reserve x,y,z for set,
  k for Element of NAT;
reserve J,J1,K for Element of Segm 13,
  a for Element of NAT,
  b,b1,b2,c,c1,c2 for Element of SCM-Data-Loc,
  f,f1,f2 for Element of SCM+FSA-Data*-Loc;

theorem Th2:
  for I being Element of SCM+FSA-Instr st I`1_3 <= 8 holds I in SCM-Instr
proof
  let I be Element of SCM+FSA-Instr such that
A1: I`1_3 <= 8;
A2: now
    assume I in { [K,{},<*c1,f1*>] : K in {11,12} };
    then consider K,c,f such that
A3: I = [K,{},<*c,f*>] and
A4: K in {11,12};
    I`1_3 = K by A3;
    then I`1_3 = 11 or I`1_3 = 12 by A4,TARSKI:def 2;
    hence contradiction by A1;
  end;
A5: now
    assume I in { [J,{},<*c,f,b*>] : J in {9,10} };
    then consider J,b,c,f such that
A6: I = [J,{},<*c,f,b*>] and
A7: J in {9,10};
    I`1_3 = J by A6;
    then I`1_3 = 9 or I`1_3 = 10 by A7,TARSKI:def 2;
    hence contradiction by A1;
  end;
A8: now
    assume I in the set of all  [13,{},<*b1*>] ;
    then consider b1 such that
A9: I = [13,{},<*b1*>];
    I`1_3 = 13 by A9;
    hence contradiction by A1;
  end;
  I in SCM-Instr \/ { [J,{},<*c,f,b*>] : J in {9,10} }
   or I in { [K,{},<*c1,f1*>] : K in {11,12} }
   or I in the set of all  [13,{},<*b1*>]
    by XBOOLE_0:def 3;
  hence thesis by A2,A5,A8,XBOOLE_0:def 3;
end;
