**Answer**
The statement you tried to prove is **not provable as it is**.
In Isabelle/HOL the induction rule that is available by default works only
on *inductive datatypes* (lists, trees, natural numbers, …).
The real numbers form no inductive datatype, so the rule
```isabelle
apply (induction arbitrary: a b rule: less_induct)
```
has no meaning for the goal you wrote. The rule `less_induct` works
on a pair of natural numbers, not on real‑valued coordinates.
In addition the hypothesis
```isabelle
f(x) = f(y) ⟶ dist x y ≥ 2/3
```
contains no definition of the function `f`. For an arbitrary function
`f : 'real × 'real ⇒ color` the implication is false – you can simply
choose `f` to be constant and then the distance of two points with the
same colour can be arbitrarily small. Therefore the theorem is
true **only** for the special colouring function `colour` that is defined
in the theory `Coloring_Of_The_Plane`. That function is not defined by a
recursive function on reals; it is defined by case‑analysis on the two
coordinates of a point. The lemma that is already present in the
theory states
```isabelle
lemma colour_plane [simp]:
assumes "colour x = colour y"
shows "dist x y ≥ 2/3"
```
and can be used directly:
```isabelle
apply (simp add: colour_plane)
```
or, in a structured proof
```isabelle
apply (simp add: colour_plane)
done
```
If you want to prove a similar lemma for your own colouring function,
you must first give a **definition** of that function. For example:
```isabelle
fun my_colour :: "real × real ⇒ color" where
"my_colour (x, y) = (if floor x + floor y mod 4 = 0 then A
else if floor x + floor y mod 4 = 1 then B
else if floor x + floor y mod 4 = 2 then C
else D)"
```
Once `my_colour` is defined you can prove the desired property by a
finite case analysis:
```isabelle
lemma my_colour_ok:
assumes "my_colour (x, y) = my_colour (u, v)"
shows "dist (x, y) (u, v) ≥ 2/3"
proof (cases "floor x + floor y mod 4")
case 0
then have "floor u + floor v mod 4 = 0" using assms by simp
then show ?thesis
by (auto simp add: dist_norm)
next
case 1
then have "floor u + floor v mod 4 = 1" using assms by simp
then show ?thesis
by (auto simp add: dist_norm)
next
case 2
then have "floor u + floor v mod 4 = 2" using assms by simp
then show ?thesis
by (auto simp add: dist_norm)
next
case 3
then have "floor u + floor v mod 4 = 3" using assms by simp
then show ?thesis
by (auto simp add: dist_norm)
qed
```
The proof above is a *finite* case‑analysis, not a structural
induction on the real numbers. It is the correct way to establish the
lower bound on the distance between points of the same colour for a
concrete colouring function.
**Bottom line** – the theorem as written cannot be proved in
Isabelle/HOL without a concrete definition of `f`. Use the lemma
`colour_plane` from the library or provide your own definition and prove
the property by case analysis.