theorem Th18: :: Step_eq0_0:
  (StepWhile=0(a,I,p,s).k).a <> 0 implies
   DataPart StepWhile=0(a,I,p,s).(k+1) = DataPart StepWhile=0(a,I,p,s).k
proof
  assume
A1: (StepWhile=0(a,I,p,s).k).a <> 0;
  set SW = StepWhile=0(a,I,p,s),
      PW = p +* while=0(a,I);
A2: while=0(a,I) c= PW by FUNCT_4:25;
A3: DataPart(Initialize(SW.k)) = DataPart SW.k
by MEMSTR_0:79;
  then
A4: SW.k.a = (Initialize(SW.k)).a by SCMFSA_M:2;
  thus DataPart SW.(k+1) = DataPart Comput(PW +* while=0(a,I),
  (Initialize(SW.k)), (LifeSpan(PW +* I,Initialize(SW.k)) + 2))
   by SCMFSA_9:def 4
    .= DataPart StepWhile=0(a,I,p,s).k by A1,A3,A4,Th16,A2;
end;
