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<span class="kw">let </span><font color="Maroon" title="c1">F</font> be   non 2 <a href="ring_3.html#V3" title="RING_3:attr.3">-characteristic</a>  <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">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">F</font> holds <br/> (  <a href="field_9.html#K5" title="FIELD_9:func.5">X^2-</a> <font color="Olive" title="b1">a</font> is  <a href="ring_2.html#NV6" title="RING_2:NV.6">reducible</a>  iff <font color="Olive" title="b1">a</font> is  <a href="realalg1.html#NV8" title="REALALG1:NV.8">square</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</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="field_9.html#K5" title="FIELD_9:func.5">X^2-</a> <font color="Maroon" title="c2">a</font> is  <a href="ring_2.html#NV6" title="RING_2:NV.6">reducible</a>  iff <font color="Maroon" title="c2">a</font> is  <a href="realalg1.html#NV8" title="REALALG1:NV.8">square</a>  )</span><br/>



<a NAME="E1:78"/><span class="lab"><font color="Green" title="E38">H</font></span>: 
( 2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">F</font> &amp; 4 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">F</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="field_9.html#T5" target="_self" onmouseover="rs('field_9/T5')" onmouseout="rh()">ch2</a>, <a class="ref" href="field_9.html#T6" target="_self" onmouseover="rs('field_9/T6')" onmouseout="rh()">ch4</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E39">B</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="78_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"> (  <a href="field_9.html#K5" title="FIELD_9:func.5">X^2-</a> <font color="Maroon" title="c2">a</font> is  <a href="ring_2.html#NV6" title="RING_2:NV.6">reducible</a>  implies <font color="Maroon" title="c2">a</font> is  <a href="realalg1.html#NV8" title="REALALG1:NV.8">square</a>  )</span></a><div class="add"><span class="kw">assume </span><a NAME="E1:78_1"/>
 <a href="field_9.html#K5" title="FIELD_9:func.5">X^2-</a> <font color="Maroon" title="c2">a</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"> <font color="Maroon" title="c2">a</font> is  <a href="realalg1.html#NV8" title="REALALG1:NV.8">square</a> </span><br/><a NAME="E2:78_1"/><span class="kw">then </span>
 <a href="field_9.html#NK8" title="FIELD_9:NK.8">DC</a> <span class="p1"><a href="field_9.html#K4" title="FIELD_9:func.4">&lt;%</a><span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c2">a</font></span>)</span>,<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">F</font></span>)</span>,<span class="p2">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">F</font></span>)</span></span><a href="field_9.html#K4" title="FIELD_9:func.4">%&gt;</a></span> is  <a href="realalg1.html#NV8" title="REALALG1:NV.8">square</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="field_9.html#T32" target="_self" onmouseover="rs('field_9/T32')" onmouseout="rh()">naH</a></span>;<br/><a NAME="E3:78_1"/><span class="kw">then </span>
 <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default">4 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span> is  <a href="realalg1.html#NV8" title="REALALG1:NV.8">square</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="field_9.html#D11" target="_self" onmouseover="rs('field_9/D11')" onmouseout="rh()">defDCpq</a></span>;<br/><a NAME="E4:78_1"/><span class="kw">then </span>
4 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span> is  <a href="realalg1.html#NV8" title="REALALG1:NV.8">square</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="realalg2.html#T6" onmouseover="rs('realalg2/T6')" onmouseout="rh()">REALALG2:6</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">w</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="E6:78_1"/><span class="lab"><font color="Green" title="E40">C</font></span>: 
<font color="Maroon" title="c3">w</font> <a href="o_ring_1.html#K1" title="O_RING_1:func.1">^2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <font color="Maroon" title="c2">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="o_ring_1.html#D2" onmouseover="rs('o_ring_1/D2')" onmouseout="rh()">O_RING_1:def 2</a></span>;<br/><font color="Maroon" title="c3">w</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">w</font> = 
4 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c2">a</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:78_1"><span class="lab"><font color="Green" title="E40">C</font></span></a>, <a class="ref" href="o_ring_1.html#D1" onmouseover="rs('o_ring_1/D1')" onmouseout="rh()">O_RING_1:def 1</a></span>
<br/>.= 
<span class="p1">(<span class="default">4 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p2">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">F</font></span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c2">a</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="realalg2.html#T5" onmouseover="rs('realalg2/T5')" onmouseout="rh()">REALALG2:5</a></span>
<br/>.= 
<font color="Maroon" title="c2">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default">4 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p2">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">F</font></span>)</span></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">then </span><span class="p1">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">w</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">4 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p3">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">F</font></span>)</span></span>)</span> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span> = 
<font color="Maroon" title="c2">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">4 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p3">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">F</font></span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">4 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p4">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">F</font></span>)</span></span>)</span> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D3" onmouseover="rs('group_1/D3')" onmouseout="rh()">GROUP_1:def 3</a></span>
<br/>.= 
<font color="Maroon" title="c2">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">4 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p4">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">F</font></span>)</span></span>)</span> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default">4 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p3">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">F</font></span>)</span></span>)</span></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/>.= 
<font color="Maroon" title="c2">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">F</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:78"><span class="lab"><font color="Green" title="E38">H</font></span></a>, <a class="ref" href="vectsp_1.html#D10" onmouseover="rs('vectsp_1/D10')" onmouseout="rh()">VECTSP_1:def 10</a></span>
;<br/><span class="kw">then </span><font color="Maroon" title="c2">a</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">w</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 2</span>)</span> <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p3">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">F</font></span>)</span></span>)</span> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">w</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p4">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">F</font></span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p3">(<span class="default">2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p4">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">F</font></span>)</span></span>)</span></span>)</span> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ring_3.html#T67" onmouseover="rs('ring_3/T67')" onmouseout="rh()">RING_3:67</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">w</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p4">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">F</font></span>)</span></span>)</span> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p4">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">F</font></span>)</span></span>)</span> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:78"><span class="lab"><font color="Green" title="E38">H</font></span></a>, <a class="ref" href="vectsp_2.html#T11" onmouseover="rs('vectsp_2/T11')" onmouseout="rh()">VECTSP_2:11</a></span>
<br/>.= 
<font color="Maroon" title="c3">w</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p5">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">F</font></span>)</span></span>)</span> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p5">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">F</font></span>)</span></span>)</span> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D3" onmouseover="rs('group_1/D3')" onmouseout="rh()">GROUP_1:def 3</a></span>
<br/>.= 
<font color="Maroon" title="c3">w</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p5">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">F</font></span>)</span></span>)</span> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p4">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">F</font></span>)</span></span>)</span> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D3" onmouseover="rs('group_1/D3')" onmouseout="rh()">GROUP_1:def 3</a></span>
<br/>.= 
<font color="Maroon" title="c3">w</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p4">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">F</font></span>)</span></span>)</span> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p5">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">F</font></span>)</span></span>)</span> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span></span>)</span></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="p1">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p4">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">F</font></span>)</span></span>)</span> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p4">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">F</font></span>)</span></span>)</span> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D3" onmouseover="rs('group_1/D3')" onmouseout="rh()">GROUP_1:def 3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p4">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">F</font></span>)</span></span>)</span> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span></span>)</span> <a href="o_ring_1.html#K1" title="O_RING_1:func.1">^2</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="o_ring_1.html#D1" onmouseover="rs('o_ring_1/D1')" onmouseout="rh()">O_RING_1:def 1</a></span>
;<br/><span class="kw">hence </span><a NAME="E10:78_1"/>
<font color="Maroon" title="c2">a</font> is  <a href="realalg1.html#NV8" title="REALALG1:NV.8">square</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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="78_2"><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"> ( <font color="Maroon" title="c2">a</font> is  <a href="realalg1.html#NV8" title="REALALG1:NV.8">square</a>  implies  <a href="field_9.html#K5" title="FIELD_9:func.5">X^2-</a> <font color="Maroon" title="c2">a</font> is  <a href="ring_2.html#NV6" title="RING_2:NV.6">reducible</a>  )</span></a><div class="add"><span class="kw">assume </span><a NAME="E1:78_2"/>
<font color="Maroon" title="c2">a</font> is  <a href="realalg1.html#NV8" title="REALALG1:NV.8">square</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">  <a href="field_9.html#K5" title="FIELD_9:func.5">X^2-</a> <font color="Maroon" title="c2">a</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">w</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="E3:78_2"/><span class="lab"><font color="Green" title="E40">C</font></span>: 
<font color="Maroon" title="c3">w</font> <a href="o_ring_1.html#K1" title="O_RING_1:func.1">^2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="o_ring_1.html#D2" onmouseover="rs('o_ring_1/D2')" onmouseout="rh()">O_RING_1:def 2</a></span>;<br/><span class="p1">(<span class="default">2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <font color="Maroon" title="c3">w</font></span>)</span> <a href="o_ring_1.html#K1" title="O_RING_1:func.1">^2</a>  = 
<span class="p1">(<span class="default">2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <font color="Maroon" title="c3">w</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default">2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <font color="Maroon" title="c3">w</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="o_ring_1.html#D1" onmouseover="rs('o_ring_1/D1')" onmouseout="rh()">O_RING_1:def 1</a></span>
<br/>.= 
2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default">2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <font color="Maroon" title="c3">w</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="realalg2.html#T5" onmouseover="rs('realalg2/T5')" onmouseout="rh()">REALALG2:5</a></span>
<br/>.= 
2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <font color="Maroon" title="c3">w</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">w</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/>.= 
2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p1">(<span class="default">2 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">w</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="realalg2.html#T5" onmouseover="rs('realalg2/T5')" onmouseout="rh()">REALALG2:5</a></span>
<br/>.= 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 2</span>)</span> <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">w</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">w</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ring_3.html#T65" onmouseover="rs('ring_3/T65')" onmouseout="rh()">RING_3:65</a></span>
<br/>.= 
4 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <font color="Maroon" title="c2">a</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:78_2"><span class="lab"><font color="Green" title="E40">C</font></span></a>, <a class="ref" href="o_ring_1.html#D1" onmouseover="rs('o_ring_1/D1')" onmouseout="rh()">O_RING_1:def 1</a></span>
;<br/><a NAME="E5:78_2"/><span class="kw">then </span>
4 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span> is  <a href="realalg1.html#NV8" title="REALALG1:NV.8">square</a> 
 ;<br/><a NAME="E6:78_2"/><span class="kw">then </span>
 <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default">4 <a href="ring_3.html#K1" title="RING_3:func.1">'*'</a> <span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span> is  <a href="realalg1.html#NV8" title="REALALG1:NV.8">square</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="realalg2.html#T6" onmouseover="rs('realalg2/T6')" onmouseout="rh()">REALALG2:6</a></span>;<br/><a NAME="E7:78_2"/><span class="kw">then </span>
 <a href="field_9.html#NK8" title="FIELD_9:NK.8">DC</a> <span class="p1"><a href="field_9.html#K4" title="FIELD_9:func.4">&lt;%</a><span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c2">a</font></span>)</span>,<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">F</font></span>)</span>,<span class="p2">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">F</font></span>)</span></span><a href="field_9.html#K4" title="FIELD_9:func.4">%&gt;</a></span> is  <a href="realalg1.html#NV8" title="REALALG1:NV.8">square</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="field_9.html#D11" target="_self" onmouseover="rs('field_9/D11')" onmouseout="rh()">defDCpq</a></span>;<br/><span class="kw">hence </span><a NAME="E8:78_2"/>
 <a href="field_9.html#K5" title="FIELD_9:func.5">X^2-</a> <font color="Maroon" title="c2">a</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="ref" href="field_9.html#T32" target="_self" onmouseover="rs('field_9/T32')" onmouseout="rh()">naH</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>
<span class="kw">hence </span><a NAME="E4:78"/>
(  <a href="field_9.html#K5" title="FIELD_9:func.5">X^2-</a> <font color="Maroon" title="c2">a</font> is  <a href="ring_2.html#NV6" title="RING_2:NV.6">reducible</a>  iff <font color="Maroon" title="c2">a</font> is  <a href="realalg1.html#NV8" title="REALALG1:NV.8">square</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:78"><span class="lab"><font color="Green" title="E39">B</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/>


</div>
