let p be Instruction-Sequence of SCM+FSA; :: thesis: for s being State of SCM+FSA
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,p & Ig is_closed_on Initialized s,p ) ) holds
(IExec ((Ig ';' j),p,s)) . a = (Exec (j,(IExec (Ig,p,s)))) . a
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,p & Ig is_closed_on Initialized s,p ) ) holds
(IExec ((Ig ';' j),p,s)) . a = (Exec (j,(IExec (Ig,p,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,p & Ig is_closed_on Initialized s,p ) ) holds
(IExec ((Ig ';' j),p,s)) . a = (Exec (j,(IExec (Ig,p,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,p & Ig is_closed_on Initialized s,p ) ) holds
(IExec ((Ig ';' j),p,s)) . a = (Exec (j,(IExec (Ig,p,s)))) . a
let a be Int-Location ; :: thesis: ( ( Ig is parahalting or ( Ig is_halting_on Initialized s,p & Ig is_closed_on Initialized s,p ) ) implies (IExec ((Ig ';' j),p,s)) . a = (Exec (j,(IExec (Ig,p,s)))) . a )
set I = Ig;
set Mj = Macro j;
a in Int-Locations by SCMFSA_2:2;
then A1: a in Data-Locations by SCMFSA_2:100, XBOOLE_0:def 3;
assume A2: ( Ig is parahalting or ( Ig is_halting_on Initialized s,p & Ig is_closed_on Initialized s,p ) ) ; :: thesis: (IExec ((Ig ';' j),p,s)) . a = (Exec (j,(IExec (Ig,p,s)))) . a
A3: DataPart (Initialized (IExec (Ig,p,s))) = DataPart (IExec (Ig,p,s)) by A2, Th11;
thus (IExec ((Ig ';' j),p,s)) . a = (IExec ((Macro j),p,(IExec (Ig,p,s)))) . a by A2, Th8
.= (IExec ((Macro j),p,(IExec (Ig,p,s)))) . a
.= (Exec (j,(Initialized (IExec (Ig,p,s))))) . a by SCMFSA6C:5
.= (Exec (j,(Initialized (IExec (Ig,p,s))))) . a
.= (DataPart (Exec (j,(Initialized (IExec (Ig,p,s)))))) . a by A1, FUNCT_1:49
.= (DataPart (Exec (j,(IExec (Ig,p,s))))) . a by A3, SCMFSA6C:4
.= (Exec (j,(IExec (Ig,p,s)))) . a by A1, FUNCT_1:49 ; :: thesis: verum