let P be Instruction-Sequence of SCM+FSA; :: thesis: for s being State of SCM+FSA
for I being InitHalting good really-closed MacroInstruction of SCM+FSA
for f being FinSeq-Location
for a being read-write Int-Location st s . a > 0 & while>0 (a,I) is InitHalting holds
(IExec ((while>0 (a,I)),P,s)) . f = (IExec ((while>0 (a,I)),P,(IExec (I,P,s)))) . f

let s be State of SCM+FSA; :: thesis: for I being InitHalting good really-closed MacroInstruction of SCM+FSA
for f being FinSeq-Location
for a being read-write Int-Location st s . a > 0 & while>0 (a,I) is InitHalting holds
(IExec ((while>0 (a,I)),P,s)) . f = (IExec ((while>0 (a,I)),P,(IExec (I,P,s)))) . f

let I be InitHalting good really-closed MacroInstruction of SCM+FSA ; :: thesis: for f being FinSeq-Location
for a being read-write Int-Location st s . a > 0 & while>0 (a,I) is InitHalting holds
(IExec ((while>0 (a,I)),P,s)) . f = (IExec ((while>0 (a,I)),P,(IExec (I,P,s)))) . f

let f be FinSeq-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)),P,s)) . f = (IExec ((while>0 (a,I)),P,(IExec (I,P,s)))) . f

let a be read-write Int-Location; :: thesis: ( s . a > 0 & while>0 (a,I) is InitHalting implies (IExec ((while>0 (a,I)),P,s)) . f = (IExec ((while>0 (a,I)),P,(IExec (I,P,s)))) . f )
set D = Data-Locations ;
assume that
A1: s . a > 0 and
A2: while>0 (a,I) is InitHalting ; :: thesis: (IExec ((while>0 (a,I)),P,s)) . f = (IExec ((while>0 (a,I)),P,(IExec (I,P,s)))) . f
f in FinSeq-Locations by SCMFSA_2:def 5;
then A3: f in Data-Locations by ;
then (IExec ((while>0 (a,I)),P,s)) . f = (DataPart (IExec ((while>0 (a,I)),P,s))) . f by FUNCT_1:49
.= (DataPart (IExec ((while>0 (a,I)),P,(IExec (I,P,s))))) . f by ;
hence (IExec ((while>0 (a,I)),P,s)) . f = (IExec ((while>0 (a,I)),P,(IExec (I,P,s)))) . f by ; :: thesis: verum