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<span class="kw">let </span><font color="Maroon" title="c4">f1</font>, <font color="Maroon" title="c5">f2</font> be   <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <font color="Maroon" title="c1">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  ex <font color="Olive" title="b1">t</font> being   <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Maroon" title="c2">p</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="polynom3.html#K13" title="POLYNOM3:func.13">*'</a> <font color="Maroon" title="c3">s</font></span>)</span> <a href="polynom3.html#K8" title="POLYNOM3:func.8">+</a> <font color="Olive" title="b1">t</font> &amp;  <a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</a> <font color="Olive" title="b1">t</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</a> <font color="Maroon" title="c3">s</font> ) &amp;  ex <font color="Olive" title="b1">t</font> being   <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Maroon" title="c2">p</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f2</font> <a href="polynom3.html#K13" title="POLYNOM3:func.13">*'</a> <font color="Maroon" title="c3">s</font></span>)</span> <a href="polynom3.html#K8" title="POLYNOM3:func.8">+</a> <font color="Olive" title="b1">t</font> &amp;  <a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</a> <font color="Olive" title="b1">t</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</a> <font color="Maroon" title="c3">s</font> ) implies <font color="Maroon" title="c4">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f2</font> )</span><br/>

<span class="kw">given </span><font color="Maroon" title="c6">t1</font> being   <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <font color="Maroon" title="c1">L</font><span class="kw"> such that </span><a NAME="E2:48_1_2"/><span class="lab"><font color="Green" title="E44">A53</font></span>: 
<font color="Maroon" title="c2">p</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="polynom3.html#K13" title="POLYNOM3:func.13">*'</a> <font color="Maroon" title="c3">s</font></span>)</span> <a href="polynom3.html#K8" title="POLYNOM3:func.8">+</a> <font color="Maroon" title="c6">t1</font>
 <span class="kw">and </span><br/><a NAME="E3:48_1_2"/><span class="lab"><font color="Green" title="E45">A54</font></span>: 
 <a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</a> <font color="Maroon" title="c6">t1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</a> <font color="Maroon" title="c3">s</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  for <font color="Olive" title="b1">t</font> being   <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <font color="Maroon" title="c1">L</font>  holds <br/>(  not <font color="Maroon" title="c2">p</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f2</font> <a href="polynom3.html#K13" title="POLYNOM3:func.13">*'</a> <font color="Maroon" title="c3">s</font></span>)</span> <a href="polynom3.html#K8" title="POLYNOM3:func.8">+</a> <font color="Olive" title="b1">t</font> or  not  <a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</a> <font color="Olive" title="b1">t</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</a> <font color="Maroon" title="c3">s</font> ) or <font color="Maroon" title="c4">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f2</font> )</span><br/>

<span class="kw">given </span><font color="Maroon" title="c7">t2</font> being   <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <font color="Maroon" title="c1">L</font><span class="kw"> such that </span><a NAME="E5:48_1_2"/><span class="lab"><font color="Green" title="E46">A55</font></span>: 
<font color="Maroon" title="c2">p</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f2</font> <a href="polynom3.html#K13" title="POLYNOM3:func.13">*'</a> <font color="Maroon" title="c3">s</font></span>)</span> <a href="polynom3.html#K8" title="POLYNOM3:func.8">+</a> <font color="Maroon" title="c7">t2</font>
 <span class="kw">and </span><br/><a NAME="E6:48_1_2"/><span class="lab"><font color="Green" title="E47">A56</font></span>: 
 <a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</a> <font color="Maroon" title="c7">t2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</a> <font color="Maroon" title="c3">s</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f2</font></span><br/>

<span class="lab"><font color="Green" title="E48">A57</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c5">f2</font> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">-</a> <font color="Maroon" title="c4">f1</font></span>)</span> <a href="polynom3.html#K13" title="POLYNOM3:func.13">*'</a> <font color="Maroon" title="c3">s</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">f2</font> <a href="polynom3.html#K13" title="POLYNOM3:func.13">*'</a> <font color="Maroon" title="c3">s</font></span>)</span> <a href="polynom3.html#K8" title="POLYNOM3:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="vfunct_1.html#K5" title="VFUNCT_1:func.5">-</a> <font color="Maroon" title="c4">f1</font></span>)</span> <a href="polynom3.html#K13" title="POLYNOM3:func.13">*'</a> <font color="Maroon" title="c3">s</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom3.html#T32" title="POLYNOM3:th.32">POLYNOM3:32</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">f2</font> <a href="polynom3.html#K13" title="POLYNOM3:func.13">*'</a> <font color="Maroon" title="c3">s</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="polynom3.html#K13" title="POLYNOM3:func.13">*'</a> <font color="Maroon" title="c3">s</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="hurwitz.html#T12" target="_self" title="HURWITZ:th.12">Th12</a></span>
;<br/>
<font color="Maroon" title="c5">f2</font> <a href="polynom3.html#K13" title="POLYNOM3:func.13">*'</a> <font color="Maroon" title="c3">s</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">f2</font> <a href="polynom3.html#K13" title="POLYNOM3:func.13">*'</a> <font color="Maroon" title="c3">s</font></span>)</span> <a href="polynom3.html#K8" title="POLYNOM3:func.8">+</a> <span class="p1">(<span class="default"><a href="polynom3.html#K9" title="POLYNOM3:func.9">0_.</a> <font color="Maroon" title="c1">L</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom3.html#T28" title="POLYNOM3:th.28">POLYNOM3:28</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">f2</font> <a href="polynom3.html#K13" title="POLYNOM3:func.13">*'</a> <font color="Maroon" title="c3">s</font></span>)</span> <a href="polynom3.html#K8" title="POLYNOM3:func.8">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t2</font> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">-</a> <font color="Maroon" title="c7">t2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom3.html#T29" title="POLYNOM3:th.29">POLYNOM3:29</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="polynom3.html#K13" title="POLYNOM3:func.13">*'</a> <font color="Maroon" title="c3">s</font></span>)</span> <a href="polynom3.html#K8" title="POLYNOM3:func.8">+</a> <font color="Maroon" title="c6">t1</font></span>)</span> <a href="polynom3.html#K8" title="POLYNOM3:func.8">+</a> <span class="p1">(<span class="default"><a href="vfunct_1.html#K5" title="VFUNCT_1:func.5">-</a> <font color="Maroon" title="c7">t2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1_2"><span class="lab"><font color="Green" title="E44">A53</font></span></a>, <a class="txt" href="#E5:48_1_2"><span class="lab"><font color="Green" title="E46">A55</font></span></a>, <a class="ref" href="polynom3.html#T26" title="POLYNOM3:th.26">POLYNOM3:26</a></span>
;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E49">A58</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c5">f2</font> <a href="polynom3.html#K13" title="POLYNOM3:func.13">*'</a> <font color="Maroon" title="c3">s</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="polynom3.html#K13" title="POLYNOM3:func.13">*'</a> <font color="Maroon" title="c3">s</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="polynom3.html#K13" title="POLYNOM3:func.13">*'</a> <font color="Maroon" title="c3">s</font></span>)</span> <a href="polynom3.html#K8" title="POLYNOM3:func.8">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">t1</font> <a href="polynom3.html#K8" title="POLYNOM3:func.8">+</a> <span class="p3">(<span class="default"><a href="vfunct_1.html#K5" title="VFUNCT_1:func.5">-</a> <font color="Maroon" title="c7">t2</font></span>)</span></span>)</span></span>)</span> <a href="polynom3.html#K8" title="POLYNOM3:func.8">+</a> <span class="p1">(<span class="default"><a href="vfunct_1.html#K5" title="VFUNCT_1:func.5">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="polynom3.html#K13" title="POLYNOM3:func.13">*'</a> <font color="Maroon" title="c3">s</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom3.html#T26" title="POLYNOM3:th.26">POLYNOM3:26</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">t1</font> <a href="polynom3.html#K8" title="POLYNOM3:func.8">+</a> <span class="p2">(<span class="default"><a href="vfunct_1.html#K5" title="VFUNCT_1:func.5">-</a> <font color="Maroon" title="c7">t2</font></span>)</span></span>)</span> <a href="polynom3.html#K8" title="POLYNOM3:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="polynom3.html#K13" title="POLYNOM3:func.13">*'</a> <font color="Maroon" title="c3">s</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="polynom3.html#K13" title="POLYNOM3:func.13">*'</a> <font color="Maroon" title="c3">s</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom3.html#T26" title="POLYNOM3:th.26">POLYNOM3:26</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">t1</font> <a href="polynom3.html#K8" title="POLYNOM3:func.8">+</a> <span class="p2">(<span class="default"><a href="vfunct_1.html#K5" title="VFUNCT_1:func.5">-</a> <font color="Maroon" title="c7">t2</font></span>)</span></span>)</span> <a href="polynom3.html#K8" title="POLYNOM3:func.8">+</a> <span class="p1">(<span class="default"><a href="polynom3.html#K9" title="POLYNOM3:func.9">0_.</a> <font color="Maroon" title="c1">L</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom3.html#T29" title="POLYNOM3:th.29">POLYNOM3:29</a></span>
<br/>.= 
<font color="Maroon" title="c6">t1</font> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">-</a> <font color="Maroon" title="c7">t2</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom3.html#T28" title="POLYNOM3:th.28">POLYNOM3:28</a></span>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_2_4"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:48_1_2_4"/><span class="lab"><font color="Green" title="E50">A59</font></span>: 
<font color="Maroon" title="c4">f1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">f2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div><span class="lab"><font color="Green" title="E51">A60</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_2_4_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:48_1_2_4_1"/>
<font color="Maroon" title="c5">f2</font> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">-</a> <font color="Maroon" title="c4">f1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="polynom3.html#K9" title="POLYNOM3:func.9">0_.</a> <font color="Maroon" title="c1">L</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">then </span><font color="Maroon" title="c4">f1</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">f2</font> <a href="polynom3.html#K8" title="POLYNOM3:func.8">+</a> <span class="p2">(<span class="default"><a href="vfunct_1.html#K5" title="VFUNCT_1:func.5">-</a> <font color="Maroon" title="c4">f1</font></span>)</span></span>)</span> <a href="polynom3.html#K8" title="POLYNOM3:func.8">+</a> <font color="Maroon" title="c4">f1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom3.html#T28" title="POLYNOM3:th.28">POLYNOM3:28</a></span>
<br/>.= 
<font color="Maroon" title="c5">f2</font> <a href="polynom3.html#K8" title="POLYNOM3:func.8">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">-</a> <font color="Maroon" title="c4">f1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom3.html#T26" title="POLYNOM3:th.26">POLYNOM3:26</a></span>
<br/>.= 
<font color="Maroon" title="c5">f2</font> <a href="polynom3.html#K8" title="POLYNOM3:func.8">+</a> <span class="p1">(<span class="default"><a href="polynom3.html#K9" title="POLYNOM3:func.9">0_.</a> <font color="Maroon" title="c1">L</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom3.html#T29" title="POLYNOM3:th.29">POLYNOM3:29</a></span>
;<br/><span class="kw">hence </span><a NAME="E3:48_1_2_4_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_1_2_4"><span class="lab"><font color="Green" title="E50">A59</font></span></a>, <a class="ref" href="polynom3.html#T28" title="POLYNOM3:th.28">POLYNOM3:28</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">d</font> =  <a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f2</font> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">-</a> <font color="Maroon" title="c4">f1</font></span>)</span>, <font color="Maroon" title="c9">degs</font> =  <a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</a> <font color="Maroon" title="c3">s</font> as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48"><span class="lab"><font color="Green" title="E43">A1</font></span></a>, <a class="txt" href="hurwitz.html#E31"><span class="lab"><font color="Green" title="E29">Lm9</font></span></a></span>;<br/><a NAME="E4:48_1_2_4"/>
 <a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">t1</font> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">-</a> <font color="Maroon" title="c7">t2</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<span class="p1">(<span class="default"><a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</a> <font color="Maroon" title="c6">t1</font></span>)</span>,<span class="p1">(<span class="default"><a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</a> <span class="p2">(<span class="default"><a href="vfunct_1.html#K5" title="VFUNCT_1:func.5">-</a> <font color="Maroon" title="c7">t2</font></span>)</span></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="hurwitz.html#T22" target="_self" title="HURWITZ:th.22">Th22</a></span>;<br/><a NAME="E5:48_1_2_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A61</font></span>: 
 <a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">t1</font> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">-</a> <font color="Maroon" title="c7">t2</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<span class="p1">(<span class="default"><a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</a> <font color="Maroon" title="c6">t1</font></span>)</span>,<span class="p1">(<span class="default"><a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</a> <font color="Maroon" title="c7">t2</font></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom4.html#T8" title="POLYNOM4:th.8">POLYNOM4:8</a></span>;<br/><a NAME="E6:48_1_2_4"/><span class="lab"><font color="Green" title="E53">A62</font></span>: 
 <a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">t1</font> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">-</a> <font color="Maroon" title="c7">t2</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">degs</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_2_4_2"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_2_4_2_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:48_1_2_4_2_1"/>
(  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<span class="p1">(<span class="default"><a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</a> <font color="Maroon" title="c6">t1</font></span>)</span>,<span class="p1">(<span class="default"><a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</a> <font color="Maroon" title="c7">t2</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</a> <font color="Maroon" title="c6">t1</font> or  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<span class="p1">(<span class="default"><a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</a> <font color="Maroon" title="c6">t1</font></span>)</span>,<span class="p1">(<span class="default"><a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</a> <font color="Maroon" title="c7">t2</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</a> <font color="Maroon" title="c7">t2</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T16" title="XXREAL_0:th.16">XXREAL_0:16</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_2_4_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:48_1_2_4_2_1_1"/>
 <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<span class="p1">(<span class="default"><a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</a> <font color="Maroon" title="c6">t1</font></span>)</span>,<span class="p1">(<span class="default"><a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</a> <font color="Maroon" title="c7">t2</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</a> <font color="Maroon" title="c6">t1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">t1</font> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">-</a> <font color="Maroon" title="c7">t2</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">degs</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:48_1_2_4_2_1_1"/>
 <a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">t1</font> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">-</a> <font color="Maroon" title="c7">t2</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">degs</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48_1_2"><span class="lab"><font color="Green" title="E45">A54</font></span></a>, <a class="txt" href="#E5:48_1_2_4"><span class="lab"><font color="Green" title="E52">A61</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_2_4_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:48_1_2_4_2_1_2"/>
 <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<span class="p1">(<span class="default"><a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</a> <font color="Maroon" title="c6">t1</font></span>)</span>,<span class="p1">(<span class="default"><a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</a> <font color="Maroon" title="c7">t2</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</a> <font color="Maroon" title="c7">t2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">t1</font> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">-</a> <font color="Maroon" title="c7">t2</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">degs</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:48_1_2_4_2_1_2"/>
 <a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">t1</font> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">-</a> <font color="Maroon" title="c7">t2</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">degs</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:48_1_2"><span class="lab"><font color="Green" title="E47">A56</font></span></a>, <a class="txt" href="#E5:48_1_2_4"><span class="lab"><font color="Green" title="E52">A61</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><a NAME="E7:48_1_2_4"/>
 <a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">t1</font> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">-</a> <font color="Maroon" title="c7">t2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">d</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">degs</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48"><span class="lab"><font color="Green" title="E43">A1</font></span></a>, <a class="txt" href="#E9:48_1_2"><span class="lab"><font color="Green" title="E49">A58</font></span></a>, <a class="txt" href="#E7:48_1_2"><span class="lab"><font color="Green" title="E48">A57</font></span></a>, <a class="txt" href="#E2:48_1_2_4"><span class="lab"><font color="Green" title="E51">A60</font></span></a>, <a class="ref" href="hurwitz.html#T23" target="_self" title="HURWITZ:th.23">Th23</a></span>;<br/><span class="kw">hence </span><a NAME="E8:48_1_2_4"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:48_1_2_4"><span class="lab"><font color="Green" title="E53">A62</font></span></a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E11:48_1_2"/>
<font color="Maroon" title="c4">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
