reserve x1,x2,z for set;
reserve A,B for non empty set;
reserve f,g,h for Element of Funcs(A,REAL);
reserve a,b for Real;

theorem Th11:
  for A being set, f be Element of Funcs(A,REAL) holds
    (RealFuncAdd A).(f,(RealFuncExtMult A).[-1,f]) = RealFuncZero(A)
proof
  let A be set, f be Element of Funcs(A,REAL);
  per cases;
  suppose
A1: A = {};
    then (RealFuncAdd A).(f,(RealFuncExtMult A).[-jj,f]) = {}
      .= RealFuncZero(A) by A1;
   hence thesis;
  end;
  suppose
    A <> {};
    then reconsider A as non empty set;
    reconsider f as Element of Funcs(A,REAL);
    now
      let x be Element of A;
      set y=f.x;
      thus ((RealFuncAdd A).(f,(RealFuncExtMult A).[-jj,f])).x = f.x + ((
      RealFuncExtMult A).[-jj,f]).x by Th1
        .= f.x + ((-1)*y) by Th4
        .= (RealFuncZero(A)).x;
    end;
    hence thesis by FUNCT_2:63;
  end;
end;
