reserve s for State of SCM+FSA,
  a, c for read-write Int-Location,
  aa, bb, cc,
  dd, x for Int-Location,
  f for FinSeq-Location,
  I, J for MacroInstruction of SCM+FSA,
  Ig for good MacroInstruction of SCM+FSA,
  i, k for Nat,
  p for Instruction-Sequence of SCM+FSA;

theorem Th10:
  a <> bb implies StepForUp(a,bb,cc,I,p,s).0.bb = s.bb
proof
  set aux = 1-stRWNotIn ({a, bb, cc} \/ UsedILoc I);
  set S = s+*(aux, s.cc-s.bb+1)+*(a, s.bb);
  bb in {a, bb, cc} by ENUMSET1:def 1;
  then bb in {a, bb, cc} \/ UsedILoc I by XBOOLE_0:def 3;
  then
A1: bb <> aux by SCMFSA_M:25;
  assume a <> bb;
  then S.bb = (s+*(aux, s.cc-s.bb+1)).bb by FUNCT_7:32
    .= s.bb by A1,FUNCT_7:32;
  hence thesis by SCMFSA_9:def 5;
end;
