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

<span class="kw">let </span><font color="Maroon" title="c1">R</font> be   <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Ring</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">p</font> being   <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <font color="Maroon" title="c1">R</font>  st  <a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</a> <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 holds <br/> for <font color="Olive" title="b2">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font>  ex <font color="Olive" title="b3">y</font>, <font color="Olive" title="b4">z</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> st <font color="Olive" title="b1">p</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><font color="Olive" title="b3">y</font>,<font color="Olive" title="b4">z</font></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">p</font> be   <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <font color="Maroon" title="c1">R</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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> <font color="Maroon" title="c2">p</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 implies  for <font color="Olive" title="b1">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font>  ex <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">z</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> st <font color="Maroon" title="c2">p</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><font color="Olive" title="b2">y</font>,<font color="Olive" title="b3">z</font></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span> )</span><br/>



<span class="kw">assume </span><a NAME="E1:33"/><span class="lab"><font color="Green" title="E13">A</font></span>: 
 <a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</a> <font color="Maroon" title="c2">p</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font>  ex <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">z</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> st <font color="Maroon" title="c2">p</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><font color="Olive" title="b2">y</font>,<font color="Olive" title="b3">z</font></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span></span><br/>

<span class="kw">let </span><font color="Maroon" title="c3">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font>, <font color="Olive" title="b2">z</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> st <font color="Maroon" title="c2">p</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><font color="Olive" title="b1">y</font>,<font color="Olive" title="b2">z</font></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span></span><br/>

<span class="kw">take </span><font color="Maroon" title="c4">y</font> = <font color="Maroon" title="c2">p</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">z</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> st <font color="Maroon" title="c2">p</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c4">y</font>,<font color="Olive" title="b1">z</font></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span></span><br/>

<span class="kw">take </span><font color="Maroon" title="c5">z</font> = <font color="Maroon" title="c2">p</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> 1; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">p</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">z</font></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span></span><br/>

<span class="kw">set </span><font color="Maroon" title="c6">q</font> = <span class="p1"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">z</font></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="33_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  holds  <font color="Maroon" title="c2">p</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">z</font></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">i</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">p</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">z</font></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:33_1_1"/>
( <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or ( <font color="Maroon" title="c7">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Maroon" title="c7">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:33_1_1_1"/>
<font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">p</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">z</font></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:33_1_1_1"/>
<font color="Maroon" title="c2">p</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">z</font></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c7">i</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom5.html#T38" onmouseover="rs('polynom5/T38')" onmouseout="rh()">POLYNOM5:38</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="33_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:33_1_1_2"/>
<font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">p</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">z</font></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:33_1_1_2"/>
<font color="Maroon" title="c2">p</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">z</font></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c7">i</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom5.html#T38" onmouseover="rs('polynom5/T38')" onmouseout="rh()">POLYNOM5:38</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="33_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:33_1_1_3"/>
( <font color="Maroon" title="c7">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Maroon" title="c7">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">p</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">z</font></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><a NAME="E2:33_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E14">B</font></span>: 
 not <font color="Maroon" title="c7">i</font> is  <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_2.html#D1" onmouseover="rs('nat_2/D1')" onmouseout="rh()">NAT_2:def 1</a></span>;<br/><a NAME="E3:33_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c7">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_2.html#T29" onmouseover="rs('nat_2/T29')" onmouseout="rh()">NAT_2:29</a></span>;<br/><a NAME="E4:33_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c7">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</a> <font color="Maroon" title="c2">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:33"><span class="lab"><font color="Green" title="E13">A</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E5:33_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c7">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="hurwitz.html#D2" onmouseover="rs('hurwitz/D2')" onmouseout="rh()">HURWITZ:def 2</a></span>;<br/><a NAME="E6:33_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c7">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T7" onmouseover="rs('int_1/T7')" onmouseout="rh()">INT_1:7</a></span>;<br/><span class="kw">hence </span><font color="Maroon" title="c2">p</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c7">i</font> = 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="algseq_1.html#T8" onmouseover="rs('algseq_1/T8')" onmouseout="rh()">ALGSEQ_1:8</a></span>
<br/>.= 
<span class="p1"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">z</font></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c7">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:33_1_1_3"><span class="lab"><font color="Green" title="E14">B</font></span></a>, <a class="ref" href="nat_2.html#T29" onmouseover="rs('nat_2/T29')" onmouseout="rh()">NAT_2:29</a>, <a class="ref" href="polynom5.html#T38" onmouseover="rs('polynom5/T38')" onmouseout="rh()">POLYNOM5:38</a></span>
;<br/> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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>
<span class="kw">hence </span><a NAME="E3:33"/>
<font color="Maroon" title="c2">p</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">z</font></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
