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<a NAME="E1:121_1_1"/><span class="lab"><font color="Green" title="E54">A</font></span>: 
<span class="p1"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <font color="Maroon" title="c2">a</font></span>)</span></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<span class="p2"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><span class="p3">(<span class="default"><a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <font color="Maroon" title="c2">a</font></span>)</span></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="field_6.html#T35" onmouseover="rs('field_6/T35')" onmouseout="rh()">FIELD_6:35</a></span>;<br/>
<a NAME="E2:121_1_1"/>
 <a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <font color="Maroon" title="c2">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <font color="Maroon" title="c2">a</font></span>)</span></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:121_1_1"/>
 <a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <font color="Maroon" title="c2">a</font> is  <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<span class="p1"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <font color="Maroon" title="c2">a</font></span>)</span></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>) <a href="field_7.html#V1" title="FIELD_7:attr.1">-membered</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:121_1_1"><span class="lab"><font color="Green" title="E54">A</font></span></a>, <a class="ref" href="field_7.html#D5" onmouseover="rs('field_7/D5')" onmouseout="rh()">FIELD_7:def 5</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">  not  <a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <font color="Maroon" title="c2">a</font> is <font color="Maroon" title="c1">F</font> <a href="field_7.html#V1" title="FIELD_7:attr.1">-membered</a> </span><br/>

<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="121_1_1_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">  not  <a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <font color="Maroon" title="c2">a</font> is <font color="Maroon" title="c1">F</font> <a href="field_7.html#V1" title="FIELD_7:attr.1">-membered</a> </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:121_1_1_1"/><span class="lab"><font color="Green" title="E55">B</font></span>: 
 <a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <font color="Maroon" title="c2">a</font> is <font color="Maroon" title="c1">F</font> <a href="field_7.html#V1" title="FIELD_7:attr.1">-membered</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"> contradiction</span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c3">sqrta</font> =  <a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <font color="Maroon" title="c2">a</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">F</font> <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 NAME="E3:121_1_1_1"/><span class="lab"><font color="Green" title="E56">C</font></span>: 
 <a href="field_1.html#K1" title="FIELD_1:func.1">deg</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> (1,<font color="Maroon" title="c3">sqrta</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <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/><a NAME="E4:121_1_1_1"/><span class="lab"><font color="Green" title="E57">D</font></span>: 
 <a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> (1,<font color="Maroon" title="c3">sqrta</font>) <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> (1,<span class="p1">(<span class="default"><a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <font color="Maroon" title="c2">a</font></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="field_4.html#T21" onmouseover="rs('field_4/T21')" onmouseout="rh()">FIELD_4:21</a></span>;<br/><a NAME="E5:121_1_1_1"/>
 <a href="field_6.html#NK9" title="FIELD_6:NK.9">MinPoly</a> (<span class="p1">(<span class="default"><a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <font color="Maroon" title="c2">a</font></span>)</span>,<font color="Maroon" title="c1">F</font>) <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> (1,<span class="p1">(<span class="default"><a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <font color="Maroon" title="c2">a</font></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:121_1_1_1"><span class="lab"><font color="Green" title="E55">B</font></span></a>, <a class="ref" href="field_7.html#D5" onmouseover="rs('field_7/D5')" onmouseout="rh()">FIELD_7:def 5</a>, <a class="ref" href="field_6.html#T54" onmouseover="rs('field_6/T54')" onmouseout="rh()">FIELD_6:54</a></span>;<br/><a NAME="E6:121_1_1_1"/><span class="kw">then </span>
 <a href="field_4.html#K3" title="FIELD_4:func.3">deg</a> (<span class="p1">(<span class="default"><a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<span class="p2"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><span class="p3">(<span class="default"><a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <font color="Maroon" title="c2">a</font></span>)</span></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span>)</span>,<font color="Maroon" title="c1">F</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="field_1.html#K1" title="FIELD_1:func.1">deg</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> (1,<font color="Maroon" title="c3">sqrta</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:121_1_1_1"><span class="lab"><font color="Green" title="E57">D</font></span></a>, <a class="ref" href="field_6.html#T67" onmouseover="rs('field_6/T67')" onmouseout="rh()">FIELD_6:67</a></span>;<br/><span class="kw">hence </span><a NAME="E7:121_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:121_1_1_1"><span class="lab"><font color="Green" title="E56">C</font></span></a>, <a class="txt" href="field_9.html#E84"><span class="lab"><font color="Green" title="E53">dega</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><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E5:121_1_1"/>
 not  <a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <font color="Maroon" title="c2">a</font> is <font color="Maroon" title="c1">F</font> <a href="field_7.html#V1" title="FIELD_7:attr.1">-membered</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/>


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