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<span class="kw">let </span><font color="Maroon" title="c1">R</font> be   non  <a href="struct_0.html#V6" title="STRUCT_0:attr.6">degenerated</a>  <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Ring</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="ring_4.html#NV4" title="RING_4:NV.4">monic</a>  <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <font color="Maroon" title="c1">R</font> holds <br/> ( <font color="Olive" title="b1">p</font> is  <a href="field_9.html#V1" title="FIELD_9:attr.1">quadratic</a>  iff  ex <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> st <font color="Olive" title="b1">p</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1"><a href="field_9.html#K4" title="FIELD_9:func.4">&lt;%</a><span class="default"><font color="Olive" title="b3">c</font>,<font color="Olive" title="b2">b</font>,<span class="p2">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">R</font></span>)</span></span><a href="field_9.html#K4" title="FIELD_9:func.4">%&gt;</a></span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">p</font> be   <a href="ring_4.html#NV4" title="RING_4:NV.4">monic</a>  <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <font color="Maroon" title="c1">R</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="field_9.html#V1" title="FIELD_9:attr.1">quadratic</a>  iff  ex <font color="Olive" title="b1">b</font>, <font color="Olive" title="b2">c</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> st <font color="Maroon" title="c2">p</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1"><a href="field_9.html#K4" title="FIELD_9:func.4">&lt;%</a><span class="default"><font color="Olive" title="b2">c</font>,<font color="Olive" title="b1">b</font>,<span class="p2">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">R</font></span>)</span></span><a href="field_9.html#K4" title="FIELD_9:func.4">%&gt;</a></span> )</span><br/>



<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="54_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"> ( <font color="Maroon" title="c2">p</font> is  <a href="field_9.html#V1" title="FIELD_9:attr.1">quadratic</a>  implies  ex <font color="Olive" title="b1">b</font>, <font color="Olive" title="b2">c</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> st <font color="Maroon" title="c2">p</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1"><a href="field_9.html#K4" title="FIELD_9:func.4">&lt;%</a><span class="default"><font color="Olive" title="b2">c</font>,<font color="Olive" title="b1">b</font>,<span class="p2">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">R</font></span>)</span></span><a href="field_9.html#K4" title="FIELD_9:func.4">%&gt;</a></span> )</span></a><div class="add"><span class="kw">assume </span><a NAME="E1:54_1"/>
<font color="Maroon" title="c2">p</font> is  <a href="field_9.html#V1" title="FIELD_9:attr.1">quadratic</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">  ex <font color="Olive" title="b1">b</font>, <font color="Olive" title="b2">c</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> st <font color="Maroon" title="c2">p</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1"><a href="field_9.html#K4" title="FIELD_9:func.4">&lt;%</a><span class="default"><font color="Olive" title="b2">c</font>,<font color="Olive" title="b1">b</font>,<span class="p2">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">R</font></span>)</span></span><a href="field_9.html#K4" title="FIELD_9:func.4">%&gt;</a></span></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">a</font> being   non  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a>  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font>, <font color="Maroon" title="c4">b</font>, <font color="Maroon" title="c5">c</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font><span class="kw"> such that </span><br/><a NAME="E3:54_1"/><span class="lab"><font color="Green" title="E23">A</font></span>: 
<font color="Maroon" title="c2">p</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1"><a href="field_9.html#K4" title="FIELD_9:func.4">&lt;%</a><span class="default"><font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c3">a</font></span><a href="field_9.html#K4" title="FIELD_9:func.4">%&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="field_9.html#T21" target="_self" onmouseover="rs('field_9/T21')" onmouseout="rh()">qua5</a></span>;<br/><a NAME="E4:54_1"/><span class="lab"><font color="Green" title="E24">H</font></span>: 
( 3 <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 &amp; <font color="Maroon" title="c3">a</font> <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">R</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D2" onmouseover="rs('xreal_0/D2')" onmouseout="rh()">XREAL_0:def 2</a></span>;<br/> <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">R</font> = 
 <a href="ratfunc1.html#NK3" title="RATFUNC1:NK.3">LC</a> <font color="Maroon" title="c2">p</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ratfunc1.html#D7" onmouseover="rs('ratfunc1/D7')" onmouseout="rh()">RATFUNC1:def 7</a></span>
<br/>.= 
<font color="Maroon" title="c2">p</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ratfunc1.html#D6" onmouseover="rs('ratfunc1/D6')" onmouseout="rh()">RATFUNC1:def 6</a></span>
<br/>.= 
<font color="Maroon" title="c2">p</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> 2
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:54_1"><span class="lab"><font color="Green" title="E24">H</font></span></a>, <a class="txt" href="#E3:54_1"><span class="lab"><font color="Green" title="E23">A</font></span></a>, <a class="txt" href="field_9.html#E26"><span class="lab"><font color="Green" title="E17">qua3</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c3">a</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:54_1"><span class="lab"><font color="Green" title="E23">A</font></span></a>, <a class="txt" href="field_9.html#E22"><span class="lab"><font color="Green" title="E15">qua1</font></span></a></span>
;<br/><span class="kw">hence </span><a NAME="E6:54_1"/>
 ex <font color="Olive" title="b1">b</font>, <font color="Olive" title="b2">c</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> st <font color="Maroon" title="c2">p</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1"><a href="field_9.html#K4" title="FIELD_9:func.4">&lt;%</a><span class="default"><font color="Olive" title="b2">c</font>,<font color="Olive" title="b1">b</font>,<span class="p2">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">R</font></span>)</span></span><a href="field_9.html#K4" title="FIELD_9:func.4">%&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:54_1"><span class="lab"><font color="Green" title="E23">A</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="E2:54"/>
( <font color="Maroon" title="c2">p</font> is  <a href="field_9.html#V1" title="FIELD_9:attr.1">quadratic</a>  iff  ex <font color="Olive" title="b1">b</font>, <font color="Olive" title="b2">c</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> st <font color="Maroon" title="c2">p</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1"><a href="field_9.html#K4" title="FIELD_9:func.4">&lt;%</a><span class="default"><font color="Olive" title="b2">c</font>,<font color="Olive" title="b1">b</font>,<span class="p2">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">R</font></span>)</span></span><a href="field_9.html#K4" title="FIELD_9:func.4">%&gt;</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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