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<span class="kw">let </span><font color="Maroon" title="c1">n</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</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">i</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a>  st <font color="Olive" title="b1">i</font> is   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> holds <br/> <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p1">(<span class="default"><a href="uniroots.html#K4" title="UNIROOTS:func.4">unital_poly</a> (<a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Olive" title="b1">i</font>) is   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a></span><br/><span class="kw">let </span><font color="Maroon" title="c2">i</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</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">i</font> is   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> implies  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p1">(<span class="default"><a href="uniroots.html#K4" title="UNIROOTS:func.4">unital_poly</a> (<a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c2">i</font>) is   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> )</span><br/>



<span class="kw">assume </span><a NAME="E1:57"/><span class="lab"><font color="Green" title="E48">A1</font></span>: 
<font color="Maroon" title="c2">i</font> is   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p1">(<span class="default"><a href="uniroots.html#K4" title="UNIROOTS:func.4">unital_poly</a> (<a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c2">i</font>) is   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c3">j</font> = <font color="Maroon" title="c2">i</font> as   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:57"><span class="lab"><font color="Green" title="E48">A1</font></span></a></span>;<br/>
<a NAME="E3:57"/>
<font color="Maroon" title="c2">i</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:57"><span class="lab"><font color="Green" title="E48">A1</font></span></a>, <a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></span>;<br/>
<a NAME="E4:57"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">i</font> &amp;  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p1">(<span class="default"><a href="uniroots.html#K4" title="UNIROOTS:func.4">unital_poly</a> (<a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Olive" title="b1">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">j</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="uniroots.html#T44" target="_self" title="UNIROOTS:th.44">Th48</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:57"/>
 <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p1">(<span class="default"><a href="uniroots.html#K4" title="UNIROOTS:func.4">unital_poly</a> (<a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c2">i</font>) is   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>
 ; <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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