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

hereby :: thesis: verum

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

suppose A3: s . a > 0 ; :: thesis: if>0 (a,I,J) is_halting_on s,P

thus if>0 (a,I,J) is_halting_on s,P by A1, A3, Th30; :: thesis: verum

end;

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

thus if>0 (a,I,J) is_halting_on s,P by A2, A4, Th34; :: thesis: verum

end;

end;

end;