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

theorem Th18:
  for a being Int-Location, I,J being MacroInstruction of SCM+FSA
   holds
  if=0(a,I,J). 0 = a =0_goto  (card J + 3) & if=0(a,I,J). 1 =
goto  2 & if>0(a,I,J). 0 = a >0_goto  (card J + 3) & if>0(a,I
  ,J). 1 = goto  2
proof
  let a be Int-Location;
  let I,J be MacroInstruction of SCM+FSA;
  set i = a =0_goto  (card J + 3);
A1: 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)
  ));
A2: dom Macro i = { 0,  1} by COMPOS_1:61;
  then  0 in dom Macro i by TARSKI:def 2;
  hence if=0(a,I,J). 0 = (Directed Macro i). 0 by A1,SCMFSA8A:14
    .= i by SCMFSA7B:1;
   1 in dom Macro i by A2,TARSKI:def 2;
  hence if=0(a,I,J). 1 = (Directed Macro i). 1 by A1,SCMFSA8A:14
    .= goto  2 by SCMFSA7B:2;
  set i = a >0_goto  (card J + 3);
A3: 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)
  ));
A4: dom Macro i = { 0,  1} by COMPOS_1:61;
  then  0 in dom Macro i by TARSKI:def 2;
  hence if>0(a,I,J). 0 = (Directed Macro i). 0 by A3,SCMFSA8A:14
    .= i by SCMFSA7B:1;
   1 in dom Macro i by A4,TARSKI:def 2;
  hence if>0(a,I,J). 1 = (Directed Macro i). 1 by A3,SCMFSA8A:14
    .= goto  2 by SCMFSA7B:2;
end;
