT holds IC (Computation (s +* (I +* (Start-At (insloc 0 )))),k) in dom I by A1, SCMFSA6B:def 2;
hence I is_closed_on s by Def7; :: thesis: verum

end;

assume A2: for s being State of SCM+FSA holds I is_closed_on s ; :: thesis: I is paraclosed

now

let s be State of SCM+FSA ; :: thesis: for k being Element of NAT st I +* (Start-At (insloc 0 )) c= s holds
IC (Computation s,k) in dom I
let k be Element of NAT ; :: thesis: ( I +* (Start-At (insloc 0 )) c= s implies IC (Computation s,k) in dom I )
assume I +* (Start-At (insloc 0 )) c= s ; :: thesis: IC (Computation s,k) in dom I
then ( I is_closed_on s & s = s +* (I +* (Start-At (insloc 0 ))) ) by A2, FUNCT_4:79;
hence IC (Computation s,k) in dom I by Def7; :: thesis: verum

end;

hence I is paraclosed by SCMFSA6B:def 2; :: thesis: verum