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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"> ( <font color="Maroon" title="c1">R</font> is  <a href="realalg1.html#V13" title="REALALG1:attr.13">preordered</a>  implies  not <font color="Maroon" title="c1">R</font> is  <a href="field_9.html#V3" title="FIELD_9:attr.3">quadratic_complete</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:107_2_1"/>
<font color="Maroon" title="c1">R</font> is  <a href="realalg1.html#V13" title="REALALG1:attr.13">preordered</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">  not <font color="Maroon" title="c1">R</font> is  <a href="field_9.html#V3" title="FIELD_9:attr.3">quadratic_complete</a> </span><br/>

<a NAME="E2:107_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A</font></span>: 
 not  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</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">R</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="realalg1.html#NK5" title="REALALG1:NK.5">QS</a> <font color="Maroon" title="c1">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="realalg2.html#D3" onmouseover="rs('realalg2/D3')" onmouseout="rh()">REALALG2:def 3</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="107_2_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 <font color="Maroon" title="c1">R</font> is  <a href="field_9.html#V3" title="FIELD_9:attr.3">quadratic_complete</a> </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:107_2_1_1"/>
<font color="Maroon" title="c1">R</font> is  <a href="field_9.html#V3" title="FIELD_9:attr.3">quadratic_complete</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/><a NAME="E2:107_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">B</font></span>: 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="realalg1.html#NK4" title="REALALG1:NK.4">SQ</a> <font color="Maroon" title="c1">R</font>
 ;<br/><a NAME="E3:107_2_1_1"/>
 <a href="realalg1.html#NK4" title="REALALG1:NK.4">SQ</a> <font color="Maroon" title="c1">R</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="realalg1.html#NK5" title="REALALG1:NK.5">QS</a> <font color="Maroon" title="c1">R</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="realalg1.html#T18" onmouseover="rs('realalg1/T18')" onmouseout="rh()">REALALG1:18</a></span>;<br/><span class="kw">hence </span><a NAME="E4:107_2_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:107_2_1"><span class="lab"><font color="Green" title="E52">A</font></span></a>, <a class="txt" href="#E2:107_2_1_1"><span class="lab"><font color="Green" title="E53">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><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:107_2_1"/>
 not <font color="Maroon" title="c1">R</font> is  <a href="field_9.html#V3" title="FIELD_9:attr.3">quadratic_complete</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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