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<span class="kw">let </span><font color="Maroon" title="c1">n0</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">zero</a>   <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">r</font> being   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/><font color="Maroon" title="c1">n0</font> <a href="polyeq_5.html#K1" title="POLYEQ_5:func.1">-root</a> <font color="Olive" title="b1">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n0</font> <a href="polyeq_5.html#NK1" title="POLYEQ_5:NK.1">-real-root</a> <font color="Olive" title="b1">r</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">r</font> be   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">r</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Maroon" title="c1">n0</font> <a href="polyeq_5.html#K1" title="POLYEQ_5:func.1">-root</a> <font color="Maroon" title="c2">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n0</font> <a href="polyeq_5.html#NK1" title="POLYEQ_5:NK.1">-real-root</a> <font color="Maroon" title="c2">r</font> )</span><br/>



<span class="kw">reconsider </span><font color="Maroon" title="c3">r9</font> = <font color="Maroon" title="c2">r</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:10"/><span class="lab"><font color="Green" title="E9">A1</font></span>: 
<font color="Maroon" title="c2">r</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">n0</font> <a href="polyeq_5.html#K1" title="POLYEQ_5:func.1">-root</a> <font color="Maroon" title="c2">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n0</font> <a href="polyeq_5.html#NK1" title="POLYEQ_5:NK.1">-real-root</a> <font color="Maroon" title="c2">r</font></span><br/>

<a NAME="E3:10"/><span class="kw">then </span>
 <a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Maroon" title="c3">r9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="comptrig.html#T35" title="COMPTRIG:th.35">COMPTRIG:35</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:10"/>
<font color="Maroon" title="c1">n0</font> <a href="polyeq_5.html#K1" title="POLYEQ_5:func.1">-root</a> <font color="Maroon" title="c2">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n0</font> <a href="polyeq_5.html#NK1" title="POLYEQ_5:NK.1">-real-root</a> <font color="Maroon" title="c2">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:10"><span class="lab"><font color="Green" title="E9">A1</font></span></a>, <a class="ref" href="complex1.html#T43" title="COMPLEX1:th.43">COMPLEX1:43</a>, <a class="ref" href="sin_cos.html#T31" title="SIN_COS:th.31">SIN_COS:31</a></span>; <a class="txt" 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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