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
  for I being really-closed MacroInstruction of SCM+FSA holds
  DataPart s1 = DataPart s2 & ProperBodyWhile>0 a,I,s1,p1 implies
  ProperBodyWhile>0 a,I,s2,p2
proof let I be really-closed MacroInstruction of SCM+FSA;
  assume that
A1: DataPart s1 = DataPart s2 and
A2: ProperBodyWhile>0 a,I,s1,p1;
  let k be Nat such that
A3: StepWhile>0(a,I,p2,s2).k.a > 0;
A4: DataPart StepWhile>0(a,I,p2,s2).k
     = DataPart StepWhile>0(a,I,p1,s1).k by A1,A2,Th34;
  then StepWhile>0(a,I,p1,s1).k.a > 0 by A3,SCMFSA_M:2;
  then I is_halting_on StepWhile>0(a,I,p1,s1).k, p1+*while>0(a,I) by A2;
  hence thesis by A4,SCMFSA8B:5;
end;
