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_onInit s & I 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 & I is_closed_onInit s & I 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 & I is_closed_onInit s & I 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 I is_closed_onInit s or not I 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: I is_closed_onInit s ; :: thesis: ( not I 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: I 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: I is_closed_on Initialize s by A2, Th40;
I 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:16;
hence ( if=0 a,I,J is_closed_onInit s & if=0 a,I,J is_halting_onInit s ) by Th40, Th41; :: thesis: verum