Was Hans Holbein the Younger a Pupil of Lucas Horenbout?

Lucas Horenbout (or Hornebolte) was a renowned Flemish artist (illuminator), who was employed by King Henry VIII of England in 1525 as a court miniaturist and later appointed to the office of King's Painter in 1534 (this position was renewed in 1544).

 

Twenty-two miniatures have been attributed to Horenbout, dating from c. 1525-c. 1543/4. His later work was overshadowed by the output of his brilliant pupil in the art of miniature painting, Hans Holbein the Younger. 

 

My interpretation of the embroidery pattern found on Katherine of Aragon's miniature portrait by Horenbout:

 

 

My interpretation of the embroidery pattern found on the Portrait Miniature possibly of Anne Boleyn:

 

 

My interpretation of the embroidery pattern found on Henry Fitzroy's nightcap.