set IF = Int-Locations \/ FinSeq-Locations;
let I be InitHalting keepInt0_1 Program of SCM+FSA; :: thesis: for s being State of SCM+FSA holds DataPart (Initialized (IExec (I,s))) = DataPart (IExec (I,s))
let s be State of SCM+FSA; :: thesis: DataPart (Initialized (IExec (I,s))) = DataPart (IExec (I,s))
set IE = IExec (I,s);

now

A1: dom (Initialized (IExec (I,s))) = the carrier of SCM+FSA by PARTFUN1:def 4;
then A2: dom (Initialized (IExec (I,s))) = (Int-Locations \/ FinSeq-Locations) \/ ({(IC SCM+FSA)} \/ NAT) by SCMFSA_2:8, XBOOLE_1:4;
A3: dom (IExec (I,s)) = the carrier of SCM+FSA by PARTFUN1:def 4;
hence dom (DataPart (Initialized (IExec (I,s)))) = (dom (IExec (I,s))) /\ (Int-Locations \/ FinSeq-Locations) by A1, RELAT_1:90, SCMFSA_2:127; :: thesis: for x being set st x in dom (DataPart (Initialized (IExec (I,s)))) holds
(DataPart (Initialized (IExec (I,s)))) . b₂ = (IExec (I,s)) . b₂
then A4: dom (DataPart (Initialized (IExec (I,s)))) = Int-Locations \/ FinSeq-Locations by A1, A3, A2, XBOOLE_1:21;
let x be set ; :: thesis: ( x in dom (DataPart (Initialized (IExec (I,s)))) implies (DataPart (Initialized (IExec (I,s)))) . b₁ = (IExec (I,s)) . b₁ )
assume A5: x in dom (DataPart (Initialized (IExec (I,s)))) ; :: thesis: (DataPart (Initialized (IExec (I,s)))) . b₁ = (IExec (I,s)) . b₁

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

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

then reconsider x9 = x as Int-Location by SCMFSA_2:11;

hereby :: thesis: verum

per cases ( not x9 is read-only or x9 is read-only ) ;

suppose A6: not x9 is read-only ; :: thesis: (DataPart (Initialized (IExec (I,s)))) . x = (IExec (I,s)) . x

thus (DataPart (Initialized (IExec (I,s)))) . x = (Initialized (IExec (I,s))) . x by A5, A4, FUNCT_1:72, SCMFSA_2:127
.= (IExec (I,s)) . x by A6, SCMFSA6C:3 ; :: thesis: verum

end;

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

then A7: x9 = intloc 0 by SF_MASTR:def 5;
thus (DataPart (Initialized (IExec (I,s)))) . x = (Initialized (IExec (I,s))) . x9 by A5, A4, FUNCT_1:72, SCMFSA_2:127
.= 1 by A7, SCMFSA6C:3
.= (IExec (I,s)) . x by A7, Th17 ; :: thesis: verum

end;

end;

end;

end;

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

then reconsider x9 = x as FinSeq-Location by SCMFSA_2:12;
thus (DataPart (Initialized (IExec (I,s)))) . x = (Initialized (IExec (I,s))) . x9 by A5, A4, FUNCT_1:72, SCMFSA_2:127
.= (IExec (I,s)) . x by SCMFSA6C:3 ; :: thesis: verum

end;

end;

end;

hence DataPart (Initialized (IExec (I,s))) = DataPart (IExec (I,s)) by FUNCT_1:68, SCMFSA_2:127; :: thesis: verum