let s be State of SCM+FSA; :: thesis: for Ig being good Program of SCM+FSA
for j being parahalting Instruction of SCM+FSA
for a being Int-Location st ( Ig is parahalting or ( Ig is_halting_on Initialized s & Ig is_closed_on Initialized s ) ) holds
(IExec ((Ig ';' j),s)) . a = (Exec (j,(IExec (Ig,s)))) . a
let Ig be good Program of SCM+FSA; :: thesis: for j being parahalting Instruction of SCM+FSA
for a being Int-Location st ( Ig is parahalting or ( Ig is_halting_on Initialized s & Ig is_closed_on Initialized s ) ) holds
(IExec ((Ig ';' j),s)) . a = (Exec (j,(IExec (Ig,s)))) . a
let j be parahalting Instruction of SCM+FSA; :: thesis: for a being Int-Location st ( Ig is parahalting or ( Ig is_halting_on Initialized s & Ig is_closed_on Initialized s ) ) holds
(IExec ((Ig ';' j),s)) . a = (Exec (j,(IExec (Ig,s)))) . a
let a be Int-Location ; :: thesis: ( ( Ig is parahalting or ( Ig is_halting_on Initialized s & Ig is_closed_on Initialized s ) ) implies (IExec ((Ig ';' j),s)) . a = (Exec (j,(IExec (Ig,s)))) . a )
set I = Ig;
set Mj = Macro j;
a in Int-Locations by SCMFSA_2:9;
then A1: a in Int-Locations \/ FinSeq-Locations by XBOOLE_0:def 3;
assume A2: ( Ig is parahalting or ( Ig is_halting_on Initialized s & Ig is_closed_on Initialized s ) ) ; :: thesis: (IExec ((Ig ';' j),s)) . a = (Exec (j,(IExec (Ig,s)))) . a
A3: DataPart (Initialized (IExec (Ig,s))) = DataPart (IExec (Ig,s)) by A2, Th11;
thus (IExec ((Ig ';' j),s)) . a = (IExec ((Macro j),(IExec (Ig,s)))) . a by A2, Th8
.= (Exec (j,(Initialized (IExec (Ig,s))))) . a by SCMFSA6C:6
.= (DataPart (Exec (j,(Initialized (IExec (Ig,s)))))) . a by A1, FUNCT_1:72, SCMFSA_2:127
.= (DataPart (Exec (j,(IExec (Ig,s))))) . a by A3, SCMFSA6C:5
.= (Exec (j,(IExec (Ig,s)))) . a by A1, FUNCT_1:72, SCMFSA_2:127 ; :: thesis: verum