<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="vectsp_1.html#V8">left_unital</a> non <a href="vectsp_1.html#V10">degenerated</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="nat_1.html#NM1">Nat</a>;<br/>



<a NAME="E1:50"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K23">+</a> 1 holds <br/><span class="p1">(<span class="default"><a href="uniroots.html#K4" target="_self">unital_poly</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:50_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K23">+</a> 1
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:50_1_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><a NAME="E2:50_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K23">+</a> 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K23">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><b>hence </b><a NAME="E3:50_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="uniroots.html#E1:50_1"><i><font color="Green">E44</font></i></a>, <a class="txt" href="uniroots.html#E1:50_1_1"><i><font color="Green">E45</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E3:50_1"/><b>then </b>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 
<b>by </b><i><a class="txt" href="uniroots.html#E1:50_1"><i><font color="Green">E44</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:50_1"/>
<span class="p1">(<span class="default"><a href="uniroots.html#K4" target="_self">unital_poly</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="uniroots.html#T43" target="_self">Th43</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:50"/><b>then </b><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K23">+</a> 1 <a href="algseq_1.html#R1">is_at_least_length_of</a>  <a href="uniroots.html#K4" target="_self">unital_poly</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="algseq_1.html#D3">ALGSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:50"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="algseq_1.html#R1">is_at_least_length_of</a>  <a href="uniroots.html#K4" target="_self">unital_poly</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:50_2"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="algseq_1.html#R1">is_at_least_length_of</a>  <a href="uniroots.html#K4" target="_self">unital_poly</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:50_2_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K23">+</a> 1
 ;<br/><a NAME="E2:50_2_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:50_2_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="uniroots.html#K4" target="_self">unital_poly</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font> &amp; <span class="p1">(<span class="default"><a href="uniroots.html#K4" target="_self">unital_poly</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="uniroots.html#E1:50_2"><i><font color="Green">E45</font></i></a>, <a class="ref" href="uniroots.html#T42" target="_self">Th42</a>, <a class="ref" href="algseq_1.html#D3">ALGSEQ_1:def 3</a></i>;<br/><b>hence </b><a NAME="E4:50_2_1"/>
contradiction
 <b>by </b><i><a class="ref" href="vectsp_1.html#D21">VECTSP_1:def 21</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:50_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:50"/>
 <a href="algseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="uniroots.html#K4" target="_self">unital_poly</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K23">+</a> 1
 <b>by </b><i><a class="txt" href="uniroots.html#E2:50"><i><font color="Green">E44</font></i></a>, <a class="ref" href="algseq_1.html#D4">ALGSEQ_1:def 4</a></i>;<br/>


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