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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a>  <a href="struct_0.html#L2">ZeroStr</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<a NAME="E1:46"/>
( not the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> is <a href="xboole_0.html#V1">empty</a> &amp; 0 <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  )
 
;<br/>
<a NAME="E2:46"/><b>then </b><i><font color="Green">E35</font></i>: 
0 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="polynom3.html#K12">0_.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E3:46"/>
( not the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> is <a href="xboole_0.html#V1">empty</a> &amp; 1 <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  )
 
;<br/>
<a NAME="E4:46"/><b>then </b><i><font color="Green">E36</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom3.html#K12">0_.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="polynom1.html#K1">+*</a> 0,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>thus </b><span class="p1"><a href="polynom5.html#K4" target="_self">&lt;%</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span> <a href="normsp_1.html#K2">.</a> 0 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom3.html#K12">0_.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="polynom1.html#K1">+*</a> 0,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="polynom1.html#K1">+*</a> 1,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="normsp_1.html#K2">.</a> 0

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom3.html#K12">0_.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="polynom1.html#K1">+*</a> 0,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="normsp_1.html#K2">.</a> 0
<b>by </b><i><a class="ref" href="funct_7.html#T34">FUNCT_7:34</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="polynom5.html#E2:46"><i><font color="Green">E35</font></i></a>, <a class="ref" href="funct_7.html#T33">FUNCT_7:33</a></i>
;<br/>

<b>thus </b><span class="p1"><a href="polynom5.html#K4" target="_self">&lt;%</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span> <a href="normsp_1.html#K2">.</a> 1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom3.html#K12">0_.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="polynom1.html#K1">+*</a> 0,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="polynom1.html#K1">+*</a> 1,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="normsp_1.html#K2">.</a> 1

<br/>.= 
<font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="polynom5.html#E4:46"><i><font color="Green">E36</font></i></a>, <a class="ref" href="funct_7.html#T33">FUNCT_7:33</a></i>
;<br/>

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E7:46"/>
<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2
 ;<br/>

<a NAME="E8:46"/><b>then </b><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="nat_1.html#K1">+</a> 1
 
;<br/>
<a NAME="E9:46"/><b>then </b><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E10:46"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="polynom5.html#E8:46"><i><font color="Green">E37</font></i></a></i>;<br/>
<b>thus </b><span class="p1"><a href="polynom5.html#K4" target="_self">&lt;%</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="polynom5.html#K4" target="_self">%&gt;</a></span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom3.html#K12">0_.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="polynom1.html#K1">+*</a> 0,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="polynom1.html#K1">+*</a> 1,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom3.html#K12">0_.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="polynom1.html#K1">+*</a> 0,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="polynom5.html#E9:46"><i><font color="Green">E38</font></i></a>, <a class="ref" href="funct_7.html#T34">FUNCT_7:34</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom3.html#K12">0_.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="polynom5.html#E10:46"><i><font color="Green">E39</font></i></a>, <a class="ref" href="funct_7.html#T34">FUNCT_7:34</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
;<br/>


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