let s be State of SCM+FSA; :: thesis: for I, J being Program of SCM+FSA
for a being read-write Int-Location st Directed I is_pseudo-closed_on s & Directed J is_pseudo-closed_on s holds
( if=0 (a,I,J) is_closed_on s & if=0 (a,I,J) is_halting_on s )
let I, J be Program of SCM+FSA; :: thesis: for a being read-write Int-Location st Directed I is_pseudo-closed_on s & Directed J is_pseudo-closed_on s holds
( if=0 (a,I,J) is_closed_on s & if=0 (a,I,J) is_halting_on s )
let a be read-write Int-Location ; :: thesis: ( Directed I is_pseudo-closed_on s & Directed J is_pseudo-closed_on s implies ( if=0 (a,I,J) is_closed_on s & if=0 (a,I,J) is_halting_on s ) )
assume A1: Directed I is_pseudo-closed_on s ; :: thesis: ( not Directed J is_pseudo-closed_on s or ( if=0 (a,I,J) is_closed_on s & if=0 (a,I,J) is_halting_on s ) )
assume A2: Directed J is_pseudo-closed_on s ; :: thesis: ( if=0 (a,I,J) is_closed_on s & if=0 (a,I,J) is_halting_on s )

hereby :: thesis: verum

per cases ( s . a = 0 or s . a <> 0 ) ;

suppose A3: s . a = 0 ; :: thesis: ( if=0 (a,I,J) is_closed_on s & if=0 (a,I,J) is_halting_on s )

hence if=0 (a,I,J) is_closed_on s by A1, Th66; :: thesis: if=0 (a,I,J) is_halting_on s
thus if=0 (a,I,J) is_halting_on s by A1, A3, Th66; :: thesis: verum

end;

suppose A4: s . a <> 0 ; :: thesis: ( if=0 (a,I,J) is_closed_on s & if=0 (a,I,J) is_halting_on s )

hence if=0 (a,I,J) is_closed_on s by A2, Th70; :: thesis: if=0 (a,I,J) is_halting_on s
thus if=0 (a,I,J) is_halting_on s by A2, A4, Th70; :: thesis: verum

end;

end;

end;