let p be the Instructions of SCM+FSA -valued ManySortedSet of NAT ; :: thesis: for s being State of SCM+FSA
for I being good InitHalting Program of SCM+FSA
for b being Int-Location
for a being read-write Int-Location st not I destroys a & s . a > 0 holds
(IExec ((Times (a,I)),p,s)) . b = (IExec ((Times (a,I)),p,(IExec ((I ';' (SubFrom (a,(intloc 0)))),p,s)))) . b
set D = Data-Locations SCM+FSA;
let s be State of SCM+FSA; :: thesis: for I being good InitHalting Program of SCM+FSA
for b being Int-Location
for a being read-write Int-Location st not I destroys a & s . a > 0 holds
(IExec ((Times (a,I)),p,s)) . b = (IExec ((Times (a,I)),p,(IExec ((I ';' (SubFrom (a,(intloc 0)))),p,s)))) . b
let I be good InitHalting Program of SCM+FSA; :: thesis: for b being Int-Location
for a being read-write Int-Location st not I destroys a & s . a > 0 holds
(IExec ((Times (a,I)),p,s)) . b = (IExec ((Times (a,I)),p,(IExec ((I ';' (SubFrom (a,(intloc 0)))),p,s)))) . b
let b be Int-Location ; :: thesis: for a being read-write Int-Location st not I destroys a & s . a > 0 holds
(IExec ((Times (a,I)),p,s)) . b = (IExec ((Times (a,I)),p,(IExec ((I ';' (SubFrom (a,(intloc 0)))),p,s)))) . b
let a be read-write Int-Location ; :: thesis: ( not I destroys a & s . a > 0 implies (IExec ((Times (a,I)),p,s)) . b = (IExec ((Times (a,I)),p,(IExec ((I ';' (SubFrom (a,(intloc 0)))),p,s)))) . b )
assume A1: ( not I destroys a & s . a > 0 ) ; :: thesis: (IExec ((Times (a,I)),p,s)) . b = (IExec ((Times (a,I)),p,(IExec ((I ';' (SubFrom (a,(intloc 0)))),p,s)))) . b
set IT = IExec ((Times (a,I)),p,(IExec ((I ';' (SubFrom (a,(intloc 0)))),p,s)));
b in Int-Locations by SCMFSA_2:9;
then A2: b in Data-Locations SCM+FSA by SCMFSA_2:127, XBOOLE_0:def 3;
hence (IExec ((Times (a,I)),p,s)) . b = (DataPart (IExec ((Times (a,I)),p,s))) . b by FUNCT_1:72
.= (DataPart (IExec ((Times (a,I)),p,(IExec ((I ';' (SubFrom (a,(intloc 0)))),p,s))))) . b by A1, Th79
.= (IExec ((Times (a,I)),p,(IExec ((I ';' (SubFrom (a,(intloc 0)))),p,s)))) . b by A2, FUNCT_1:72 ;
:: thesis: verum