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 Th10:
 for Ig being good really-closed Program of SCM+FSA holds
  Ig is parahalting or Ig is_halting_on Initialized s,p
  implies IExec(Ig ";" j,p,s).a = Exec(j, IExec(Ig,p,s)).a
proof let Ig be good really-closed Program of SCM+FSA;
  set I = Ig;
  set Mj = Macro j;
  a in Int-Locations by AMI_2:def 16;
  then
A1: a in D by SCMFSA_2:100,XBOOLE_0:def 3;
  assume that
A2: I is parahalting or I is_halting_on Initialized s,p;
A3: DataPart Initialized IExec(I,p,s)
       = DataPart IExec(I,p,s) by A2,Th9;
  thus IExec(I ";" j,p,s).a = IExec(Mj,p,IExec(I,p,s)).a by A2,Th6
    .= ( IExec(Mj,p,IExec(I,p,s))).a
    .= ( Exec(j, Initialized IExec(I,p,s))).a by SCMFSA6C:5
    .= Exec(j, Initialized IExec(I,p,s)).a
    .= (DataPart Exec(j, Initialized IExec(I,p,s))).a by A1,FUNCT_1:49

    .= (DataPart Exec(j, IExec(I,p,s))).a by A3,SCMFSA6C:4
    .= Exec(j, IExec(I,p,s)).a by A1,FUNCT_1:49;
end;
