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 & J is_halting_onInit s holds
( if>0 a,I,J is_closed_onInit s & if>0 a,I,J is_halting_onInit s )
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 & J is_halting_onInit s holds
( if>0 a,I,J is_closed_onInit s & if>0 a,I,J is_halting_onInit s )
let a be read-write Int-Location ; :: thesis: ( s . a <= 0 & J is_closed_onInit s & J is_halting_onInit s implies ( if>0 a,I,J is_closed_onInit s & if>0 a,I,J is_halting_onInit s ) )
assume A1: s . a <= 0 ; :: thesis: ( not J is_closed_onInit s or not J is_halting_onInit s or ( if>0 a,I,J is_closed_onInit s & if>0 a,I,J is_halting_onInit s ) )
assume A2: J is_closed_onInit s ; :: thesis: ( not J is_halting_onInit s or ( if>0 a,I,J is_closed_onInit s & if>0 a,I,J is_halting_onInit s ) )
assume A3: J is_halting_onInit s ; :: thesis: ( if>0 a,I,J is_closed_onInit s & if>0 a,I,J is_halting_onInit s )
set Is = Initialize s;
A4: (Initialize s) . a <= 0 by A1, SCMFSA6C:3;
A5: J is_closed_on Initialize s by A2, Th40;
J is_halting_on Initialize s by A3, Th41;
then ( if>0 a,I,J is_closed_on Initialize s & if>0 a,I,J is_halting_on Initialize s ) by A4, A5, SCMFSA8B:24;
hence ( if>0 a,I,J is_closed_onInit s & if>0 a,I,J is_halting_onInit s ) by Th40, Th41; :: thesis: verum