let p be Instruction-Sequence of SCM+FSA; :: thesis: for s being State of SCM+FSA
for I being really-closed good InitHalting MacroInstruction of SCM+FSA
for f being FinSeq-Location
for a being read-write Int-Location st not I destroys a & s . a > 0 holds
(IExec ((Times (a,I)),p,s)) . f = (IExec ((Times (a,I)),p,(IExec ((I ";" (SubFrom (a,()))),p,s)))) . f

set D = Data-Locations ;
let s be State of SCM+FSA; :: thesis: for I being really-closed good InitHalting MacroInstruction of SCM+FSA
for f being FinSeq-Location
for a being read-write Int-Location st not I destroys a & s . a > 0 holds
(IExec ((Times (a,I)),p,s)) . f = (IExec ((Times (a,I)),p,(IExec ((I ";" (SubFrom (a,()))),p,s)))) . f

let I be really-closed good InitHalting MacroInstruction of SCM+FSA ; :: thesis: for f being FinSeq-Location
for a being read-write Int-Location st not I destroys a & s . a > 0 holds
(IExec ((Times (a,I)),p,s)) . f = (IExec ((Times (a,I)),p,(IExec ((I ";" (SubFrom (a,()))),p,s)))) . f

let f be FinSeq-Location ; :: thesis: for a being read-write Int-Location st not I destroys a & s . a > 0 holds
(IExec ((Times (a,I)),p,s)) . f = (IExec ((Times (a,I)),p,(IExec ((I ";" (SubFrom (a,()))),p,s)))) . f

let a be read-write Int-Location; :: thesis: ( not I destroys a & s . a > 0 implies (IExec ((Times (a,I)),p,s)) . f = (IExec ((Times (a,I)),p,(IExec ((I ";" (SubFrom (a,()))),p,s)))) . f )
assume A1: ( not I destroys a & s . a > 0 ) ; :: thesis: (IExec ((Times (a,I)),p,s)) . f = (IExec ((Times (a,I)),p,(IExec ((I ";" (SubFrom (a,()))),p,s)))) . f
set IT = IExec ((Times (a,I)),p,(IExec ((I ";" (SubFrom (a,()))),p,s)));
f in FinSeq-Locations by SCMFSA_2:def 5;
then A2: f in Data-Locations by ;
hence (IExec ((Times (a,I)),p,s)) . f = (DataPart (IExec ((Times (a,I)),p,s))) . f by FUNCT_1:49
.= (DataPart (IExec ((Times (a,I)),p,(IExec ((I ";" (SubFrom (a,()))),p,s))))) . f by
.= (IExec ((Times (a,I)),p,(IExec ((I ";" (SubFrom (a,()))),p,s)))) . f by ;
:: thesis: verum