1) Define the complex number 0 (base 10) : http://us.metamath.org/mpegif/df-0.html

2) Define the complex number 1 (base 10) : http://us.metamath.org/mpegif/df-1.html (Note that the numbers 0 and 1 are constants defined as primitives of the complex number axiom system vs. the definition of the number 2 below)

3) 1 and 0 are distinct : http://us.metamath.org/mpegif/ax1ne0.html Axiom 14 of 25 for real and complex numbers (cf. http://us.metamath.org/mpegif/mmcomplex.html ), derived from ZF set theory (cf. http://us.metamath.org/mpegif/mmset.html#axioms ).

4) Define the unit interval with the Euclidean topology : http://us.metamath.org/mpegif/df-ii.html

5) 0 is an identity element for addition : http://us.metamath.org/mpegif/ax0id.html Axiom 15 of 25 for real and complex numbers (i.e. 0 + 1 = 1)

6) Define the number 2 as 2 = 1 + 1 : http://us.metamath.org/mpegif/df-2.html

QED

Grossly misleading headline.

The real title of the question is: "I need mathematical proof that the distance from zero to 1 is the equal to the distance from 1 to 2"

either way, i'm finding this thread highly entertaining and victorian. here is a gem -

No matter what I said to him, he just disregarded offhandedly. I am so upset that I want to punch him in the face, but I would rather get the proofs and show him that he is completely wrong. To me this would be the equivalent of slapping him on the face with a gauntlet

80 character limit makes it difficult to fit all that in.

"I need a proof that the distances between '0 and 1' and '1 and 2' are equal"

Why in the world would you continue discourse with a troll?

"Troll" is the most abused word in internet forums.