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 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 ;

let s be State of SCM+FSA; :: thesis: for I being really-closed good InitHalting MacroInstruction 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 really-closed good InitHalting MacroInstruction 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 AMI_2:def 16;

then A2: b in Data-Locations by SCMFSA_2:100, XBOOLE_0:def 3;

hence (IExec ((Times (a,I)),p,s)) . b = (DataPart (IExec ((Times (a,I)),p,s))) . b by FUNCT_1:49

.= (DataPart (IExec ((Times (a,I)),p,(IExec ((I ";" (SubFrom (a,(intloc 0)))),p,s))))) . b by A1, Th48

.= (IExec ((Times (a,I)),p,(IExec ((I ";" (SubFrom (a,(intloc 0)))),p,s)))) . b by A2, FUNCT_1:49 ;

:: thesis: verum

for I being really-closed good InitHalting MacroInstruction 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 ;

let s be State of SCM+FSA; :: thesis: for I being really-closed good InitHalting MacroInstruction 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 really-closed good InitHalting MacroInstruction 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 AMI_2:def 16;

then A2: b in Data-Locations by SCMFSA_2:100, XBOOLE_0:def 3;

hence (IExec ((Times (a,I)),p,s)) . b = (DataPart (IExec ((Times (a,I)),p,s))) . b by FUNCT_1:49

.= (DataPart (IExec ((Times (a,I)),p,(IExec ((I ";" (SubFrom (a,(intloc 0)))),p,s))))) . b by A1, Th48

.= (IExec ((Times (a,I)),p,(IExec ((I ";" (SubFrom (a,(intloc 0)))),p,s)))) . b by A2, FUNCT_1:49 ;

:: thesis: verum