reserve s for State of SCM+FSA,
  I for MacroInstruction of SCM+FSA,
  a for read-write Int-Location;
reserve i,j,k,n for Nat;
reserve P,P1,P2,Q for Instruction-Sequence of SCM+FSA;

theorem
  StepWhile>0(a,P,s,I).(k+1) =StepWhile>0(a,P,StepWhile>0(a,P,s,I).k,I).1
proof
  set sk=StepWhile>0(a,P,s,I).k;
  set sk0=StepWhile>0(a,P,sk,I).0;
  sk0=sk by Def1;
  hence
  StepWhile>0(a,P,s,I).(k+1)
   = Comput(P +* while>0(a,I),Initialized sk0,
      LifeSpan(P +* while>0(a,I) +* I,Initialized sk0) + 2) by Def1
    .=StepWhile>0(a,P,sk,I).(0+1) by Def1
    .=StepWhile>0(a,P,sk,I).1;
end;
