How did MM realize Fujiwara was still seeing his first wife in The Kagero Diary: A Woman’s Autobiographical Text from Tenth-century?

After MM gave birth to Michitsuna, Fujiwara abandoned them for weeks at a time. On one of his infrequent visits back to them, MM found a letter amid his belongings that was a love note written for Fujiawara. He claimed he was merely testing her feelings, which proved to be false when MM had him followed and discovered he was having an affair.