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 Th10:
  for A being set, f being Element of Funcs(A,REAL) holds
    (RealFuncAdd A).(RealFuncZero(A),f) = f
proof
  let A be set, f be Element of Funcs(A,REAL);
  per cases;
  suppose A = {}; then
A1: f = {};
    thus (RealFuncAdd A).(RealFuncZero(A),f) = addreal.:(RealFuncZero(A),f) by
Def1
      .= f by A1;
  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;
      thus ((RealFuncAdd A).(RealFuncZero(A),f)).x = (RealFuncZero(A)).x + f.x
      by Th1
        .= 0 + f.x
        .= f.x;
    end;
    hence thesis;
  end;
end;
