reserve p,P,P1,P2 for Instruction-Sequence of SCM+FSA;
reserve s, S for State of SCM+FSA,
  I, J for Program of SCM+FSA,
  Ig for good Program of SCM+FSA,
  i for good sequential Instruction of SCM+FSA,
  j for sequential Instruction of SCM+FSA,
  a, b for Int-Location,
  f for FinSeq-Location;

theorem Th9:
 for Ig being good really-closed Program of SCM+FSA holds
  Ig is parahalting or Ig is_halting_on Initialized s,p implies
DataPart Initialized IExec(Ig,p,s)= DataPart IExec(Ig,p,s)
proof let Ig be good really-closed Program of SCM+FSA;
  set I = Ig;
  set IE = IExec(I,p,s);
  assume that
A1: I is parahalting or I is_halting_on Initialized s,p;
A2: I is_halting_on Initialized s,p by A1,SCMFSA7B:19;
  now
A3: dom ( Initialized IE) = the carrier of SCM+FSA by PARTFUN1:def 2;
A4: dom ( Initialized IE) = D \/ {IC SCM+FSA} by MEMSTR_0:13
      .= D \/ ({IC SCM+FSA});
A5: dom IE = the carrier of SCM+FSA by PARTFUN1:def 2;
    hence dom DataPart  Initialized IE = dom IE /\ D by A3,RELAT_1:61;
    then
A6: dom DataPart  Initialized IE = D by A3,A5,A4,XBOOLE_1:21;
    let x be object;
    assume
A7: x in dom DataPart  Initialized IE;
    per cases by A7,A6,SCMFSA_2:100,XBOOLE_0:def 3;
    suppose
    x in Int-Locations;
      then reconsider x9 = x as Int-Location by AMI_2:def 16;
      hereby
        per cases;
        suppose
A8:       x9 is read-write;
          thus (DataPart  Initialized IE).x =
          ( Initialized IE).x by A7,A6,FUNCT_1:49
            .= IE.x by A8,SCMFSA_M:37;
        end;
        suppose
          x9 is read-only;
          then
A9:      x9 = intloc 0;
          thus (DataPart  Initialized IE).x =
          ( Initialized IE).x9 by A7,A6,FUNCT_1:49
            .= 1 by A9,SCMFSA_M:9
            .= IE.x by A2,A9,SCMFSA8C:67;
        end;
      end;
    end;
    suppose
    x in FinSeq-Locations;
      then reconsider x9 = x as FinSeq-Location by SCMFSA_2:def 5;
      thus (DataPart  Initialized IE).x = ( Initialized IE).x9
      by A7,A6,FUNCT_1:49
        .= IE.x by SCMFSA_M:37;
    end;
  end;
  hence thesis by FUNCT_1:46;
end;
