let s be State of SCM+FSA ; :: thesis: for I, J being Program of SCM+FSA
for a being read-write Int-Location st s . a < 0 & I is_closed_on s & I is_halting_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 s . a < 0 & I is_closed_on s & I is_halting_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: ( s . a < 0 & I is_closed_on s & I is_halting_on s implies ( if<0 a,I,J is_closed_on s & if<0 a,I,J is_halting_on s ) )
assume A1: s . a < 0 ; :: thesis: ( not I is_closed_on s or not I is_halting_on s or ( if<0 a,I,J is_closed_on s & if<0 a,I,J is_halting_on s ) )
assume ( I is_closed_on s & I is_halting_on s ) ; :: thesis: ( if<0 a,I,J is_closed_on s & if<0 a,I,J is_halting_on s )
then ( if>0 a,J,I is_closed_on s & if>0 a,J,I is_halting_on s ) by A1, Th24;
hence ( if<0 a,I,J is_closed_on s & if<0 a,I,J is_halting_on s ) by A1, Th18; :: thesis: verum