theorem Th3:
  QC-pred_symbols(Al) c= QC-pred_symbols(Al2)
proof
  for Q being object st Q in QC-pred_symbols(Al)
       holds Q in QC-pred_symbols(Al2)
  proof
    let Q be object such that
A1: Q in QC-pred_symbols(Al);
    set preds = { [k,b] : 7 <= k };
    set preds2 = { [k,b2] where b2 is QC-symbol of Al2 : 7 <= k };
    consider k,b such that
A2: Q=[k,b] & 7 <= k by A1;
    b in QC-symbols(Al2) by Th2,TARSKI:def 3;
    hence Q in QC-pred_symbols(Al2) by A2;
  end;
  hence thesis;
end;
