On page 41 you can read, that the evidence is written in a document from the year 1541:
"... Johann Bidennbach ..., sein Altvater mütterlicherseits solle Graf Eberhards im Barte lediger Bruder gewesen sein."
"... Johann Bidennbach ..., his maternal grandfather is said to be the brother of duke Eberhard im Barte."
and about the beard:
"Seinen Beinamen „im Bart“ soll er einem auf der Pilgerreise geleisteten Gelübde verdanken, dem zufolge er sich den Bart in Zukunft nicht mehr schneiden werde."
... Sorry, could you please put the 1541 document in context, what it was about and said? ...
The father Johann Bidennbach writes on 16 September 1541, that he was born in Dagersheim, lives in Brackenheim, had worked 5 years as "Vogt", but was dismissed. [From other sources we know, that he was Vogt in Brackenheim from fall 1534 until 23 April 1540]. This letter he writes, is a petition, because of his son, who needs money for his studying in Tübingen. The son Johann Bidenbach began to study on 16 February 1540. Unfortunatly the son lost all his belongings in a fire, and has now to live in his fathers home in Brackenheim.
In an other document the high administrative official (an "Obervogt"), who received this petition, made some remarks about. The Obervogt writes, that it is said, that the father Johann Bidennbach had a maternal grandfather who was the brother of duke Eberhard.
About your other question about the "cousin" Hanne (or Hans) Wirtemberger, named "goldsmith". He was not a goldsmith, he was just named "goldsmith" because his father seems to has been worked as goldsmith.
Hanne Wirtemberger died in 1529. In 1534 Johann Bidembach in Hessia wrote he would go to Württemberg because he received a job as Vogt there, and he has received a heritage from a cousin in Württemberg. We could explain the 5 years waiting time between the death of Hanne Wirtemberger in 1529 and the migration to Württemberg in 1534 with the political situation: the duke Ulrich left his exile and came back to Württemberg just in 1534.
But there is no evidence that this unnamed cousin of Johann Bidembach really was Hanne Wirtemberger.
Johann Bidembach in Hessia and Johann Bidennbach in Brackenheim were identical persons. The evidence is the list of the names of his sons. In Hessia in 1534 are these named as "Johann, Eberhard and Balthasar". Later in Brackenheim are these the same names of his 3 sons.
... the "cousin" Hanne (or Hans) Wirtemberger, named "goldsmith". He was not a goldsmith, he was just named "goldsmith" because his father seems to has been worked as goldsmith ...
This statement was not correct. The "cousin", named goldsmith, seems to really had worked as goldsmith. Lonhard writes in his article, that he had been a goldsmith in 1519, and later he had worked as a government employee, and then was nicknamed "goldsmith". His father, Hans Wirtemberger from Dagersheim, was never titled or called "goldsmith".
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