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<span class="kw">let </span><font color="Maroon" title="c1">F</font> be   <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Field</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">F</font><br/>  for <font color="Olive" title="b2">q</font> being   non  <a href="ratfunc1.html#NV1" title="RATFUNC1:NV.1">zero</a>  <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <font color="Maroon" title="c1">F</font>  holds  <a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="hurwitz.html#K6" title="HURWITZ:func.6">mod</a> <font color="Olive" title="b2">q</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="field_1.html#K1" title="FIELD_1:func.1">deg</a> <font color="Olive" title="b2">q</font></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">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">  for <font color="Olive" title="b1">q</font> being   non  <a href="ratfunc1.html#NV1" title="RATFUNC1:NV.1">zero</a>  <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <font color="Maroon" title="c1">F</font>  holds  <a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="hurwitz.html#K6" title="HURWITZ:func.6">mod</a> <font color="Olive" title="b1">q</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="field_1.html#K1" title="FIELD_1:func.1">deg</a> <font color="Olive" title="b1">q</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">q</font> be   non  <a href="ratfunc1.html#NV1" title="RATFUNC1:NV.1">zero</a>  <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <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">  <a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="hurwitz.html#K6" title="HURWITZ:func.6">mod</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="field_1.html#K1" title="FIELD_1:func.1">deg</a> <font color="Maroon" title="c3">q</font></span><br/>





<a NAME="E1:18"/>
<font color="Maroon" title="c3">q</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>
 
;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">t</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="E3:18"/><span class="lab"><font color="Green" title="E12">C</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"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="hurwitz.html#K5" title="HURWITZ:func.5">div</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="polynom3.html#K7" title="POLYNOM3:func.7">+</a> <font color="Maroon" title="c4">t</font> &amp;  <a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</a> <font color="Maroon" title="c4">t</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="field_1.html#K1" title="FIELD_1:func.1">deg</a> <font color="Maroon" title="c3">q</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="hurwitz.html#D5" onmouseover="rs('hurwitz/D5')" onmouseout="rh()">HURWITZ:def 5</a></span>;<br/>
<font color="Maroon" title="c2">p</font> <a href="hurwitz.html#K6" title="HURWITZ:func.6">mod</a> <font color="Maroon" title="c3">q</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="hurwitz.html#K5" title="HURWITZ:func.5">div</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="polynom3.html#K7" title="POLYNOM3:func.7">+</a> <font color="Maroon" title="c4">t</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="hurwitz.html#K5" title="HURWITZ:func.5">div</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <font color="Maroon" title="c3">q</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:18"><span class="lab"><font color="Green" title="E12">C</font></span></a>, <a class="ref" href="hurwitz.html#D6" onmouseover="rs('hurwitz/D6')" onmouseout="rh()">HURWITZ:def 6</a></span>
<br/>.= 
<font color="Maroon" title="c4">t</font> <a href="polynom3.html#K7" title="POLYNOM3:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="hurwitz.html#K5" title="HURWITZ:func.5">div</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="hurwitz.html#K5" title="HURWITZ:func.5">div</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <font color="Maroon" title="c3">q</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom3.html#T26" onmouseover="rs('polynom3/T26')" onmouseout="rh()">POLYNOM3:26</a></span>
<br/>.= 
<font color="Maroon" title="c4">t</font> <a href="polynom3.html#K7" title="POLYNOM3:func.7">+</a> <span class="p1">(<span class="default"><a href="polynom3.html#K8" title="POLYNOM3:func.8">0_.</a> <font color="Maroon" title="c1">F</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom3.html#T29" onmouseover="rs('polynom3/T29')" onmouseout="rh()">POLYNOM3:29</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E5:18"/>
 <a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="hurwitz.html#K6" title="HURWITZ:func.6">mod</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="field_1.html#K1" title="FIELD_1:func.1">deg</a> <font color="Maroon" title="c3">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:18"><span class="lab"><font color="Green" title="E12">C</font></span></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/>


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