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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="rlvect_1.html#V3">Abelian</a> <a href="rlvect_1.html#V4">add-associative</a> <a href="rlvect_1.html#V5">right_zeroed</a> <a href="rlvect_1.html#V6">right_complementable</a> <a href="group_1.html#V2">unital</a> <a href="group_1.html#V4">associative</a> <a href="vectsp_1.html#V7">distributive</a> non <a href="realset2.html#V3">trivial</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a> ;<br/>

<b>set </b><font color="Maroon">c<sub>2</sub></font> =  <a href="xboole_0.html#K1">{}</a> ;<br/>
<b>set </b><font color="Maroon">c<sub>3</sub></font> =  <a href="polynom1.html#K30">Polynom-Ring</a> <a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">c<sub>1</sub></font>;<br/>
<a NAME="E1:16"/><i><font color="Green">E5</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="polynom1.html#NM1">bag</a> of  <a href="xboole_0.html#K1">{}</a>  holds <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="polynom1.html#NM1">bag</a> of  <a href="xboole_0.html#K1">{}</a> ;<br/>

<a NAME="E1:16_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K14">Bags</a> <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="polynom1.html#D14">POLYNOM1:def 14</a></i>;<br/>
<b>hence </b><a NAME="E2:16_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="polynom1.html#T55">POLYNOM1:55</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:16"/><b>then </b><i><font color="Green">E6</font></i>: 
 <a href="polynom1.html#K16">EmptyBag</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> =  <a href="xboole_0.html#K1">{}</a>  as   <a href="polynom1.html#NM1">bag</a> of  <a href="xboole_0.html#K1">{}</a>  ;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] means ex <font color="Olive">b<sub>1</sub></font> being  <a href="polynom1.html#NM4">Polynomial</a> of <a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>2</sub></font> );<br/>
<a NAME="E4:16"/><i><font color="Green">E7</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">c<sub>1</sub></font></span>)</span>ex <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>

<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="polynom1.html#NM4">Polynomial</a> of <a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a> 
;<br/>

<a NAME="E2:16_2"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="polynom1.html#K14">Bags</a> <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E3:16_2"/><b>then </b><i><font color="Green">E8</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="polynom7.html#E2:16"><i><font color="Green">E6</font></i></a>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<a NAME="E4:16_2"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/>
<b>hence </b><a NAME="E5:16_2"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a> ] )
 <b>by </b><i><a class="txt" href="polynom7.html#E3:16_2"><i><font color="Green">E8</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">c<sub>1</sub></font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E6:16"/><i><font color="Green">E8</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">c<sub>1</sub></font></span>)</span> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>(<a class="txt" href="polynom7.html#E4:16"><i><font color="Green">E7</font></i></a>);<br/></i>
<a NAME="E7:16"/><i><font color="Green">E9</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <a href="xboole_0.html#K1">{}</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/>
<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="struct_0.html#NM7">Function</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E9:16"/>
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 <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">c<sub>1</sub></font></span>)</span> holds <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>

<b>consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="polynom1.html#NM4">Polynomial</a> of <a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E2:16_3"/><i><font color="Green">E10</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="txt" href="polynom7.html#E6:16"><i><font color="Green">E8</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="polynom1.html#NM4">Polynomial</a> of <a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E4:16_3"/><i><font color="Green">E11</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> )
 <b>by </b><i><a class="txt" href="polynom7.html#E6:16"><i><font color="Green">E8</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>11</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E6:16_3"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <a href="xboole_0.html#K1">{}</a> </span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="polynom7.html#E7"><i><font color="Green">Lemma4</font></i></a></i>;<br/>
<a NAME="E7:16_3"/><i><font color="Green">E13</font></i>: 
 <a href="polynom1.html#K16">EmptyBag</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <a href="xboole_0.html#K1">{}</a> </span>)</span></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E8:16_3"/><b>then </b><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="txt" href="polynom7.html#E2:16"><i><font color="Green">E6</font></i></a>, <a class="txt" href="polynom7.html#E6:16_3"><i><font color="Green">E12</font></i></a>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>12</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E10:16_3"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <a href="xboole_0.html#K1">{}</a> </span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="polynom7.html#E7"><i><font color="Green">Lemma4</font></i></a></i>;<br/>
<a NAME="E11:16_3"/>
<font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="txt" href="polynom7.html#E2:16"><i><font color="Green">E6</font></i></a>, <a class="txt" href="polynom7.html#E7:16_3"><i><font color="Green">E13</font></i></a>, <a class="txt" href="polynom7.html#E10:16_3"><i><font color="Green">E15</font></i></a>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/>
<a NAME="E12:16_3"/><b>then </b><i><font color="Green">E16</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="txt" href="polynom7.html#E2:16"><i><font color="Green">E6</font></i></a>, <a class="txt" href="polynom7.html#E2:16_3"><i><font color="Green">E10</font></i></a>, <a class="txt" href="polynom7.html#E4:16_3"><i><font color="Green">E11</font></i></a>, <a class="txt" href="polynom7.html#E6:16_3"><i><font color="Green">E12</font></i></a>, <a class="txt" href="polynom7.html#E7:16_3"><i><font color="Green">E13</font></i></a>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>13</sub></font> being   <a href="polynom1.html#NM4">Polynomial</a> of <a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E14:16_3"/><i><font color="Green">E17</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="polynom7.html#E6:16"><i><font color="Green">E8</font></i></a></i>;<br/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="polynom1.html#K23">+</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#D21">POLYNOM1:def 21</a></i>
<br/>.= 
<font color="Maroon">c<sub>11</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>12</sub></font>
<b>by </b><i><a class="txt" href="polynom7.html#E2:16"><i><font color="Green">E6</font></i></a>, <a class="txt" href="polynom7.html#E7:16_3"><i><font color="Green">E13</font></i></a>, <a class="txt" href="polynom7.html#E8:16_3"><i><font color="Green">E14</font></i></a>, <a class="txt" href="polynom7.html#E10:16_3"><i><font color="Green">E15</font></i></a>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
;<br/>
<b>hence </b><a NAME="E16:16_3"/>
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="polynom7.html#E2:16_3"><i><font color="Green">E10</font></i></a>, <a class="txt" href="polynom7.html#E4:16_3"><i><font color="Green">E11</font></i></a>, <a class="txt" href="polynom7.html#E12:16_3"><i><font color="Green">E16</font></i></a>, <a class="txt" href="polynom7.html#E14:16_3"><i><font color="Green">E17</font></i></a>, <a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:16"/><b>then </b><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>6</sub></font> is <a href="grcat_1.html#V1">additive</a>
 
<b>by </b><i><a class="ref" href="grcat_1.html#D13">GRCAT_1:def 13</a></i>;<br/>
<a NAME="E11:16"/>
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 <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">c<sub>1</sub></font></span>)</span> holds <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="group_1.html#K1">*</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>

<b>consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="polynom1.html#NM4">Polynomial</a> of <a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E2:16_4"/><i><font color="Green">E11</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="txt" href="polynom7.html#E6:16"><i><font color="Green">E8</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="polynom1.html#NM4">Polynomial</a> of <a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E4:16_4"/><i><font color="Green">E12</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> )
 <b>by </b><i><a class="txt" href="polynom7.html#E6:16"><i><font color="Green">E8</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>11</sub></font> being   <a href="polynom1.html#NM4">Polynomial</a> of <a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E6:16_4"/><i><font color="Green">E13</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="polynom7.html#E6:16"><i><font color="Green">E8</font></i></a></i>;<br/>
<a NAME="E7:16_4"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_4_1"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">c<sub>12</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>;<br/>
<a NAME="E1:16_4_1"/><i><font color="Green">E14</font></i>: 
 <a href="polynom1.html#K21">decomp</a> <span class="p1">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <a href="xboole_0.html#K1">{}</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <a href="xboole_0.html#K1">{}</a> </span>)</span>,<span class="p3">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <a href="xboole_0.html#K1">{}</a> </span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span></span><a href="finseq_1.html#K9">*&gt;</a></span>
 
<b>by </b><i><a class="ref" href="polynom1.html#T77">POLYNOM1:77</a></i>;<br/>
<a NAME="E2:16_4_1"/><b>then </b><i><font color="Green">E15</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <span class="p2">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <a href="xboole_0.html#K1">{}</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>13</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E4:16_4_1"/><i><font color="Green">E16</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="polynom1.html#K15">.</a> <span class="p1">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <a href="xboole_0.html#K1">{}</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">c<sub>13</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <span class="p2">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <a href="xboole_0.html#K1">{}</a> </span>)</span></span>)</span> &amp; ( 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="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>13</sub></font> holds <br/>ex <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being  <a href="polynom1.html#NM1">bag</a> of  <a href="xboole_0.html#K1">{}</a>  st <br/>( <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <span class="p2">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <a href="xboole_0.html#K1">{}</a> </span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font></span><a href="finseq_1.html#K10">*&gt;</a></span> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="polynom1.html#K15">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="polynom1.html#K15">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> ) ) )
 <b>by </b><i><a class="ref" href="polynom1.html#D26">POLYNOM1:def 26</a></i>;<br/>
<a NAME="E5:16_4_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="txt" href="polynom7.html#E1:16_4_1"><i><font color="Green">E14</font></i></a>, <a class="txt" href="polynom7.html#E4:16_4_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>;<br/>
<a NAME="E6:16_4_1"/><b>then </b>
 <a href="finseq_1.html#K2">Seg</a> 1 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>13</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E7:16_4_1"/><b>then </b><i><font color="Green">E17</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>13</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>14</sub></font>, <font color="Maroon">c<sub>15</sub></font> being   <a href="polynom1.html#NM1">bag</a> of  <a href="xboole_0.html#K1">{}</a> <b> such that </b><br/><a NAME="E9:16_4_1"/><i><font color="Green">E18</font></i>: 
( <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <span class="p2">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <a href="xboole_0.html#K1">{}</a> </span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font></span><a href="finseq_1.html#K10">*&gt;</a></span> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="polynom7.html#E4:16_4_1"><i><font color="Green">E16</font></i></a></i>;<br/>
<font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> 1 = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="polynom7.html#E7:16_4_1"><i><font color="Green">E17</font></i></a>, <a class="txt" href="polynom7.html#E9:16_4_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="polynom7.html#E1:16"><i><font color="Green">E5</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="polynom7.html#E1:16"><i><font color="Green">E5</font></i></a></i>
;<br/>
<a NAME="E11:16_4_1"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="polynom7.html#E2:16_4_1"><i><font color="Green">E15</font></i></a>, <a class="txt" href="polynom7.html#E4:16_4_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/>
<a NAME="E12:16_4_1"/><b>then </b>
 <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#T61">RLVECT_1:61</a></i>;<br/>
<b>hence </b><a NAME="E13:16_4_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="polynom7.html#E1:16"><i><font color="Green">E5</font></i></a>, <a class="txt" href="polynom7.html#E4:16_4_1"><i><font color="Green">E16</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:16_4"/>
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="polynom7.html#E2:16_4"><i><font color="Green">E11</font></i></a>, <a class="txt" href="polynom7.html#E4:16_4"><i><font color="Green">E12</font></i></a>, <a class="txt" href="polynom7.html#E6:16_4"><i><font color="Green">E13</font></i></a>, <a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E12:16"/><b>then </b><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>6</sub></font> is <a href="group_6.html#V1">multiplicative</a>
 
<b>by </b><i><a class="ref" href="group_6.html#D7">GROUP_6:def 7</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="polynom1.html#NM4">Polynomial</a> of <a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E14:16"/><i><font color="Green">E12</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">c<sub>1</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <span class="p2">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="polynom7.html#E6:16"><i><font color="Green">E8</font></i></a></i>;<br/>
<i><font color="Green">E13</font></i>: <font color="Maroon">c<sub>7</sub></font> = 
 <a href="polynom1.html#K27">1_</a> <a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="txt" href="polynom7.html#E14:16"><i><font color="Green">E12</font></i></a>, <a class="ref" href="polynom1.html#T90">POLYNOM1:90</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom1.html#K26">0_</a> <a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="polynom1.html#K1">+*</a> <span class="p1">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <a href="xboole_0.html#K1">{}</a> </span>)</span>,<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#D25">POLYNOM1:def 25</a></i>
;<br/>
<a NAME="E16:16"/><i><font color="Green">E14</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K26">0_</a> <a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="polynom1.html#K14">Bags</a> <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E17:16"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <a href="xboole_0.html#K1">{}</a> </span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <a href="xboole_0.html#K1">{}</a> </span>)</span></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="cqc_lang.html#T5">CQC_LANG:5</a></i>;<br/>
<a NAME="E18:16"/><b>then </b><i><font color="Green">E15</font></i>: 
 <a href="polynom1.html#K16">EmptyBag</a> <a href="xboole_0.html#K1">{}</a>  <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="polynom1.html#K16">EmptyBag</a> <a href="xboole_0.html#K1">{}</a> </span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<font color="Maroon">c<sub>7</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>4</sub></font> = 
<font color="Maroon">c<sub>7</sub></font> <a href="polynom1.html#K15">.</a> <span class="p1">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <a href="xboole_0.html#K1">{}</a> </span>)</span>
<b>by </b><i><a class="txt" href="polynom7.html#E1:16"><i><font color="Green">E5</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K26">0_</a> <a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <a href="xboole_0.html#K1">{}</a> </span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p3">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <a href="xboole_0.html#K1">{}</a> </span>)</span>
<b>by </b><i><a class="txt" href="polynom7.html#E15:16"><i><font color="Green">E13</font></i></a>, <a class="txt" href="polynom7.html#E16:16"><i><font color="Green">E14</font></i></a>, <a class="ref" href="funct_7.html#D3">FUNCT_7:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <a href="xboole_0.html#K1">{}</a> </span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <a href="xboole_0.html#K1">{}</a> </span>)</span>
<b>by </b><i><a class="txt" href="polynom7.html#E18:16"><i><font color="Green">E15</font></i></a>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
<br/>.= 
 <a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="ref" href="cqc_lang.html#T6">CQC_LANG:6</a></i>
;<br/>
<a NAME="E20:16"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> is <a href="endalg.html#V1">unity-preserving</a>
 
<b>by </b><i><a class="txt" href="polynom7.html#E14:16"><i><font color="Green">E12</font></i></a>, <a class="ref" href="endalg.html#D8">ENDALG:def 8</a></i>;<br/>
<a NAME="E21:16"/><b>then </b><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>6</sub></font> is <a href="quofield.html#V1">RingHomomorphism</a>
 
<b>by </b><i><a class="txt" href="polynom7.html#E10:16"><i><font color="Green">E10</font></i></a>, <a class="txt" href="polynom7.html#E12:16"><i><font color="Green">E11</font></i></a>, <a class="ref" href="quofield.html#D21">QUOFIELD:def 21</a></i>;<br/>
<a NAME="E22:16"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:16_7"/><i><font color="Green">E17</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>9</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">c<sub>1</sub></font></span>)</span> ;<br/>
<b>consider </b><font color="Maroon">c<sub>12</sub></font> being   <a href="polynom1.html#NM4">Polynomial</a> of <a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E4:16_7"/><i><font color="Green">E18</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="txt" href="polynom7.html#E6:16"><i><font color="Green">E8</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>13</sub></font> being   <a href="polynom1.html#NM4">Polynomial</a> of <a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E6:16_7"/><i><font color="Green">E19</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>11</sub></font> )
 <b>by </b><i><a class="txt" href="polynom7.html#E6:16"><i><font color="Green">E8</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>14</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E8:16_7"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <a href="xboole_0.html#K1">{}</a> </span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="polynom7.html#E7"><i><font color="Green">Lemma4</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>15</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E10:16_7"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <a href="xboole_0.html#K1">{}</a> </span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="polynom7.html#E7"><i><font color="Green">Lemma4</font></i></a></i>;<br/>
<a NAME="E11:16_7"/>
 <a href="polynom1.html#K16">EmptyBag</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <a href="xboole_0.html#K1">{}</a> </span>)</span></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E12:16_7"/><b>then </b><i><font color="Green">E22</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="polynom1.html#K15">.</a> <span class="p1">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <a href="xboole_0.html#K1">{}</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="polynom1.html#K15">.</a> <span class="p1">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <a href="xboole_0.html#K1">{}</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> )
 
<b>by </b><i><a class="txt" href="polynom7.html#E8:16_7"><i><font color="Green">E20</font></i></a>, <a class="txt" href="polynom7.html#E10:16_7"><i><font color="Green">E21</font></i></a>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/>
<a NAME="E13:16_7"/>
( <font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="polynom1.html#K15">.</a> <span class="p1">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <a href="xboole_0.html#K1">{}</a> </span>)</span> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="polynom1.html#K15">.</a> <span class="p1">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <a href="xboole_0.html#K1">{}</a> </span>)</span> )
 
<b>by </b><i><a class="txt" href="polynom7.html#E1:16"><i><font color="Green">E5</font></i></a></i>;<br/>
<b>hence </b><a NAME="E14:16_7"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="polynom7.html#E1:16_7"><i><font color="Green">E17</font></i></a>, <a class="txt" href="polynom7.html#E4:16_7"><i><font color="Green">E18</font></i></a>, <a class="txt" href="polynom7.html#E6:16_7"><i><font color="Green">E19</font></i></a>, <a class="txt" href="polynom7.html#E8:16_7"><i><font color="Green">E20</font></i></a>, <a class="txt" href="polynom7.html#E10:16_7"><i><font color="Green">E21</font></i></a>, <a class="txt" href="polynom7.html#E12:16_7"><i><font color="Green">E22</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E23:16"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/>
<a NAME="E24:16"/><b>then </b><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>6</sub></font> is <a href="quofield.html#V3">RingMonomorphism</a>
 
<b>by </b><i><a class="txt" href="polynom7.html#E21:16"><i><font color="Green">E16</font></i></a>, <a class="ref" href="quofield.html#D23">QUOFIELD:def 23</a></i>;<br/>
<a NAME="E25:16"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/>
<a NAME="E26:16"/><b>then </b><i><font color="Green">E18</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font> 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>1</sub></font>
 
;<br/>
<a NAME="E27:16"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <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>1</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:16_8"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<b>set </b><font color="Maroon">c<sub>10</sub></font> = <span class="p1">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <a href="xboole_0.html#K1">{}</a> </span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>9</sub></font>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <span class="p1">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <a href="xboole_0.html#K1">{}</a> </span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>9</sub></font> as   <a href="funct_1.html#NM1">Function</a> ;<br/>
<a NAME="E4:16_8"/>
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <a href="xboole_0.html#K1">{}</a> </span>)</span></span><a href="domain_1.html#K6">}</a></span> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>9</sub></font></span><a href="domain_1.html#K6">}</a></span> )
 
<b>by </b><i><a class="ref" href="cqc_lang.html#T5">CQC_LANG:5</a></i>;<br/>
<a NAME="E5:16_8"/><b>then </b>
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="polynom1.html#K14">Bags</a> <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>11</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="polynom1.html#T55">POLYNOM1:55</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>11</sub></font> as   <a href="funct_2.html#NM1">Function</a> of  <a href="polynom1.html#K14">Bags</a> <a href="xboole_0.html#K1">{}</a> ,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="funct_2.html#T4">FUNCT_2:4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>12</sub></font> as   <a href="struct_0.html#NM6">Function</a> of  <a href="polynom1.html#K14">Bags</a> <a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">c<sub>1</sub></font> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>13</sub></font> as   <a href="polynom1.html#NM3">Series</a> of <a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">c<sub>1</sub></font> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_8_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_8_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:16_8_1_1"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font> or <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_8_1_1_1"><b>case </b></a><a NAME="E1:16_8_1_1_1"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><div class="add"><a NAME="E2:16_8_1_1_1"/>
 <a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_8_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:16_8_1_1_1_1"/><i><font color="Green">E20</font></i>: 
 <a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>15</sub></font> being    <a href="subset_1.html#M1">Element</a> of  <a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>14</sub></font>;<br/>
<a NAME="E3:16_8_1_1_1_1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="txt" href="polynom7.html#E1:16_8_1_1_1_1"><i><font color="Green">E20</font></i></a></i>;<br/>
<a NAME="E4:16_8_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>15</sub></font> is   <a href="polynom1.html#NM1">bag</a> of  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="polynom1.html#D14">POLYNOM1:def 14</a></i>;<br/>
<b>then </b><font color="Maroon">c<sub>14</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font> = 
<font color="Maroon">c<sub>14</sub></font> <a href="polynom1.html#K15">.</a> <span class="p1">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <a href="xboole_0.html#K1">{}</a> </span>)</span>
<b>by </b><i><a class="txt" href="polynom7.html#E1:16"><i><font color="Green">E5</font></i></a>, <a class="txt" href="polynom7.html#E2:16"><i><font color="Green">E6</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="ref" href="cqc_lang.html#T6">CQC_LANG:6</a></i>
;<br/>
<b>hence </b><a NAME="E6:16_8_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="polynom7.html#E1:16_8_1_1_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="polynom7.html#E3:16_8_1_1_1_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="polynom1.html#D9">POLYNOM1:def 9</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:16_8_1_1_1"/>
 <a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>14</sub></font> is <a href="finset_1.html#V1">finite</a>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_8_1_1_2"><b>case </b></a><a NAME="E1:16_8_1_1_2"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><div class="add"><a NAME="E2:16_8_1_1_2"/><i><font color="Green">E20</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>14</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <a href="xboole_0.html#K1">{}</a> </span>)</span></span><a href="domain_1.html#K6">}</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_8_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>15</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:16_8_1_1_2_1"/>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>14</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>15</sub></font> as   <a href="polynom1.html#NM1">bag</a> of  <a href="xboole_0.html#K1">{}</a>  <b>by </b><i><a class="ref" href="polynom1.html#D14">POLYNOM1:def 14</a></i>;<br/>
<a NAME="E3:16_8_1_1_2_1"/>
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a>  <a href="polynom1.html#K16">EmptyBag</a> <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="polynom7.html#E1:16"><i><font color="Green">E5</font></i></a>, <a class="txt" href="polynom7.html#E2:16"><i><font color="Green">E6</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:16_8_1_1_2_1"/>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <a href="xboole_0.html#K1">{}</a> </span>)</span></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:16_8_1_1_2"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <a href="xboole_0.html#K1">{}</a> </span>)</span></span><a href="domain_1.html#K6">}</a></span> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>14</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_8_1_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>15</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:16_8_1_1_2_2"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <a href="xboole_0.html#K1">{}</a> </span>)</span></span><a href="domain_1.html#K6">}</a></span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>15</sub></font> as    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <a href="xboole_0.html#K1">{}</a>  ;<br/>
<font color="Maroon">c<sub>14</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>16</sub></font> = 
<font color="Maroon">c<sub>14</sub></font> <a href="polynom1.html#K15">.</a> <span class="p1">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <a href="xboole_0.html#K1">{}</a> </span>)</span>
<b>by </b><i><a class="txt" href="polynom7.html#E1:16_8_1_1_2_2"><i><font color="Green">E21</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>
<br/>.= 
<font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="ref" href="cqc_lang.html#T6">CQC_LANG:6</a></i>
;<br/>
<b>hence </b><a NAME="E4:16_8_1_1_2_2"/>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="polynom7.html#E1:16_8_1_1_2"><i><font color="Green">E19</font></i></a>, <a class="ref" href="polynom1.html#D9">POLYNOM1:def 9</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E4:16_8_1_1_2"/>
 <a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>14</sub></font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="txt" href="polynom7.html#E2:16_8_1_1_2"><i><font color="Green">E20</font></i></a>, <a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>14</sub></font> as   <a href="polynom1.html#NM4">Polynomial</a> of <a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="polynom1.html#D10">POLYNOM1:def 10</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>15</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>17</sub></font> being   <a href="polynom1.html#NM4">Polynomial</a> of <a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E13:16_8"/><i><font color="Green">E19</font></i>: 
( <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>16</sub></font> )
 <b>by </b><i><a class="txt" href="polynom7.html#E6:16"><i><font color="Green">E8</font></i></a></i>;<br/>
<font color="Maroon">c<sub>17</sub></font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  = 
<font color="Maroon">c<sub>15</sub></font> <a href="polynom1.html#K15">.</a> <span class="p1">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <a href="xboole_0.html#K1">{}</a> </span>)</span>
<b>by </b><i><a class="txt" href="polynom7.html#E1:16"><i><font color="Green">E5</font></i></a>, <a class="txt" href="polynom7.html#E13:16_8"><i><font color="Green">E19</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="ref" href="cqc_lang.html#T6">CQC_LANG:6</a></i>
;<br/>
<b>hence </b><a NAME="E15:16_8"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="polynom7.html#E7:16"><i><font color="Green">E9</font></i></a>, <a class="txt" href="polynom7.html#E13:16_8"><i><font color="Green">E19</font></i></a>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E28:16"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="polynom7.html#E26:16"><i><font color="Green">E18</font></i></a>, <a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>
<a NAME="E29:16"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> is <a href="quofield.html#V2">RingEpimorphism</a>
 
<b>by </b><i><a class="txt" href="polynom7.html#E21:16"><i><font color="Green">E16</font></i></a>, <a class="ref" href="quofield.html#D22">QUOFIELD:def 22</a></i>;<br/>
<a NAME="E30:16"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> is <a href="quofield.html#V4">RingIsomorphism</a>
 
<b>by </b><i><a class="txt" href="polynom7.html#E24:16"><i><font color="Green">E17</font></i></a>, <a class="ref" href="quofield.html#D24">QUOFIELD:def 24</a></i>;<br/>
<b>hence </b><a NAME="E31:16"/>
 <a href="polynom1.html#K30">Polynom-Ring</a> <a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">c<sub>1</sub></font> <a href="quofield.html#R2">is_ringisomorph_to</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="quofield.html#D26">QUOFIELD:def 26</a></i>;<br/>


</div>
