let p be Instruction-Sequence of SCM+FSA; :: thesis: for I being InitHalting keepInt0_1 Program of SCM+FSA
for s being State of SCM+FSA holds DataPart (Initialized (IExec (I,p,s))) = DataPart (IExec (I,p,s))
set IF = Data-Locations ;
let I be InitHalting keepInt0_1 Program of SCM+FSA; :: thesis: for s being State of SCM+FSA holds DataPart (Initialized (IExec (I,p,s))) = DataPart (IExec (I,p,s))
let s be State of SCM+FSA; :: thesis: DataPart (Initialized (IExec (I,p,s))) = DataPart (IExec (I,p,s))
set IE = IExec (I,p,s);

now :: thesis: ( dom (DataPart (Initialized (IExec (I,p,s)))) = (dom (IExec (I,p,s))) /\ (Data-Locations ) & ( for x being object st x in dom (DataPart (Initialized (IExec (I,p,s)))) holds
(DataPart (Initialized (IExec (I,p,s)))) . x = (IExec (I,p,s)) . x ) )

A1: dom (Initialized (IExec (I,p,s))) = the carrier of SCM+FSA by PARTFUN1:def 2;
A2: dom (Initialized (IExec (I,p,s))) = (Data-Locations ) \/ {(IC )} by MEMSTR_0:13;
A3: dom (IExec (I,p,s)) = the carrier of SCM+FSA by PARTFUN1:def 2;
hence dom (DataPart (Initialized (IExec (I,p,s)))) = (dom (IExec (I,p,s))) /\ (Data-Locations ) by A1, RELAT_1:61; :: thesis: for x being object st x in dom (DataPart (Initialized (IExec (I,p,s)))) holds
(DataPart (Initialized (IExec (I,p,s)))) . b₂ = (IExec (I,p,s)) . b₂
then A4: dom (DataPart (Initialized (IExec (I,p,s)))) = Data-Locations by A1, A3, A2, XBOOLE_1:21;
let x be object ; :: thesis: ( x in dom (DataPart (Initialized (IExec (I,p,s)))) implies (DataPart (Initialized (IExec (I,p,s)))) . b₁ = (IExec (I,p,s)) . b₁ )
assume A5: x in dom (DataPart (Initialized (IExec (I,p,s)))) ; :: thesis: (DataPart (Initialized (IExec (I,p,s)))) . b₁ = (IExec (I,p,s)) . b₁

per cases ( x in Int-Locations or x in FinSeq-Locations ) by A5, A4, SCMFSA_2:100, XBOOLE_0:def 3;

suppose x in Int-Locations ; :: thesis: (DataPart (Initialized (IExec (I,p,s)))) . b₁ = (IExec (I,p,s)) . b₁

then reconsider x9 = x as Int-Location by AMI_2:def 16;

hereby :: thesis: verum

per cases ( x9 is read-write or x9 is read-only ) ;

suppose A6: x9 is read-write ; :: thesis: (DataPart (Initialized (IExec (I,p,s)))) . x = (IExec (I,p,s)) . x

thus (DataPart (Initialized (IExec (I,p,s)))) . x = (Initialized (IExec (I,p,s))) . x by A5, A4, FUNCT_1:49
.= (IExec (I,p,s)) . x by A6, SCMFSA_M:37 ; :: thesis: verum

end;

suppose x9 is read-only ; :: thesis: (DataPart (Initialized (IExec (I,p,s)))) . x = (IExec (I,p,s)) . x

then A7: x9 = intloc 0 ;
thus (DataPart (Initialized (IExec (I,p,s)))) . x = (Initialized (IExec (I,p,s))) . x9 by A5, A4, FUNCT_1:49
.= 1 by A7, SCMFSA_M:9
.= (IExec (I,p,s)) . x by A7, Th7 ; :: thesis: verum

end;

end;

end;

end;

suppose x in FinSeq-Locations ; :: thesis: (DataPart (Initialized (IExec (I,p,s)))) . b₁ = (IExec (I,p,s)) . b₁

then reconsider x9 = x as FinSeq-Location by SCMFSA_2:def 5;
thus (DataPart (Initialized (IExec (I,p,s)))) . x = (Initialized (IExec (I,p,s))) . x9 by A5, A4, FUNCT_1:49
.= (IExec (I,p,s)) . x by SCMFSA_M:37 ; :: thesis: verum

end;

end;

end;

hence DataPart (Initialized (IExec (I,p,s))) = DataPart (IExec (I,p,s)) by FUNCT_1:46; :: thesis: verum