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Contradiction was due to a misstake in the definition in sequence.
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False can be proven with
iff_sequence, codom_of_emptyset_can_be_anything,sequence,
emptyset_is_function_on_emptyset,id_dom,in_irrefl,
suc_subseteq_implies_in,emptyset_subseteq
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Finalised the proof output and goals.
The Local Function definition now producces a Function f with the right
domain and range, together with the rules presented in cases.
Then proof goal of this local definition is set to
for all x we have x is element of dom(f) if and only if x is in exactly one of the subdomains.
This suffices as welldefindness check on f, besides the right range.
Further checks that should be implemented are the correct range of the function.
And optional subproof such that the presented goal can be check easily.
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since now some could define a function with overlapping
and worng subdomains
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But the Proof that the domain of the local function
is not right. Also if in the definition of our local
function we just use f(x) = x then we get a technical
ambigus parse
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