reserve x,y,z for set;
reserve I,J,K for Element of Segm 9,
  i,a,a1,a2 for Nat,
  b,b1,b2,c,c1 for Element of SCM-Data-Loc;

theorem Th9:
 for x being Element of SCM-Instr holds
  x in {[SCM-Halt,{},{}] } & InsCode x = 0 or
  x in { [J,<*a*>,{}] : J = 6 } & InsCode x = 6 or
  x in { [K,<*a1*>,<*b1*>] : K in { 7,8 } }
        & (InsCode x = 7 or InsCode x = 8) or
  x in { [I,{},<*b,c*>] : I in { 1,2,3,4,5} }
    & (InsCode x = 1 or InsCode x = 2 or InsCode x = 3
      or InsCode x = 4 or InsCode x = 5)
proof
 let x be Element of SCM-Instr;
 x in {[SCM-Halt,{},{}] } \/ { [J,<*a*>,{}] : J = 6 }
   \/ { [K,<*a1*>,<*b1*>] : K in { 7,8 } } or
    x in { [I,{},<*b,c*>] : I in { 1,2,3,4,5} } by XBOOLE_0:def 3;
 then x in{[SCM-Halt,{},{}] } \/ { [J,<*a*>,{}] : J = 6 } or
    x in  { [K,<*a1*>,<*b1*>] : K in { 7,8 } } or
    x in { [I,{},<*b,c*>] : I in { 1,2,3,4,5} } by XBOOLE_0:def 3;
 then per cases by XBOOLE_0:def 3;
 case x in {[SCM-Halt,{},{}] };
   then x = [SCM-Halt,{},{}] by TARSKI:def 1;
  hence thesis;
 end;
 case x in { [J,<*a*>,{}] : J = 6 };
   then ex J,a st x = [J,<*a*>,{}] & J = 6;
  hence thesis;
 end;
 case x in { [K,<*a1*>,<*b1*>] : K in { 7,8 } };
   then consider K,a1,b1 such that
A1: x = [K,<*a1*>,<*b1*>] and
A2: K in { 7,8 };
   InsCode x = K by A1;
  hence thesis by A2,TARSKI:def 2;
 end;
 case x in { [I,{},<*b,c*>] : I in { 1,2,3,4,5} };
   then consider I,b,c such that
A3: x = [I,{},<*b,c*>] and
A4: I in { 1,2,3,4,5};
   InsCode x = I by A3;
  hence thesis by A4,ENUMSET1:def 3;
 end;
end;
