let f be FinSeq-Location ; :: thesis: for s being State of SCM+FSA
for I being parahalting keeping_0 Program of SCM+FSA
for j being parahalting Instruction of SCM+FSA holds (IExec ((I ';' j),s)) . f = (Exec (j,(IExec (I,s)))) . f
let s be State of SCM+FSA; :: thesis: for I being parahalting keeping_0 Program of SCM+FSA
for j being parahalting Instruction of SCM+FSA holds (IExec ((I ';' j),s)) . f = (Exec (j,(IExec (I,s)))) . f
let I be parahalting keeping_0 Program of SCM+FSA; :: thesis: for j being parahalting Instruction of SCM+FSA holds (IExec ((I ';' j),s)) . f = (Exec (j,(IExec (I,s)))) . f
let j be parahalting Instruction of SCM+FSA; :: thesis: (IExec ((I ';' j),s)) . f = (Exec (j,(IExec (I,s)))) . f
set Mj = Macro j;
set SA = Start-At (((IC (IExec ((Macro j),(IExec (I,s))))) + (card I)),SCM+FSA);
A1: not f in dom (Start-At (((IC (IExec ((Macro j),(IExec (I,s))))) + (card I)),SCM+FSA)) by SCMFSA6B:10;
A2: DataPart (Initialized (IExec (I,s))) = DataPart (IExec (I,s)) by Lm3;
f in FinSeq-Locations by SCMFSA_2:10;
then A3: f in Int-Locations \/ FinSeq-Locations by XBOOLE_0:def 3;
thus (IExec ((I ';' j),s)) . f = ((IExec ((Macro j),(IExec (I,s)))) +* (Start-At (((IC (IExec ((Macro j),(IExec (I,s))))) + (card I)),SCM+FSA))) . f by SCMFSA6B:44
.= (IExec ((Macro j),(IExec (I,s)))) . f by A1, FUNCT_4:12
.= (Exec (j,(Initialized (IExec (I,s))))) . f by Th6
.= (DataPart (Exec (j,(Initialized (IExec (I,s)))))) . f by A3, FUNCT_1:72, SCMFSA_2:127
.= (DataPart (Exec (j,(IExec (I,s))))) . f by A2, Th5
.= (Exec (j,(IExec (I,s)))) . f by A3, FUNCT_1:72, SCMFSA_2:127 ; :: thesis: verum