<?xml version="1.0"?>
<div><span class="kw">theorem </span><span class="lab"><font color="Green" title="E35">Th30</font></span>: <a NAME="T27"><span class="comment"><font color="firebrick">:: HILB10_1:27</font></span><br/></a><div class="add"> for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><br/>  for <font color="Olive" title="b3">a</font> being   non  <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <span class="p1">(<span class="default"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b2">y</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="hilb10_1.html#K2" title="HILB10_1:func.2">Py</a> (<font color="Olive" title="b3">a</font>,<span class="p2"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b2">y</font></span>)</span></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>)</span>)</span>, <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="absvalue.html#K1" title="ABSVALUE:func.1">sgn</a> <font color="Olive" title="b2">y</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><a href="hilb10_1.html#K2" title="HILB10_1:func.2">Py</a> (<font color="Olive" title="b3">a</font>,<span class="p3"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Olive" title="b2">y</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>)</span>)</span></span>)</span> <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a>  <a href="hilb10_1.html#K1" title="HILB10_1:func.1">Px</a> (<font color="Olive" title="b3">a</font>,<span class="p1"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>)</div></div>
