reserve p,p1,p2,h for Instruction-Sequence of SCM+FSA;
reserve k, l, n for Nat,
  j for Integer,
  i,i1 for Instruction of SCM+FSA;
reserve s, s1, s2 for State of SCM+FSA,
  a for read-write Int-Location,
  b for Int-Location,
  I, J for MacroInstruction of SCM+FSA,
  Ig for good MacroInstruction of SCM+FSA,
  i, j, k, m, n for Nat;

theorem Th19: :: Step_eq0_1:
 for I being really-closed MacroInstruction of SCM+FSA holds
  ( I is_halting_on Initialized StepWhile=0(a,I,p,s).k ,p+*while=0(a,I)
or
    I is parahalting) & (
  StepWhile=0(a,I,p,s).k).a = 0 & (StepWhile=0(a,I,p,s).k).intloc 0 = 1 implies
DataPart StepWhile=0(a,I,p,s).(k+1) =
 DataPart IExec(I,p+*while=0(a,I),StepWhile=0(a,I,p,s).k)
proof let I be really-closed MacroInstruction of SCM+FSA;
  set Ins = NAT;
  assume that
A1: I is_halting_on Initialized StepWhile=0(a,I,p,s).k ,p+*while=0(a,I)
or
     I is parahalting and
A2: (StepWhile=0(a,I,p,s).k).a = 0 and
A3: (StepWhile=0(a,I,p,s).k).intloc 0 = 1;
  set ISWk = Initialized StepWhile=0(a,I,p,s).k;
  set SW = StepWhile=0(a,I,p,s), PW = p +* while=0(a,I);
  set SWkI = Initialized(SW.k), PWI = p +* while=0(a,I) +* I;
  DataPart ISWk = DataPart SW.k by A3,SCMFSA_M:19;
  then
A4: I is_halting_on SW.k ,PW by A1,SCMFSA7B:19,SCMFSA8B:5;
  I is_halting_on ISWk ,PW by A1,SCMFSA7B:19;
  then
A5: I is_halting_on Initialized SW.k ,PW;
     Initialized SW.k = Initialize Initialized(SW.k) by MEMSTR_0:44;
    then
A6:  PW +* I halts_on Initialized(SW.k) by A5,SCMFSA7B:def 7;
A7: PWI halts_on SWkI by A6;
  set SWkIS = Initialize(SW.k),
      PWIS = PW +* I;
A8: SWkI = SWkIS by A3,SCMFSA_M:18;
A9: SW.(k+1) = Comput(PW +* while=0(a,I),
  (Initialize(SW.k)),LifeSpan(PWIS,SWkIS)+ 2)
   by SCMFSA_9:def 4;
A10: DataPart IExec(I,PW,SW.k)
     = DataPart Result(PWI,SWkI) by SCMFSA6B:def 1
    .= DataPart Comput(PWIS,SWkIS,LifeSpan(PWIS,SWkIS)) by A8,A7,EXTPRO_1:23;
  thus DataPart StepWhile=0(a,I,p,s).(k+1)
   = DataPart Comput(PWIS,SWkIS,LifeSpan(PWIS,SWkIS)) by A2,A4,Th17,A9
   .= DataPart IExec(I,PW,StepWhile=0(a,I,p,s).k) by A10;
end;
