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 & J is_closed_onInit s,p & J is_halting_onInit s,p holds
( if>0 (a,I,J) is_closed_onInit s,p & if>0 (a,I,J) is_halting_onInit 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 & J is_closed_onInit s,p & J is_halting_onInit s,p holds
( if>0 (a,I,J) is_closed_onInit s,p & if>0 (a,I,J) is_halting_onInit s,p )
let I, J be Program of SCM+FSA; :: thesis: for a being read-write Int-Location st s . a <= 0 & J is_closed_onInit s,p & J is_halting_onInit s,p holds
( if>0 (a,I,J) is_closed_onInit s,p & if>0 (a,I,J) is_halting_onInit s,p )
let a be read-write Int-Location ; :: thesis: ( s . a <= 0 & J is_closed_onInit s,p & J is_halting_onInit s,p implies ( if>0 (a,I,J) is_closed_onInit s,p & if>0 (a,I,J) is_halting_onInit s,p ) )
set Is = Initialized s;
assume s . a <= 0 ; :: thesis: ( not J is_closed_onInit s,p or not J is_halting_onInit s,p or ( if>0 (a,I,J) is_closed_onInit s,p & if>0 (a,I,J) is_halting_onInit s,p ) )
then A1: (Initialized s) . a <= 0 by SCMFSA6C:3;
assume J is_closed_onInit s,p ; :: thesis: ( not J is_halting_onInit s,p or ( if>0 (a,I,J) is_closed_onInit s,p & if>0 (a,I,J) is_halting_onInit s,p ) )
then A2: J is_closed_on Initialized s,p by Th40;
assume J is_halting_onInit s,p ; :: thesis: ( if>0 (a,I,J) is_closed_onInit s,p & if>0 (a,I,J) is_halting_onInit s,p )
then J is_halting_on Initialized s,p by Th41;
then ( if>0 (a,I,J) is_closed_on Initialized s,p & if>0 (a,I,J) is_halting_on Initialized s,p ) by A1, A2, SCMFSA8B:21;
hence ( if>0 (a,I,J) is_closed_onInit s,p & if>0 (a,I,J) is_halting_onInit s,p ) by Th40, Th41; :: thesis: verum