let P be Instruction-Sequence of SCM+FSA; :: thesis: for s being State of SCM+FSA
for I, J being Program of SCM+FSA
for a being read-write Int-Location st s . a < 0 & I is_closed_on s,P & I is_halting_on s,P holds
( if<0 (a,I,J) is_closed_on s,P & if<0 (a,I,J) is_halting_on s,P )
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,P & I is_halting_on s,P holds
( if<0 (a,I,J) is_closed_on s,P & if<0 (a,I,J) is_halting_on s,P )
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,P & I is_halting_on s,P holds
( if<0 (a,I,J) is_closed_on s,P & if<0 (a,I,J) is_halting_on s,P )
let a be read-write Int-Location ; :: thesis: ( s . a < 0 & I is_closed_on s,P & I is_halting_on s,P implies ( if<0 (a,I,J) is_closed_on s,P & if<0 (a,I,J) is_halting_on s,P ) )
assume A1: s . a < 0 ; :: thesis: ( not I is_closed_on s,P or not I is_halting_on s,P or ( if<0 (a,I,J) is_closed_on s,P & if<0 (a,I,J) is_halting_on s,P ) )
assume that
A2: I is_closed_on s,P and
A3: I is_halting_on s,P ; :: thesis: ( if<0 (a,I,J) is_closed_on s,P & if<0 (a,I,J) is_halting_on s,P )
A4: if>0 (a,J,I) is_halting_on s,P by A1, A2, A3, Th24;
if>0 (a,J,I) is_closed_on s,P by A1, A2, A3, Th24;
hence ( if<0 (a,I,J) is_closed_on s,P & if<0 (a,I,J) is_halting_on s,P ) by A1, A4, Th18; :: thesis: verum