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 0 ,SCM+FSA ) c= s +* (I +* (Start-At 0 ,SCM+FSA )) by FUNCT_4:26;
then for k being Element of NAT holds IC (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),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 0 ,SCM+FSA ) c= s holds
IC (Comput (ProgramPart s),s,k) in dom I
let k be Element of NAT ; :: thesis: ( I +* (Start-At 0 ,SCM+FSA ) c= s implies IC (Comput (ProgramPart s),s,k) in dom I )
assume I +* (Start-At 0 ,SCM+FSA ) c= s ; :: thesis: IC (Comput (ProgramPart s),s,k) in dom I
then A3: s = s +* (I +* (Start-At 0 ,SCM+FSA )) by FUNCT_4:79;
I is_closed_on s by A2;
hence IC (Comput (ProgramPart s),s,k) in dom I by A3, Def7; :: thesis: verum

end;

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