<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">f1</font> be   <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">L</font>, <font color="Maroon">f2</font> be   <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">L</font>;<br/>

<b>given </b><font color="Maroon">t1</font> being   <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">L</font><b> such that </b><a NAME="E1:48_1_2"/><i><font color="Green">E158</font></i>: 
( <font color="Maroon">p</font> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="polynom3.html#K15">*'</a> <font color="Maroon">s</font></span>)</span> <a href="polynom3.html#K9">+</a> <font color="Maroon">t1</font> &amp;  <a href="hurwitz.html#NK3" target="_self">deg</a> <font color="Maroon">t1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="hurwitz.html#NK3" target="_self">deg</a> <font color="Maroon">s</font> )
 ;<br/>

<b>given </b><font color="Maroon">t2</font> being   <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">L</font><b> such that </b><a NAME="E2:48_1_2"/><i><font color="Green">E159</font></i>: 
( <font color="Maroon">p</font> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="polynom3.html#K15">*'</a> <font color="Maroon">s</font></span>)</span> <a href="polynom3.html#K9">+</a> <font color="Maroon">t2</font> &amp;  <a href="hurwitz.html#NK3" target="_self">deg</a> <font color="Maroon">t2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="hurwitz.html#NK3" target="_self">deg</a> <font color="Maroon">s</font> )
 ;<br/>

<font color="Maroon">f2</font> <a href="polynom3.html#K15">*'</a> <font color="Maroon">s</font> = 
<span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="polynom3.html#K15">*'</a> <font color="Maroon">s</font></span>)</span> <a href="polynom3.html#K9">+</a> <span class="p1">(<span class="default"><a href="polynom3.html#K12">0_.</a> <font color="Maroon">L</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom3.html#T29">POLYNOM3:29</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="polynom3.html#K15">*'</a> <font color="Maroon">s</font></span>)</span> <a href="polynom3.html#K9">+</a> <span class="p1">(<span class="default"><font color="Maroon">t2</font> <a href="polynom3.html#K11">-</a> <font color="Maroon">t2</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom3.html#T30">POLYNOM3:30</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="polynom3.html#K15">*'</a> <font color="Maroon">s</font></span>)</span> <a href="polynom3.html#K9">+</a> <font color="Maroon">t1</font></span>)</span> <a href="polynom3.html#K9">+</a> <span class="p1">(<span class="default"><a href="polynom3.html#K10">-</a> <font color="Maroon">t2</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="polynom3.html#T26">POLYNOM3:26</a></i>
;<br/>
<b>then </b><i><font color="Green">E172</font></i>: <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="polynom3.html#K15">*'</a> <font color="Maroon">s</font></span>)</span> <a href="polynom3.html#K11">-</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="polynom3.html#K15">*'</a> <font color="Maroon">s</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="polynom3.html#K15">*'</a> <font color="Maroon">s</font></span>)</span> <a href="polynom3.html#K9">+</a> <span class="p2">(<span class="default"><font color="Maroon">t1</font> <a href="polynom3.html#K9">+</a> <span class="p3">(<span class="default"><a href="polynom3.html#K10">-</a> <font color="Maroon">t2</font></span>)</span></span>)</span></span>)</span> <a href="polynom3.html#K9">+</a> <span class="p1">(<span class="default"><a href="polynom3.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="polynom3.html#K15">*'</a> <font color="Maroon">s</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="polynom3.html#T26">POLYNOM3:26</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">t1</font> <a href="polynom3.html#K9">+</a> <span class="p2">(<span class="default"><a href="polynom3.html#K10">-</a> <font color="Maroon">t2</font></span>)</span></span>)</span> <a href="polynom3.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="polynom3.html#K15">*'</a> <font color="Maroon">s</font></span>)</span> <a href="polynom3.html#K11">-</a> <span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="polynom3.html#K15">*'</a> <font color="Maroon">s</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="polynom3.html#T26">POLYNOM3:26</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">t1</font> <a href="polynom3.html#K9">+</a> <span class="p2">(<span class="default"><a href="polynom3.html#K10">-</a> <font color="Maroon">t2</font></span>)</span></span>)</span> <a href="polynom3.html#K9">+</a> <span class="p1">(<span class="default"><a href="polynom3.html#K12">0_.</a> <font color="Maroon">L</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom3.html#T30">POLYNOM3:30</a></i>
<br/>.= 
<font color="Maroon">t1</font> <a href="polynom3.html#K11">-</a> <font color="Maroon">t2</font>
<b>by </b><i><a class="ref" href="polynom3.html#T29">POLYNOM3:29</a></i>
;<br/>
<i><font color="Green">E173</font></i>: <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="polynom3.html#K11">-</a> <font color="Maroon">f1</font></span>)</span> <a href="polynom3.html#K15">*'</a> <font color="Maroon">s</font> = 
<span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="polynom3.html#K15">*'</a> <font color="Maroon">s</font></span>)</span> <a href="polynom3.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom3.html#K10">-</a> <font color="Maroon">f1</font></span>)</span> <a href="polynom3.html#K15">*'</a> <font color="Maroon">s</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom3.html#T33">POLYNOM3:33</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="polynom3.html#K15">*'</a> <font color="Maroon">s</font></span>)</span> <a href="polynom3.html#K11">-</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="polynom3.html#K15">*'</a> <font color="Maroon">s</font></span>)</span>
<b>by </b><i/>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_2_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:48_1_2_4"/><i><font color="Green">E175</font></i>: 
<font color="Maroon">f1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">f2</font>
 ;<br/><div><i><font color="Green">E176</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_2_4_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:48_1_2_4_1"/>
<font color="Maroon">f2</font> <a href="polynom3.html#K11">-</a> <font color="Maroon">f1</font> <a href="funct_2.html#R2">=</a>  <a href="polynom3.html#K12">0_.</a> <font color="Maroon">L</font>
 ;<br/><b>then </b><font color="Maroon">f1</font> = 
<span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="polynom3.html#K9">+</a> <span class="p2">(<span class="default"><a href="polynom3.html#K10">-</a> <font color="Maroon">f1</font></span>)</span></span>)</span> <a href="polynom3.html#K9">+</a> <font color="Maroon">f1</font>
<b>by </b><i><a class="ref" href="polynom3.html#T29">POLYNOM3:29</a></i>
<br/>.= 
<font color="Maroon">f2</font> <a href="polynom3.html#K9">+</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="polynom3.html#K11">-</a> <font color="Maroon">f1</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom3.html#T26">POLYNOM3:26</a></i>
<br/>.= 
<font color="Maroon">f2</font> <a href="polynom3.html#K9">+</a> <span class="p1">(<span class="default"><a href="polynom3.html#K12">0_.</a> <font color="Maroon">L</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom3.html#T30">POLYNOM3:30</a></i>
;<br/><b>hence </b><a NAME="E3:48_1_2_4_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="polynom3.html#T29">POLYNOM3:29</a></i>;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon">d</font> =  <a href="hurwitz.html#NK3" target="_self">deg</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="polynom3.html#K11">-</a> <font color="Maroon">f1</font></span>)</span>, <font color="Maroon">degs</font> =  <a href="hurwitz.html#NK3" target="_self">deg</a> <font color="Maroon">s</font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i>, <a class="txt" href="hurwitz.html#E1:48"><i><font color="Green">E29</font></i></a></i>;<br/><a NAME="E4:48_1_2_4"/><i><font color="Green">E177</font></i>: 
 <a href="hurwitz.html#NK3" target="_self">deg</a> <span class="p1">(<span class="default"><font color="Maroon">t1</font> <a href="polynom3.html#K11">-</a> <font color="Maroon">t2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">degs</font>
 <b>by </b><i>, <a class="ref" href="hurwitz.html#T2" target="_self">Th2</a>, <a class="ref" href="hurwitz.html#T1" target="_self">Th1</a>, <a class="ref" href="hurwitz.html#T3" target="_self">Th3</a>, </i>;<br/><a NAME="E5:48_1_2_4"/>
 <a href="hurwitz.html#NK3" target="_self">deg</a> <span class="p1">(<span class="default"><font color="Maroon">t1</font> <a href="polynom3.html#K11">-</a> <font color="Maroon">t2</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="xxreal_0.html#K4">max</a> <span class="p1">(<span class="default"><a href="hurwitz.html#NK3" target="_self">deg</a> <font color="Maroon">t1</font></span>)</span>,<span class="p1">(<span class="default"><a href="hurwitz.html#NK3" target="_self">deg</a> <span class="p2">(<span class="default"><a href="polynom3.html#K10">-</a> <font color="Maroon">t2</font></span>)</span></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E6:48_1_2_4"/><b>then </b><i><font color="Green">E178</font></i>: 
 <a href="hurwitz.html#NK3" target="_self">deg</a> <span class="p1">(<span class="default"><font color="Maroon">t1</font> <a href="polynom3.html#K11">-</a> <font color="Maroon">t2</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="xxreal_0.html#K4">max</a> <span class="p1">(<span class="default"><a href="hurwitz.html#NK3" target="_self">deg</a> <font color="Maroon">t1</font></span>)</span>,<span class="p1">(<span class="default"><a href="hurwitz.html#NK3" target="_self">deg</a> <font color="Maroon">t2</font></span>)</span>
 <b>by </b><i><a class="ref" href="polynom4.html#T11">POLYNOM4:11</a></i>;<br/><a NAME="E7:48_1_2_4"/>
 <a href="hurwitz.html#NK3" target="_self">deg</a> <span class="p1">(<span class="default"><font color="Maroon">t1</font> <a href="polynom3.html#K11">-</a> <font color="Maroon">t2</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">degs</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_2_4_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_2_4_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:48_1_2_4_2_1"/>
(  <a href="xxreal_0.html#K4">max</a> <span class="p1">(<span class="default"><a href="hurwitz.html#NK3" target="_self">deg</a> <font color="Maroon">t1</font></span>)</span>,<span class="p1">(<span class="default"><a href="hurwitz.html#NK3" target="_self">deg</a> <font color="Maroon">t2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="hurwitz.html#NK3" target="_self">deg</a> <font color="Maroon">t1</font> or  <a href="xxreal_0.html#K4">max</a> <span class="p1">(<span class="default"><a href="hurwitz.html#NK3" target="_self">deg</a> <font color="Maroon">t1</font></span>)</span>,<span class="p1">(<span class="default"><a href="hurwitz.html#NK3" target="_self">deg</a> <font color="Maroon">t2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="hurwitz.html#NK3" target="_self">deg</a> <font color="Maroon">t2</font> )
 </span><b>by </b><i><a class="ref" href="square_1.html#T49">SQUARE_1:49</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_2_4_2_1_1"><b>suppose </b></a><a NAME="E1:48_1_2_4_2_1_1"/>
 <a href="xxreal_0.html#K4">max</a> <span class="p1">(<span class="default"><a href="hurwitz.html#NK3" target="_self">deg</a> <font color="Maroon">t1</font></span>)</span>,<span class="p1">(<span class="default"><a href="hurwitz.html#NK3" target="_self">deg</a> <font color="Maroon">t2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="hurwitz.html#NK3" target="_self">deg</a> <font color="Maroon">t1</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:48_1_2_4_2_1_1"/>
 <a href="hurwitz.html#NK3" target="_self">deg</a> <span class="p1">(<span class="default"><font color="Maroon">t1</font> <a href="polynom3.html#K11">-</a> <font color="Maroon">t2</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">degs</font>
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_2_4_2_1_2"><b>suppose </b></a><a NAME="E1:48_1_2_4_2_1_2"/>
 <a href="xxreal_0.html#K4">max</a> <span class="p1">(<span class="default"><a href="hurwitz.html#NK3" target="_self">deg</a> <font color="Maroon">t1</font></span>)</span>,<span class="p1">(<span class="default"><a href="hurwitz.html#NK3" target="_self">deg</a> <font color="Maroon">t2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="hurwitz.html#NK3" target="_self">deg</a> <font color="Maroon">t2</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:48_1_2_4_2_1_2"/>
 <a href="hurwitz.html#NK3" target="_self">deg</a> <span class="p1">(<span class="default"><font color="Maroon">t1</font> <a href="polynom3.html#K11">-</a> <font color="Maroon">t2</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">degs</font>
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>hence </b><a NAME="E8:48_1_2_4"/>
contradiction
 <b>by </b><i>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:48_1_2"/>
<font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font>
 ;<br/>


</div>
