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,J being MacroInstruction of SCM+FSA, a being Int-Location holds
  UsedILoc if=0(a,I,J) = {a} \/ UsedILoc I \/ UsedILoc J &
  UsedILoc if>0(a,I,J) = {a} \/ UsedILoc I \/ UsedILoc J
proof
  let I,J be MacroInstruction of SCM+FSA, a be Int-Location;
  set g1= a=0_goto  (card J + 3), g2= Goto  (card I + 1),
  g3= a>0_goto  (card J + 3), SS=Stop SCM+FSA;
  thus UsedILoc if=0(a,I,J) =UsedILoc (g1 ";" J ";" g2 ";"I ";" SS)
    .=UsedILoc (g1 ";" J ";" g2 ";"I) \/ {} by Th3,SF_MASTR:27
    .=UsedILoc (g1 ";" J ";" g2) \/ UsedILoc I by SF_MASTR:27
    .=UsedILoc (g1 ";" J) \/ UsedILoc g2 \/ UsedILoc I by SF_MASTR:27
    .=UsedILoc (g1 ";" J) \/ {} \/ UsedILoc I by Th5
    .=UsedIntLoc g1 \/ UsedILoc J \/ UsedILoc I by SF_MASTR:29
    .={a} \/ UsedILoc J \/ UsedILoc I by SF_MASTR:16
    .={a} \/ UsedILoc I \/ UsedILoc J by XBOOLE_1:4;
  thus UsedILoc if>0(a,I,J) =UsedILoc (g3 ";" J ";" g2 ";"I ";" SS)
    .=UsedILoc (g3 ";" J ";" g2 ";"I) \/ {} by Th3,SF_MASTR:27
    .=UsedILoc (g3 ";" J ";" g2) \/ UsedILoc I by SF_MASTR:27
    .=UsedILoc (g3 ";" J) \/ UsedILoc g2 \/ UsedILoc I by SF_MASTR:27
    .=UsedILoc (g3 ";" J) \/ {} \/ UsedILoc I by Th5
    .=UsedIntLoc g3 \/ UsedILoc J \/ UsedILoc I by SF_MASTR:29
    .={a} \/ UsedILoc J \/ UsedILoc I by SF_MASTR:16
    .={a} \/ UsedILoc I \/ UsedILoc J by XBOOLE_1:4;
end;
