Ussing reCAPTCHA on HTML5 whit contact-form.php - php

Question

I have a website Html5 and try adding the reCaptcha still sending messages without activating the reCaptcha and last night I received over 300 messages a boot.
Help me please how to add so that only sent when the button is activated reCaptcha.
Send sends works well but not activation reCaptcha.
To start contact.html within my template I have put this way:
<!-- Start formulario de contacto -->
<div class="row">
<div class="col-md-9">
<h2>Formulario de contacto</h2>
<form action="php/contact-form.php" id="contact-form">
<div class="alert alert-success hidden" id="contact-alert-success">
<strong>Mensaje enviado correctamente!</strong> Muchas gracias, pronto nos pondremos en contacto con usted, normalmente nuestro tiempo de respuesta es inferior a 2 horas.
</div>
<div class="alert alert-danger hidden" id="contact-alert-error">
<strong>Error!</strong> A sucedido un error si lo desea puede contactarnos directamente en XXXX#tize.XXXX
</div>
<div class="row">
<div class="col-md-4">
<div class="form-group">
<label>Nombre <span class="required">*</span></label>
<input type="text"
value=""
data-msg-required="Por favor introduzca su nombre"
class="form-control"
name="name" id="name">
</div>
</div>
<div class="col-md-4">
<div class="form-group">
<label>eMail <span class="required">*</span></label>
<input type="email"
value=""
data-msg-required="Por favor introduzca su eMail"
data-msg-email="Por favor introduzca un eMail válido"
class="form-control"
name="email"
id="email">
</div>
</div>
<div class="col-md-4">
<div class="form-group">
<label>Asunto <span class="required">*</span></label>
<input type="text"
value=""
data-msg-required="Por favor introduzca el asunto"
class="form-control"
name="subject"
id="subject">
</div>
</div>
</div>
<div class="row">
<div class="col-md-12">
<div class="form-group">
<label>Mensaje <span class="required">*</span></label>
<textarea
data-msg-required="Por favor introduzca su mensaje"
rows="10"
class="form-control"
name="message"
id="message"></textarea>
</div>
</div>
</div>
<!-- Start Google Recaptcha -->
<div class="g-recaptcha" data-sitekey="6Lc88P4SAAAAANiT-ZXILUo-ET4xQmbivHy7uHc8"></div><br>
<!-- End Google Recaptcha -->
<div class="row">
<div class="col-md-12">
<input type="submit" value="Enviar mensaje" class="btn btn-primary" data-loading-text="Cargando...">
</div>
</div>
</form>
</div>
<!-- End formulario de contacto -->
And in php form to send the messages have this post with contact-form.php :
<?php
session_cache_limiter('nocache');
header('Expires: ' . gmdate('r', 0));
header('Content-type: application/json');
// Enter your email address
$to = 'XXXX#tize.XX';
$subject = $_POST['subject'];
if($to) {
$name = $_POST['name'];
$email = $_POST['email'];
$fields = array(
0 => array(
'text' => 'Name',
'val' => $_POST['name']
),
1 => array(
'text' => 'Email address',
'val' => $_POST['email']
),
2 => array(
'text' => 'Message',
'val' => $_POST['message']
)
);
$message = "";
foreach($fields as $field) {
$message .= $field['text'].": " . htmlspecialchars($field['val'], ENT_QUOTES) . "<br>\n";
}
$headers = '';
$headers .= 'From: ' . $name . ' <' . $email . '>' . "\r\n";
$headers .= "Reply-To: " . $email . "\r\n";
$headers .= "MIME-Version: 1.0\r\n";
$headers .= "Content-Type: text/html; charset=UTF-8\r\n";
if (mail($to, $subject, $message, $headers)){
$arrResult = array ('response'=>'success');
} else{
$arrResult = array ('response'=>'error');
}
echo json_encode($arrResult);
} else {
$arrResult = array ('response'=>'error');
echo json_encode($arrResult);
}
?>
Picture of my form, If anyone wants to see my website please let me know and send you the link. Thank you very much.
sending without activating the reCaptcha http://goo.gl/oSLQG9