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

<b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="vectsp_1.html#V3">strict</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:66_1_2"/><i><font color="Green">E41</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> iff <font color="Olive">b<sub>1</sub></font> is   <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font> )
 <b>and </b><br/><a NAME="E2:66_1_2"/><i><font color="Green">E42</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font><br/>for <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font> being  <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>4</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> <a href="polynom3.html#K8" target="_self">+</a> <font color="Olive">b<sub>4</sub></font>
 <b>and </b><br/><a NAME="E3:66_1_2"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font><br/>for <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font> being  <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>4</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="group_1.html#K1">*</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> <a href="polynom3.html#K14" target="_self">*'</a> <font color="Olive">b<sub>4</sub></font>
 <b>and </b><br/><a NAME="E4:66_1_2"/><i><font color="Green">E44</font></i>: 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="polynom3.html#K12" target="_self">0_.</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E5:66_1_2"/><i><font color="Green">E45</font></i>: 
 <a href="vectsp_1.html#K2">1_</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="polynom3.html#K13" target="_self">1_.</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E6:66_1_2"/><i><font color="Green">E46</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font> iff <font color="Olive">b<sub>1</sub></font> is   <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font> )
 <b>and </b><br/><a NAME="E7:66_1_2"/><i><font color="Green">E47</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>3</sub></font><br/>for <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font> being  <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>4</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> <a href="polynom3.html#K8" target="_self">+</a> <font color="Olive">b<sub>4</sub></font>
 <b>and </b><br/><a NAME="E8:66_1_2"/><i><font color="Green">E48</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>3</sub></font><br/>for <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font> being  <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>4</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="group_1.html#K1">*</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> <a href="polynom3.html#K14" target="_self">*'</a> <font color="Olive">b<sub>4</sub></font>
 <b>and </b><br/><a NAME="E9:66_1_2"/><i><font color="Green">E49</font></i>: 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="polynom3.html#K12" target="_self">0_.</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E10:66_1_2"/><i><font color="Green">E50</font></i>: 
 <a href="vectsp_1.html#K2">1_</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="polynom3.html#K13" target="_self">1_.</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<div><i><font color="Green">E51</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><a NAME="E1:66_1_2_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> iff <font color="Maroon">c<sub>4</sub></font> is   <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="polynom3.html#E1:66_1_2"><i><font color="Green">E41</font></i></a></i>;<br/><b>hence </b><a NAME="E2:66_1_2_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> iff <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="polynom3.html#E6:66_1_2"><i><font color="Green">E46</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E12:66_1_2"/><b>then </b><i><font color="Green">E52</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>3</sub></font>;<br/><b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="txt" href="polynom3.html#E11:66_1_2"><i><font color="Green">E51</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="polynom3.html#E11:66_1_2"><i><font color="Green">E51</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="polynom3.html#E1:66_1_2"><i><font color="Green">E41</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="polynom3.html#E6:66_1_2"><i><font color="Green">E46</font></i></a></i>;<br/><b>thus </b>the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>2</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> = 
<font color="Maroon">c<sub>4</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>7</sub></font>

<br/>.= 
<font color="Maroon">c<sub>8</sub></font> <a href="polynom3.html#K8" target="_self">+</a> <font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="txt" href="polynom3.html#E2:66_1_2"><i><font color="Green">E42</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>6</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="polynom3.html#E7:66_1_2"><i><font color="Green">E47</font></i></a></i>
<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>

;<br/></div><b>end;</b></div>
<a NAME="E14:66_1_2"/><b>then </b><i><font color="Green">E53</font></i>: 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#R1">=</a> the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="polynom3.html#E12:66_1_2"><i><font color="Green">E52</font></i></a>, <a class="ref" href="binop_1.html#T2">BINOP_1:2</a></i>;<br/>
<div><i><font color="Green">E54</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_2_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>3</sub></font>;<br/><b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="txt" href="polynom3.html#E11:66_1_2"><i><font color="Green">E51</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="polynom3.html#E11:66_1_2"><i><font color="Green">E51</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="polynom3.html#E1:66_1_2"><i><font color="Green">E41</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="polynom3.html#E6:66_1_2"><i><font color="Green">E46</font></i></a></i>;<br/><b>thus </b>the <a href="group_1.html#U1">mult</a> of <font color="Maroon">c<sub>2</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> = 
<font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>7</sub></font>

<br/>.= 
<font color="Maroon">c<sub>8</sub></font> <a href="polynom3.html#K14" target="_self">*'</a> <font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="txt" href="polynom3.html#E3:66_1_2"><i><font color="Green">E43</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>6</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="polynom3.html#E8:66_1_2"><i><font color="Green">E48</font></i></a></i>
<br/>.= 
the <a href="group_1.html#U1">mult</a> of <font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>

;<br/></div><b>end;</b></div>
<i><font color="Green">E55</font></i>: the <a href="vectsp_1.html#U1">unity</a> of <font color="Maroon">c<sub>2</sub></font> = 
 <a href="polynom3.html#K13" target="_self">1_.</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="txt" href="polynom3.html#E5:66_1_2"><i><font color="Green">E45</font></i></a></i>
<br/>.= 
the <a href="vectsp_1.html#U1">unity</a> of <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="polynom3.html#E10:66_1_2"><i><font color="Green">E50</font></i></a></i>
;<br/>
the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">c<sub>2</sub></font> = 
 <a href="polynom3.html#K12" target="_self">0_.</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="txt" href="polynom3.html#E4:66_1_2"><i><font color="Green">E44</font></i></a></i>
<br/>.= 
the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="polynom3.html#E9:66_1_2"><i><font color="Green">E49</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E18:66_1_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="polynom3.html#E12:66_1_2"><i><font color="Green">E52</font></i></a>, <a class="txt" href="polynom3.html#E14:66_1_2"><i><font color="Green">E53</font></i></a>, <a class="txt" href="polynom3.html#E15:66_1_2"><i><font color="Green">E54</font></i></a>, <a class="txt" href="polynom3.html#E16:66_1_2"><i><font color="Green">E55</font></i></a>, <a class="ref" href="binop_1.html#T2">BINOP_1:2</a></i>;<br/>


</div>
