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

<span class="kw">let </span><font color="Maroon" title="c2">p</font> be    <a href="ring_4.html#M1" title="RING_4:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K12" title="POLYNOM3:func.12">Polynom-Ring</a> <font color="Maroon" title="c1">F</font></span>)</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"> (  not <font color="Maroon" title="c2">p</font> is  <a href="field_5.html#V1" title="FIELD_5:attr.1">linear</a>  &amp; <font color="Maroon" title="c2">p</font> is  <a href="polynom5.html#V1" title="POLYNOM5:attr.1">with_roots</a>  implies <font color="Maroon" title="c2">p</font> is  <a href="ring_2.html#NV6" title="RING_2:NV.6">reducible</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:25_1_1"/><span class="lab"><font color="Green" title="E14">AS</font></span>: 
(  not <font color="Maroon" title="c2">p</font> is  <a href="field_5.html#V1" title="FIELD_5:attr.1">linear</a>  &amp; <font color="Maroon" title="c2">p</font> is  <a href="polynom5.html#V1" title="POLYNOM5:attr.1">with_roots</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> is  <a href="ring_2.html#NV6" title="RING_2:NV.6">reducible</a> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">F</font><span class="kw"> such that </span><br/><a NAME="E3:25_1_1"/><span class="lab"><font color="Green" title="E15">A</font></span>: 
<font color="Maroon" title="c3">a</font> <a href="polynom5.html#R1" title="POLYNOM5:pred.1">is_a_root_of</a> <font color="Maroon" title="c2">p</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom5.html#D8" onmouseover="rs('polynom5/D8')" onmouseout="rh()">POLYNOM5:def 8</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c4">q1</font> being   <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <font color="Maroon" title="c1">F</font><span class="kw"> such that </span><br/><a NAME="E5:25_1_1"/><span class="lab"><font color="Green" title="E16">B</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"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> (1,<font color="Maroon" title="c3">a</font>)</span>)</span> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <font color="Maroon" title="c4">q1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:25_1_1"><span class="lab"><font color="Green" title="E15">A</font></span></a>, <a class="ref" href="hurwitz.html#T33" onmouseover="rs('hurwitz/T33')" onmouseout="rh()">HURWITZ:33</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">q</font> = <font color="Maroon" title="c4">q1</font> as    <a href="ring_4.html#M1" title="RING_4:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K12" title="POLYNOM3:func.12">Polynom-Ring</a> <font color="Maroon" title="c1">F</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom3.html#D10" onmouseover="rs('polynom3/D10')" onmouseout="rh()">POLYNOM3:def 10</a></span>;<br/>
<a NAME="E7:25_1_1"/><span class="lab"><font color="Green" title="E17">C</font></span>: 
<font color="Maroon" title="c5">q</font> <a href="ring_4.html#R1" title="RING_4:pred.1">divides</a> <font color="Maroon" title="c2">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:25_1_1"><span class="lab"><font color="Green" title="E16">B</font></span></a>, <a class="ref" href="ring_4.html#T1" onmouseover="rs('ring_4/T1')" onmouseout="rh()">RING_4:1</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_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="E1:25_1_1_1"/>
( <font color="Maroon" title="c2">p</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="polynom3.html#K8" title="POLYNOM3:func.8">0_.</a> <font color="Maroon" title="c1">F</font> or <font color="Maroon" title="c2">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom3.html#K8" title="POLYNOM3:func.8">0_.</a> <font color="Maroon" title="c1">F</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:25_1_1_1_1"/>
<font color="Maroon" title="c2">p</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="polynom3.html#K8" title="POLYNOM3:func.8">0_.</a> <font color="Maroon" title="c1">F</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"> <font color="Maroon" title="c2">p</font> is  <a href="ring_2.html#NV6" title="RING_2:NV.6">reducible</a> </span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:25_1_1_1_1"/>
<font color="Maroon" title="c2">p</font> is  <a href="ring_2.html#NV6" title="RING_2:NV.6">reducible</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"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:25_1_1_1_2"/><span class="lab"><font color="Green" title="E18">L</font></span>: 
<font color="Maroon" title="c2">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom3.html#K8" title="POLYNOM3:func.8">0_.</a> <font color="Maroon" title="c1">F</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"> <font color="Maroon" title="c2">p</font> is  <a href="ring_2.html#NV6" title="RING_2:NV.6">reducible</a> </span><br/><div class="add"><a NAME="E2:25_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E19">M</font></span>: 
<font color="Maroon" title="c4">q1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom3.html#K8" title="POLYNOM3:func.8">0_.</a> <font color="Maroon" title="c1">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:25_1_1"><span class="lab"><font color="Green" title="E16">B</font></span></a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">D</font></span>:  <a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</a> <font color="Maroon" title="c2">p</font> = 
<span class="p1">(<span class="default"><a href="field_1.html#K1" title="FIELD_1:func.1">deg</a> <span class="p2">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> (1,<font color="Maroon" title="c3">a</font>)</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</a> <font color="Maroon" title="c4">q1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:25_1_1"><span class="lab"><font color="Green" title="E16">B</font></span></a>, <a class="ref" href="hurwitz.html#T23" onmouseover="rs('hurwitz/T23')" onmouseout="rh()">HURWITZ:23</a></span>
<br/>.= 
1 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</a> <font color="Maroon" title="c4">q1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="hurwitz.html#T27" onmouseover="rs('hurwitz/T27')" onmouseout="rh()">HURWITZ:27</a></span>
;<br/><span class="kw">reconsider </span><font color="Maroon" title="c6">degp</font> =  <a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</a> <font color="Maroon" title="c2">p</font>, <font color="Maroon" title="c7">degq</font> =  <a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</a> <font color="Maroon" title="c5">q</font> as    <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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:25_1_1_1_2"><span class="lab"><font color="Green" title="E19">M</font></span></a>, <a class="txt" href="#E1:25_1_1_1_2"><span class="lab"><font color="Green" title="E18">L</font></span></a>, <a class="ref" href="field_1.html#T1" onmouseover="rs('field_1/T1')" onmouseout="rh()">FIELD_1:1</a></span>;<br/><a NAME="E5:25_1_1_1_2"/>
1 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c7">degq</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:25_1_1_1_2"><span class="lab"><font color="Green" title="E20">D</font></span></a>, <a class="txt" href="#E1:25_1_1"><span class="lab"><font color="Green" title="E14">AS</font></span></a>, <a class="ref" href="nat_1.html#T23" onmouseover="rs('nat_1/T23')" onmouseout="rh()">NAT_1:23</a></span>;<br/><a NAME="E6:25_1_1_1_2"/><span class="kw">then </span>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">degq</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/><span class="kw">hence </span><a NAME="E7:25_1_1_1_2"/>
<font color="Maroon" title="c2">p</font> is  <a href="ring_2.html#NV6" title="RING_2:NV.6">reducible</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:25_1_1"><span class="lab"><font color="Green" title="E17">C</font></span></a>, <a class="txt" href="#E3:25_1_1_1_2"><span class="lab"><font color="Green" title="E20">D</font></span></a>, <a class="ref" href="nat_1.html#T19" onmouseover="rs('nat_1/T19')" onmouseout="rh()">NAT_1:19</a>, <a class="ref" href="ring_4.html#T40" onmouseover="rs('ring_4/T40')" onmouseout="rh()">RING_4:40</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><span class="kw">end;</span></div>

</div>
