The direct property of transposition is a work method by which we can move terms and operations from a side of the equality to the other one in inverted way.

As we can see in the drawing, in the transposition of terms or operations we don't have to make any type of operations (addition, subtraction, multiplication, etc.) alone we move "physically" the terms to the other side of the equation, only taking its inverse value.

When any term or operation is transposed to the contrary side of the equality, the both members of the equality are increased/decreased in the same value.

The transposition is an easy and simple method of organizing equalities and equations due to in it we don't need to make operations, alone move terms.
Contrarily, in the current method we need to make double operations and double simplification when we want to move terms.

Why the transpositions work?

In equalities and equations the both members are equivalents, and then when changing any of them we also have the change the other in the same value.
Given 25 - 8 = 17
If we eliminate the term (-8) we have to compensate the second member with the value that makes again equivalent to the equality.
And what this term is? Of course, the inverse value of -8, that is, +8, and then:
25 = 17 + 8
This occurs with any type of operations:
8*5 = 40 and then 8 = 40/5
So the transposition is moving terms between the sides of the equations in its inverse value.

Why the cross-multiplication work?

--The cross-multiplication is explained by means of the double transposition of its denominators.

--But also could be explained with the equalization of members by means mixture of the same ones.
Certainly, the cross-multiplication one members is composed by the multiplication of the bigger numerator of a fraction by the minor denominator of the other fraction, and the other member is composed by the multiplication of the minor numerator by the bigger denominator.
For instance: Given 8/4 = 6/3 ----------> 8*3 = 6*4

As we see in the drawing, in the transposition of terms and operations between the members of equations and equalities, the terms (addition, subtraction) are independent and prior and should be passed directly with the contrary sign.
Contrarily operations are included and subject to terms, and so, we must to pass firstly the terms to be able the transposition of operations later.