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

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

assume A1: I is InitClosed ; :: thesis: for s being State of SCM+FSA holds I is_closed_on Initialize s
let s be State of SCM+FSA ; :: thesis: I is_closed_on Initialize s
I is_closed_onInit s by A1, Th35;
hence I is_closed_on Initialize s by Th40; :: thesis: verum

end;

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

now

let s be State of SCM+FSA ; :: thesis: I is_closed_onInit s
I is_closed_on Initialize s by A2;
hence I is_closed_onInit s by Th40; :: thesis: verum

end;

hence I is InitClosed by Th35; :: thesis: verum