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 Initialize s & Ig is_closed_on Initialize 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 Initialize s & Ig is_closed_on Initialize 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 Initialize s & Ig is_closed_on Initialize 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 Initialize s & Ig is_closed_on Initialize s ) ) implies (IExec (Ig ';' j),s) . a = (Exec j,(IExec Ig,s)) . a )
set I = Ig;
assume A1: ( Ig is parahalting or ( Ig is_halting_on Initialize s & Ig is_closed_on Initialize s ) ) ; :: thesis: (IExec (Ig ';' j),s) . a = (Exec j,(IExec Ig,s)) . a
set Mj = Macro j;
A2: DataPart (Initialize (IExec Ig,s)) = DataPart (IExec Ig,s) by A1, Th11;
a in Int-Locations by SCMFSA_2:9;
then A3: a in Int-Locations \/ FinSeq-Locations by XBOOLE_0:def 3;
thus (IExec (Ig ';' j),s) . a = (IExec (Macro j),(IExec Ig,s)) . a by A1, Th8
.= (Exec j,(Initialize (IExec Ig,s))) . a by SCMFSA6C:6
.= (DataPart (Exec j,(Initialize (IExec Ig,s)))) . a by A3, FUNCT_1:72, SCMFSA_2:127
.= (DataPart (Exec j,(IExec Ig,s))) . a by A2, SCMFSA6C:5
.= (Exec j,(IExec Ig,s)) . a by A3, FUNCT_1:72, SCMFSA_2:127 ; :: thesis: verum