let s be State of SCM+FSA; :: thesis: for I being InitHalting good Program of SCM+FSA
for b being Int-Location
for a being read-write Int-Location st s . a > 0 & while>0 (a,I) is InitHalting holds
(IExec ((while>0 (a,I)),s)) . b = (IExec ((while>0 (a,I)),(IExec (I,s)))) . b
let I be InitHalting good Program of SCM+FSA; :: thesis: for b being Int-Location
for a being read-write Int-Location st s . a > 0 & while>0 (a,I) is InitHalting holds
(IExec ((while>0 (a,I)),s)) . b = (IExec ((while>0 (a,I)),(IExec (I,s)))) . b
let b be Int-Location ; :: thesis: for a being read-write Int-Location st s . a > 0 & while>0 (a,I) is InitHalting holds
(IExec ((while>0 (a,I)),s)) . b = (IExec ((while>0 (a,I)),(IExec (I,s)))) . b
let a be read-write Int-Location ; :: thesis: ( s . a > 0 & while>0 (a,I) is InitHalting implies (IExec ((while>0 (a,I)),s)) . b = (IExec ((while>0 (a,I)),(IExec (I,s)))) . b )
set D = Int-Locations \/ FinSeq-Locations;
assume that
A1: s . a > 0 and
A2: while>0 (a,I) is InitHalting ; :: thesis: (IExec ((while>0 (a,I)),s)) . b = (IExec ((while>0 (a,I)),(IExec (I,s)))) . b
b in Int-Locations by SCMFSA_2:9;
then A3: b in Int-Locations \/ FinSeq-Locations by XBOOLE_0:def 3;
then (IExec ((while>0 (a,I)),s)) . b = (DataPart (IExec ((while>0 (a,I)),s))) . b by FUNCT_1:72, SCMFSA_2:127
.= (DataPart (IExec ((while>0 (a,I)),(IExec (I,s))))) . b by A1, A2, Th33 ;
hence (IExec ((while>0 (a,I)),s)) . b = (IExec ((while>0 (a,I)),(IExec (I,s)))) . b by A3, FUNCT_1:72, SCMFSA_2:127; :: thesis: verum