reserve m for Nat;
reserve P,PP,P1,P2 for Instruction-Sequence of SCM+FSA;

theorem Th17:
  for a being Int-Location, I,J being MacroInstruction of SCM+FSA holds
:::   0 in dom if=0(a,I,J) &
   1 in dom if=0(a,I,J) &
:::    0 in dom  if>0(a,I,J) &
1 in dom if>0(a,I,J)
proof
  let a be Int-Location;
  let I,J be MacroInstruction of SCM+FSA;
  set i = a =0_goto  (card J + 3);
  if=0(a,I,J) = i ";" J ";" Goto  (card I + 1) ";" I ";" Stop
  SCM+FSA
    .= i ";" J ";" Goto  (card I + 1) ";" (I ";" Stop SCM+FSA) by SCMFSA6A:25
    .= i ";" J ";" (Goto  (card I + 1) ";" (I ";" Stop SCM+FSA)) by SCMFSA6A:25
    .= i ";" (J ";" (Goto  (card I + 1) ";" (I ";" Stop SCM+FSA))) by
SCMFSA6A:29
    .= Macro i ";" (J ";" (Goto  (card I + 1) ";" (I ";" Stop SCM+FSA)
  ));
  then
A1: dom Macro i c= dom if=0(a,I,J) by SCMFSA6A:17;
 dom Macro i = { 0,  1} by COMPOS_1:61;
  then
A2:  1 in dom Macro i by TARSKI:def 2;
  thus  1 in dom if=0(a,I,J) by A1,A2;
  set i = a >0_goto  (card J + 3);
  if>0(a,I,J) = i ";" J ";" Goto  (card I + 1) ";" I ";" Stop
  SCM+FSA
    .= i ";" J ";" Goto  (card I + 1) ";" (I ";" Stop SCM+FSA) by SCMFSA6A:25
    .= i ";" J ";" (Goto  (card I + 1) ";" (I ";" Stop SCM+FSA)) by SCMFSA6A:25
    .= i ";" (J ";" (Goto  (card I + 1) ";" (I ";" Stop SCM+FSA))) by
SCMFSA6A:29
    .= Macro i ";" (J ";" (Goto  (card I + 1) ";" (I ";" Stop SCM+FSA)
  ));
  then
A3: dom Macro i c= dom if>0(a,I,J) by SCMFSA6A:17;
 dom Macro i = { 0,  1} by COMPOS_1:61;
  then
A4:  1 in dom Macro i by TARSKI:def 2;
  thus thesis by A3,A4;
end;
