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

theorem Th1:
  h = (RealFuncAdd A).(f,g) iff for x being Element of A holds h.x = f.x + g.x
proof
A1: now
    assume
A2: for x being Element of A holds h.x=f.x + g.x;
    now
      let x be Element of A;
A3:   dom (addreal.:(f,g)) = A by FUNCT_2:def 1;
      thus ((RealFuncAdd A).(f,g)).x = (addreal.:(f,g)).x by Def1
        .= addreal.(f.x,g.x) by A3,FUNCOP_1:22
        .= f.x + g.x by BINOP_2:def 9
        .= h.x by A2;
    end;
    hence h = (RealFuncAdd A).(f,g);
  end;
  now
    assume
A4: h = (RealFuncAdd A).(f,g);
    let x be Element of A;
A5: dom (addreal.:(f,g)) = A by FUNCT_2:def 1;
    thus h.x = (addreal.:(f,g)).x by A4,Def1
      .= addreal.(f.x,g.x) by A5,FUNCOP_1:22
      .= f.x + g.x by BINOP_2:def 9;
  end;
  hence thesis by A1;
end;
