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

theorem
  for I,J being MacroInstruction of SCM+FSA, a,c being Int-Location st I
does not destroy c & J does not destroy c holds if=0(a,I,J) does not destroy c
  & if>0(a,I,J) does not destroy c
proof
  let I,J be MacroInstruction of SCM+FSA;
  let a,c be Int-Location;
  assume
A1: I does not destroy c;
A2: Goto  (card I + 1) does not destroy c by Th48;
  assume
A3: J does not destroy c;
  then a =0_goto  (card J + 3) ";" J does not destroy c by Th44,SCMFSA7B:12;
  then a =0_goto  (card J + 3) ";" J ";" Goto  (card I + 1)
  does not destroy c by A2,Th43;
  then
A4: a =0_goto  (card J + 3) ";" J ";" Goto  (card I + 1) ";" I
  does not destroy c by A1,Th43;
A5: Goto  (card I + 1) does not destroy c by Th48;
  a >0_goto  (card J + 3) ";" J does not destroy c by A3,Th44,SCMFSA7B:13;
  then a >0_goto  (card J + 3) ";" J ";" Goto  (card I + 1)
  does not destroy c by A5,Th43;
  then
A6: a >0_goto  (card J + 3) ";" J ";" Goto  (card I + 1) ";" I
  does not destroy c by A1,Th43;
  Stop SCM+FSA does not destroy c by Th47;
  hence if=0(a,I,J) does not destroy c by A4,Th43;
  Stop SCM+FSA does not destroy c by Th47;
  hence thesis by A6,Th43;
end;
