reserve x,y for set;
reserve i, j, k for Nat;
reserve I,J,K for Element of Segm 9,
  a,a1 for Nat,
  b,b1,c for Element of Data-Locations SCM;
reserve a, b for Data-Location,
  loc for Nat;
reserve I,J,K for Element of Segm 9,
  a,a1 for Nat,
  b,b1,c for Element of Data-Locations SCM,
  da,db for Data-Location;

theorem
  for ins being Instruction of SCM st InsCode ins = 0 holds ins = halt SCM
proof
  let ins be Instruction of SCM such that
A1: InsCode ins = 0;
A2: now
    assume ins in { [J,<*a*>,{}] : J = 6 };
    then ex J,a st ins = [J,<*a*>,{}] & J = 6;
     then InsCode ins = 6;
    hence contradiction by A1;
  end;
  now
    assume ins in { [I,{},<*b,c*>] : I in { 1,2,3,4,5} };
    then ex I,b,c st ins = [I,{},<*b,c*>] & I in { 1,2,3,4,5};
    then InsCode ins in { 1,2,3,4,5};
    hence contradiction by A1,ENUMSET1:def 3;
  end;
  then
A3: ins in { [SCM-Halt,{},{}] }
 \/ { [J,<*a*>,{}] : J = 6 } \/ { [K,<*a1*>,<*b1*>] :
  K in { 7,8 } } by AMI_3:27,XBOOLE_0:def 3;
  now
    assume ins in { [K,<*a1*>,<*b1*>] : K in { 7,8 } };
    then ex K,a1,b1 st ins = [K,<*a1*>,<*b1*>] & K in { 7,8 };
    then InsCode ins in {7,8};
    hence contradiction by A1, TARSKI:def 2;
  end;
  then ins in { [SCM-Halt,{},{}] } \/ { [J,<*a*>,{}] : J = 6 }
  by A3,XBOOLE_0:def 3;
  then ins in {[SCM-Halt,{},{}]} by A2,XBOOLE_0:def 3;
  then ins = [SCM-Halt,{},{}] by TARSKI:def 1;
  hence thesis by AMI_3:26;
end;
