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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">E</font> being   <a href="field_4.html#NM4" title="FIELD_4:NM.4">FieldExtension</a> of <font color="Maroon" title="c1">F</font><br/>  for <font color="Olive" title="b2">p</font> being   <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <font color="Maroon" title="c1">F</font>  st  <a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</a> <font color="Olive" title="b2">p</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 holds <br/> for <font color="Olive" title="b3">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b1">E</font>  ex <font color="Olive" title="b4">y</font>, <font color="Olive" title="b5">z</font> being  <font color="Maroon" title="c1">F</font> <a href="field_7.html#V1" title="FIELD_7:attr.1">-membered</a>  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b1">E</font> st  <a href="algnum_1.html#K1" title="ALGNUM_1:func.1">Ext_eval</a> (<font color="Olive" title="b2">p</font>,<font color="Olive" title="b3">a</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">y</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b5">z</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">E</font> be   <a href="field_4.html#NM4" title="FIELD_4:NM.4">FieldExtension</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">p</font> being   <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <font color="Maroon" title="c1">F</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="c2">E</font>  ex <font color="Olive" title="b3">y</font>, <font color="Olive" title="b4">z</font> being  <font color="Maroon" title="c1">F</font> <a href="field_7.html#V1" title="FIELD_7:attr.1">-membered</a>  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">E</font> st  <a href="algnum_1.html#K1" title="ALGNUM_1:func.1">Ext_eval</a> (<font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">a</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">y</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b4">z</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">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"> (  <a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</a> <font color="Maroon" title="c3">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="c2">E</font>  ex <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">z</font> being  <font color="Maroon" title="c1">F</font> <a href="field_7.html#V1" title="FIELD_7:attr.1">-membered</a>  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">E</font> st  <a href="algnum_1.html#K1" title="ALGNUM_1:func.1">Ext_eval</a> (<font color="Maroon" title="c3">p</font>,<font color="Olive" title="b1">a</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">y</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b3">z</font></span>)</span> )</span><br/>





<span class="kw">assume </span><a NAME="E1:35"/><span class="lab"><font color="Green" title="E14">A</font></span>: 
 <a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</a> <font color="Maroon" title="c3">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="c2">E</font>  ex <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">z</font> being  <font color="Maroon" title="c1">F</font> <a href="field_7.html#V1" title="FIELD_7:attr.1">-membered</a>  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">E</font> st  <a href="algnum_1.html#K1" title="ALGNUM_1:func.1">Ext_eval</a> (<font color="Maroon" title="c3">p</font>,<font color="Olive" title="b1">a</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">y</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b3">z</font></span>)</span></span><br/>

<span class="kw">let </span><font color="Maroon" title="c4">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">E</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  <font color="Maroon" title="c1">F</font> <a href="field_7.html#V1" title="FIELD_7:attr.1">-membered</a>  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">E</font> st  <a href="algnum_1.html#K1" title="ALGNUM_1:func.1">Ext_eval</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">a</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">y</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b2">z</font></span>)</span></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c5">q</font> = <font color="Maroon" title="c3">p</font> as   <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <font color="Maroon" title="c2">E</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="field_4.html#T8" onmouseover="rs('field_4/T8')" onmouseout="rh()">FIELD_4:8</a></span>;<br/>
<a NAME="E3:35"/><span class="lab"><font color="Green" title="E15">B</font></span>: 
( <font color="Maroon" title="c3">p</font> is    <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> &amp; <font color="Maroon" title="c5">q</font> is    <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="c2">E</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="E4:35"/><span class="kw">then </span><span class="lab"><font color="Green" title="E16">C</font></span>: 
 <a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</a> <font color="Maroon" title="c5">q</font> <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="c3">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="field_4.html#T20" onmouseover="rs('field_4/T20')" onmouseout="rh()">FIELD_4:20</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">yF</font>, <font color="Maroon" title="c7">zF</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="E6:35"/><span class="lab"><font color="Green" title="E17">D</font></span>: 
<font color="Maroon" title="c3">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="c6">yF</font>,<font color="Maroon" title="c7">zF</font></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35"><span class="lab"><font color="Green" title="E14">A</font></span></a>, <a class="ref" href="field_9.html#T13" target="_self" onmouseover="rs('field_9/T13')" onmouseout="rh()">deg2</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c8">y</font>, <font color="Maroon" title="c9">z</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">E</font><span class="kw"> such that </span><br/><a NAME="E8:35"/><span class="lab"><font color="Green" title="E18">E</font></span>: 
<font color="Maroon" title="c5">q</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="c8">y</font>,<font color="Maroon" title="c9">z</font></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35"><span class="lab"><font color="Green" title="E14">A</font></span></a>, <a class="txt" href="#E4:35"><span class="lab"><font color="Green" title="E16">C</font></span></a>, <a class="ref" href="field_9.html#T13" target="_self" onmouseover="rs('field_9/T13')" onmouseout="rh()">deg2</a></span>;<br/>
<font color="Maroon" title="c8">y</font> = 
<font color="Maroon" title="c5">q</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:35"><span class="lab"><font color="Green" title="E18">E</font></span></a>, <a class="ref" href="polynom5.html#T38" onmouseover="rs('polynom5/T38')" onmouseout="rh()">POLYNOM5:38</a></span>
<br/>.= 
<font color="Maroon" title="c6">yF</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:35"><span class="lab"><font color="Green" title="E17">D</font></span></a>, <a class="ref" href="polynom5.html#T38" onmouseover="rs('polynom5/T38')" onmouseout="rh()">POLYNOM5:38</a></span>
;<br/>
<a NAME="E10:35"/><span class="kw">then </span><span class="lab"><font color="Green" title="E19">G1</font></span>: 
<font color="Maroon" title="c8">y</font> is <font color="Maroon" title="c1">F</font> <a href="field_7.html#V1" title="FIELD_7:attr.1">-membered</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="field_7.html#D5" onmouseover="rs('field_7/D5')" onmouseout="rh()">FIELD_7:def 5</a></span>;<br/>
<font color="Maroon" title="c9">z</font> = 
<font color="Maroon" title="c5">q</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:35"><span class="lab"><font color="Green" title="E18">E</font></span></a>, <a class="ref" href="polynom5.html#T38" onmouseover="rs('polynom5/T38')" onmouseout="rh()">POLYNOM5:38</a></span>
<br/>.= 
<font color="Maroon" title="c7">zF</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:35"><span class="lab"><font color="Green" title="E17">D</font></span></a>, <a class="ref" href="polynom5.html#T38" onmouseover="rs('polynom5/T38')" onmouseout="rh()">POLYNOM5:38</a></span>
;<br/>
<a NAME="E12:35"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">G2</font></span>: 
<font color="Maroon" title="c9">z</font> is <font color="Maroon" title="c1">F</font> <a href="field_7.html#V1" title="FIELD_7:attr.1">-membered</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="field_7.html#D5" onmouseover="rs('field_7/D5')" onmouseout="rh()">FIELD_7:def 5</a></span>;<br/>
 <a href="algnum_1.html#K1" title="ALGNUM_1:func.1">Ext_eval</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">a</font>) = 
 <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c4">a</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35"><span class="lab"><font color="Green" title="E15">B</font></span></a>, <a class="ref" href="field_4.html#T26" onmouseover="rs('field_4/T26')" onmouseout="rh()">FIELD_4:26</a></span>
<br/>.= 
<font color="Maroon" title="c8">y</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">a</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:35"><span class="lab"><font color="Green" title="E18">E</font></span></a>, <a class="ref" href="polynom5.html#T44" onmouseover="rs('polynom5/T44')" onmouseout="rh()">POLYNOM5:44</a></span>
<br/>.= 
<font color="Maroon" title="c8">y</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c9">z</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D12" onmouseover="rs('group_1/D12')" onmouseout="rh()">GROUP_1:def 12</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E14:35"/>
 ex <font color="Olive" title="b1">y</font>, <font color="Olive" title="b2">z</font> being  <font color="Maroon" title="c1">F</font> <a href="field_7.html#V1" title="FIELD_7:attr.1">-membered</a>  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">E</font> st  <a href="algnum_1.html#K1" title="ALGNUM_1:func.1">Ext_eval</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">a</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">y</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b2">z</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:35"><span class="lab"><font color="Green" title="E19">G1</font></span></a>, <a class="txt" href="#E12:35"><span class="lab"><font color="Green" title="E20">G2</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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