<?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="rlvect_1.html#V4">add-associative</a> <a href="rlvect_1.html#V5">right_zeroed</a> <a href="group_1.html#V2">unital</a> <a href="group_1.html#V4">associative</a> <a href="group_1.html#V7">commutative</a> <a href="vectsp_1.html#V7">distributive</a> <a href="algstr_1.html#V1">left_zeroed</a> <a href="binom.html#V1">add-cancelable</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <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>3</sub></font> = <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>2</sub></font> as    <a href="struct_0.html#M1">Element</a> of <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></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/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is    <a href="ideal_1.html#M1" target="_self">LinearCombination</a> of <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span> : verum </span>}</span> ;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#K10">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : verum </span>}</span> ;<br/>
<a NAME="E2:93"/><i><font color="Green">E48</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> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <font color="Olive">b<sub>2</sub></font></span>)</span> where B is    <a href="ideal_1.html#M1" target="_self">LinearCombination</a> of <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span> : verum </span>}</span>  holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#K10">*</a> <font color="Olive">b<sub>2</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : verum </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="93_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:93_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is    <a href="ideal_1.html#M1" target="_self">LinearCombination</a> of <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span> : verum </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>8</sub></font> being    <a href="ideal_1.html#M1" target="_self">LinearCombination</a> of <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span><b> such that </b><br/><a NAME="E3:93_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/>
<b>consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> ,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="E5:93_1"/><i><font color="Green">E50</font></i>: 
 <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E6:93_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E7:93_1"/><i><font color="Green">E52</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a><br/>for <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="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> holds <br/><font color="Maroon">c<sub>9</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="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="rlvect_1.html#K2">+</a> <font color="Olive">b<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D12">RLVECT_1:def 12</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#K10">*</a> <font color="Olive">b<sub>1</sub></font>;<br/>
<a NAME="E8:93_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="ideal_1.html#E6:93_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="binom.html#T2">BINOM:2</a></i>;<br/>
<a NAME="E9:93_1"/><b>then </b><i><font color="Green">E53</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
;<br/>
<div><i><font color="Green">E54</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="93_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>10</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:93_1_1"/><i><font color="Green">E55</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>8</sub></font> )
 ;<br/><b>thus </b><a NAME="E2:93_1_1"/>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>10</sub></font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1] )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="93_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:93_1_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#K10">*</a> <font color="Olive">b<sub>1</sub></font>
 ;<br/>

<b>then 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="E3:93_1_1_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>11</sub></font>
 ;<br/>
<a NAME="E4:93_1_1_1"/><i><font color="Green">E57</font></i>: 
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E5:93_1_1_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="ideal_1.html#E1:93_1_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E6:93_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="ideal_1.html#E4:93_1_1_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E7:93_1_1_1"/><b>then </b><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E8:93_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E9:93_1_1_1"/><b>then </b><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="ideal_1.html#E7:93_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="ideal_1.html#E1:93_1_1"><i><font color="Green">E55</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>12</sub></font>, <font color="Maroon">c<sub>13</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>14</sub></font> being    <a href="struct_0.html#M1">Element</a> of <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span><b> such that </b><br/><a NAME="E11:93_1_1_1"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="ideal_1.html#E7:93_1_1_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="ideal_1.html#D9" target="_self">Def9</a></i>;<br/>
<font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="ideal_1.html#E3:93_1_1_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="ideal_1.html#E9:93_1_1_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="ideal_1.html#E11:93_1_1_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="rlvect_1.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D18">VECTSP_1:def 18</a></i>
;<br/>
<b>hence </b><a NAME="E13:93_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1]
 ;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E11:93_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>8</sub></font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="polynom2.html#S4">POLYNOM2:sch 4</a>(<a class="txt" href="ideal_1.html#E9:93_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="ideal_1.html#E10:93_1"><i><font color="Green">E54</font></i></a>);<br/></i>
<a NAME="E12:93_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#K10">*</a> <font color="Olive">b<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="ideal_1.html#E5:93_1"><i><font color="Green">E50</font></i></a></i>;<br/>
<b>hence </b><a NAME="E13:93_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#K10">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : verum </span>}</span> 
 <b>by </b><i><a class="txt" href="ideal_1.html#E3:93_1"><i><font color="Green">E49</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:93"/>
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">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#K10">*</a> <font color="Olive">b<sub>2</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : verum </span>}</span>  holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <font color="Olive">b<sub>2</sub></font></span>)</span> where B is    <a href="ideal_1.html#M1" target="_self">LinearCombination</a> of <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span> : verum </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="93_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:93_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#K10">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : verum </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>8</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="E3:93_2"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/>
<b>set </b><font color="Maroon">c<sub>9</sub></font> = <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>;<br/>
<a NAME="E4:93_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>  <a href="relset_1.html#K4">dom</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> holds <br/>ex <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>ex <font color="Olive">b<sub>4</sub></font> being   <a href="struct_0.html#M1">Element</a> of <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span> st <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></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">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="group_1.html#K10">*</a> <font color="Olive">b<sub>4</sub></font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Olive">b<sub>3</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="93_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>10</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:93_2_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 ;<br/>

<a NAME="E2:93_2_1"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/>
<a NAME="E3:93_2_1"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="txt" href="ideal_1.html#E1:93_2_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E4:93_2_1"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>then </b><span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>10</sub></font> = 
<font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="ref" href="finseq_4.html#T25">FINSEQ_4:25</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="group_1.html#K10">*</a> <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="vectsp_1.html#D13">VECTSP_1:def 13</a></i>
;<br/>
<b>hence </b><a NAME="E6:93_2_1"/>
ex <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>1</sub></font>ex <font color="Olive">b<sub>3</sub></font> being   <a href="struct_0.html#M1">Element</a> of <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span> st <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="group_1.html#K10">*</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Olive">b<sub>2</sub></font>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>10</sub></font> = <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> as    <a href="ideal_1.html#M1" target="_self">LinearCombination</a> of <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span> <b>by </b><i><a class="ref" href="ideal_1.html#D9" target="_self">Def9</a></i>;<br/>
<a NAME="E6:93_2"/>
 <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="ref" href="binom.html#T3">BINOM:3</a></i>;<br/>
<b>hence </b><a NAME="E7:93_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is    <a href="ideal_1.html#M1" target="_self">LinearCombination</a> of <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span> : verum </span>}</span> 
 <b>by </b><i><a class="txt" href="ideal_1.html#E3:93_2"><i><font color="Green">E49</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:93"/><b>then </b><i><font color="Green">E49</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is    <a href="ideal_1.html#M1" target="_self">LinearCombination</a> of <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span> : verum </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#K10">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : verum </span>}</span> 
 
<b>by </b><i><a class="txt" href="ideal_1.html#E2:93"><i><font color="Green">E48</font></i></a>, <a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="93_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>7</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:93_3_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="ideal_1.html#K7" target="_self">-Ideal</a> 
 ;<br/><a NAME="E2:93_3_1"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="ideal_1.html#K9" target="_self">-RightIdeal</a> 
 <b>by </b><i><a class="ref" href="ideal_1.html#T63" target="_self">Th63</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>8</sub></font> being    <a href="ideal_1.html#M3" target="_self">RightLinearCombination</a> of <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span><b> such that </b><br/><a NAME="E4:93_3_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="ref" href="ideal_1.html#T62" target="_self">Th62</a></i>;<br/><a NAME="E5:93_3_1"/>
<font color="Maroon">c<sub>8</sub></font> is    <a href="ideal_1.html#M1" target="_self">LinearCombination</a> of <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="ideal_1.html#T28" target="_self">Th28</a></i>;<br/><b>hence </b><a NAME="E6:93_3_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is    <a href="ideal_1.html#M1" target="_self">LinearCombination</a> of <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span> : verum </span>}</span> 
 <b>by </b><i><a class="txt" href="ideal_1.html#E4:93_3_1"><i><font color="Green">E50</font></i></a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:93_3"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is    <a href="ideal_1.html#M1" target="_self">LinearCombination</a> of <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span> : verum </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">c<sub>8</sub></font> being    <a href="ideal_1.html#M1" target="_self">LinearCombination</a> of <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span><b> such that </b><br/><a NAME="E4:93_3"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/><b>thus </b><a NAME="E5:93_3"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="ideal_1.html#K7" target="_self">-Ideal</a> 
 <b>by </b><i><a class="txt" href="ideal_1.html#E4:93_3"><i><font color="Green">E50</font></i></a>, <a class="ref" href="ideal_1.html#T60" target="_self">Th60</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:93"/>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="ideal_1.html#K7" target="_self">-Ideal</a>  <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#K10">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : verum </span>}</span> 
 <b>by </b><i><a class="txt" href="ideal_1.html#E4:93"><i><font color="Green">E49</font></i></a>, <a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>


</div>
