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I look for an argument to this statement:

$$ \int_a^x dx_1 \int_a^{x_1} K(x_1,t) dt= \int_a^xdt \int_t^x K(x_1,t) dx_1 $$ It is certainly an integration by change of variables that I can not clarify

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Do you see the domain..? – AD. Mar 19 '12 at 20:59
$K\ge0$? $\,\,\,\,$ – AD. Mar 19 '12 at 21:00

$a\leq x_1\leq x$ and $a\leq t\leq x_1$. So, one has that $t$ changes from $a$ to $x_1$ and $x_1$ changes from $a$ to $x$. It is the same that $t$ changes from $a$ to $x$ and $x_1$ changes from $t$ (it is more than $\min(a,t)=t$) to $x$(it less than $x$). You can draw a picture and everything will be evident.

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When changing the order of integration in a double definite integral, always draw a picture of the region. – Robert Israel Mar 19 '12 at 21:46

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