let s be State of SCM+FSA ; :: thesis: for Ig being good Program of SCM+FSA st ( Ig is parahalting or ( Ig is_closed_on Initialize s & Ig is_halting_on Initialize s ) ) holds
DataPart (Initialize (IExec Ig,s)) = DataPart (IExec Ig,s)
let Ig be good Program of SCM+FSA ; :: thesis: ( ( Ig is parahalting or ( Ig is_closed_on Initialize s & Ig is_halting_on Initialize s ) ) implies DataPart (Initialize (IExec Ig,s)) = DataPart (IExec Ig,s) )
set I = Ig;
set IE = IExec Ig,s;
assume A1: ( Ig is parahalting or ( Ig is_closed_on Initialize s & Ig is_halting_on Initialize s ) ) ; :: thesis: DataPart (Initialize (IExec Ig,s)) = DataPart (IExec Ig,s)
A2: Ig is_halting_on Initialize s by A1, SCMFSA7B:25;
A3: Ig is_closed_on Initialize s by A1, SCMFSA7B:24;

now

A4: dom (Initialize (IExec Ig,s)) = the carrier of SCM+FSA by PARTFUN1:def 4;
then A5: dom (Initialize (IExec Ig,s)) = (Int-Locations \/ FinSeq-Locations ) \/ ({(IC SCM+FSA )} \/ NAT ) by SCMFSA_2:8, XBOOLE_1:4;
A6: dom (IExec Ig,s) = the carrier of SCM+FSA by PARTFUN1:def 4;
hence dom (DataPart (Initialize (IExec Ig,s))) = (dom (IExec Ig,s)) /\ (Int-Locations \/ FinSeq-Locations ) by A4, RELAT_1:90, SCMFSA_2:127; :: thesis: for x being set st x in dom (DataPart (Initialize (IExec Ig,s))) holds
(DataPart (Initialize (IExec Ig,s))) . b₂ = (IExec Ig,s) . b₂
then A7: dom (DataPart (Initialize (IExec Ig,s))) = Int-Locations \/ FinSeq-Locations by A4, A6, A5, XBOOLE_1:21;
let x be set ; :: thesis: ( x in dom (DataPart (Initialize (IExec Ig,s))) implies (DataPart (Initialize (IExec Ig,s))) . b₁ = (IExec Ig,s) . b₁ )
assume A8: x in dom (DataPart (Initialize (IExec Ig,s))) ; :: thesis: (DataPart (Initialize (IExec Ig,s))) . b₁ = (IExec Ig,s) . b₁

per cases ( x in Int-Locations or x in FinSeq-Locations ) by A8, A7, XBOOLE_0:def 3;

suppose x in Int-Locations ; :: thesis: (DataPart (Initialize (IExec Ig,s))) . b₁ = (IExec Ig,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 A9: not x9 is read-only ; :: thesis: (DataPart (Initialize (IExec Ig,s))) . x = (IExec Ig,s) . x

thus (DataPart (Initialize (IExec Ig,s))) . x = (Initialize (IExec Ig,s)) . x by A8, A7, FUNCT_1:72, SCMFSA_2:127
.= (IExec Ig,s) . x by A9, SCMFSA6C:3 ; :: thesis: verum

end;

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

then A10: x9 = intloc 0 by SF_MASTR:def 5;
thus (DataPart (Initialize (IExec Ig,s))) . x = (Initialize (IExec Ig,s)) . x9 by A8, A7, FUNCT_1:72, SCMFSA_2:127
.= 1 by A10, SCMFSA6C:3
.= (IExec Ig,s) . x by A3, A2, A10, SCMFSA8C:96 ; :: thesis: verum

end;

end;

end;

end;

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

then reconsider x9 = x as FinSeq-Location by SCMFSA_2:12;
thus (DataPart (Initialize (IExec Ig,s))) . x = (Initialize (IExec Ig,s)) . x9 by A8, A7, FUNCT_1:72, SCMFSA_2:127
.= (IExec Ig,s) . x by SCMFSA6C:3 ; :: thesis: verum

end;

end;

end;

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