Transitive property of the Trinity?

Anyone remotely familiar with Christian theology has seen this diagram.

This is the typical representation of the holy Trinity. 

It is supposed to make the dogma clearer and easier to understand but it always struck me as a clarification of the contradiction.

Warning! I am about to use logic and maths on theology:

There's a wonderful little property of equality known as the Transitive Property.  If two values are each equal to a third, then the two values are equal.

if A = B and B = C

then A = C

Let us assume that by "is" in the diagram theologians mean "equals".  (note Rational Wiki does a similar exercise by interpreting "is" as "subset", but since the "is" goes both ways (i.e. God is Father and Father is God) I just skipped a step and went straight to equality).

Using standard logic, we would expect that:

if Son = God and God = Spirit

then Son = Spirit

But according to our lovely diagram, Spirit is not equal to Son.  

Is this a contradiction? I certainly think so and haven't found any explanations of how this works out.  I did however find one guy who used the transitive property to prove the Trinity.  

Anyone able to make sense of this or explain what I'm missing?  I seriously don't understand how this one gets out of the gate!

This is all just too confusing: