let a be Int-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)) . a = (Exec (j,(IExec (I,s)))) . a
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)) . a = (Exec (j,(IExec (I,s)))) . a
let I be parahalting keeping_0 Program of SCM+FSA; :: thesis: for j being parahalting Instruction of SCM+FSA holds (IExec ((I ';' j),s)) . a = (Exec (j,(IExec (I,s)))) . a
let j be parahalting Instruction of SCM+FSA; :: thesis: (IExec ((I ';' j),s)) . a = (Exec (j,(IExec (I,s)))) . a
set Mj = Macro j;
set SA = Start-At (((IC (IExec ((Macro j),(IExec (I,s))))) + (card I)),SCM+FSA);
A1: not a in dom (Start-At (((IC (IExec ((Macro j),(IExec (I,s))))) + (card I)),SCM+FSA)) by SCMFSA6B:9;
A2: DataPart (Initialized (IExec (I,s))) = DataPart (IExec (I,s)) by Lm3;
a in Int-Locations by SCMFSA_2:9;
then A3: a in Int-Locations \/ FinSeq-Locations by XBOOLE_0:def 3;
thus (IExec ((I ';' j),s)) . a = ((IExec ((Macro j),(IExec (I,s)))) +* (Start-At (((IC (IExec ((Macro j),(IExec (I,s))))) + (card I)),SCM+FSA))) . a by SCMFSA6B:44
.= (IExec ((Macro j),(IExec (I,s)))) . a by A1, FUNCT_4:12
.= (Exec (j,(Initialized (IExec (I,s))))) . a by Th6
.= (DataPart (Exec (j,(Initialized (IExec (I,s)))))) . a by A3, FUNCT_1:72, SCMFSA_2:127
.= (DataPart (Exec (j,(IExec (I,s))))) . a by A2, Th5
.= (Exec (j,(IExec (I,s)))) . a by A3, FUNCT_1:72, SCMFSA_2:127 ; :: thesis: verum