let s be State of SCM+FSA; :: thesis: for p being Instruction-Sequence of SCM+FSA
for a being Int-Location
for I being Program of SCM+FSA st ( ( I is_closed_on Initialized s,p & I is_halting_on Initialized s,p ) or I is parahalting ) & ( s . (intloc 0) = 1 or a is read-write ) & not a in UsedIntLoc I holds
(IExec (I,p,s)) . a = s . a
let p be Instruction-Sequence of SCM+FSA; :: thesis: for a being Int-Location
for I being Program of SCM+FSA st ( ( I is_closed_on Initialized s,p & I is_halting_on Initialized s,p ) or I is parahalting ) & ( s . (intloc 0) = 1 or a is read-write ) & not a in UsedIntLoc I holds
(IExec (I,p,s)) . a = s . a
let a be Int-Location; :: thesis: for I being Program of SCM+FSA st ( ( I is_closed_on Initialized s,p & I is_halting_on Initialized s,p ) or I is parahalting ) & ( s . (intloc 0) = 1 or a is read-write ) & not a in UsedIntLoc I holds
(IExec (I,p,s)) . a = s . a
let I be Program of SCM+FSA; :: thesis: ( ( ( I is_closed_on Initialized s,p & I is_halting_on Initialized s,p ) or I is parahalting ) & ( s . (intloc 0) = 1 or a is read-write ) & not a in UsedIntLoc I implies (IExec (I,p,s)) . a = s . a )
assume that
A1: ( ( I is_closed_on Initialized s,p & I is_halting_on Initialized s,p ) or I is parahalting ) and
A2: ( s . (intloc 0) = 1 or a is read-write ) and
A3: not a in UsedIntLoc I ; :: thesis: (IExec (I,p,s)) . a = s . a
A4: ( a = intloc 0 or a is read-write ) by SCMFSA_M:def 2;
( I is_closed_on Initialized s,p & I is_halting_on Initialized s,p ) by A1, SCMFSA7B:18, SCMFSA7B:19;
hence (IExec (I,p,s)) . a = (Initialized s) . a by A3, Th1
.= s . a by A2, A4, SCMFSA_M:9, SCMFSA_M:37 ;
:: thesis: verum