<?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#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#V4">associative</a> <a href="vectsp_1.html#V7">distributive</a> <a href="vectsp_1.html#V8">left_unital</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  non <a href="struct_0.html#V3">empty</a> <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="vectsp_1.html#V12">VectSp-like</a>  <a href="vectsp_1.html#L4">VectSpStr</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>





<b>assume </b><a NAME="E1:19"/><i><font color="Green">E14</font></i>: 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>thus </b><a NAME="E2:19"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">c<sub>3</sub></font> is <a href="vectsp_4.html#V1">lineary-closed</a> implies for <font color="Olive">b<sub>1</sub></font> being   <a href="vectsp_6.html#M2" target="_self">Linear_Combination</a> of <font color="Maroon">c<sub>3</sub></font> holds  <a href="vectsp_6.html#K4" target="_self">Sum</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1"><b>proof </b></a><div class="add">

<b>assume </b><b>that </b><br/><a NAME="E1:19_1"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>and </b><br/><a NAME="E2:19_1"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="vectsp_4.html#V1">lineary-closed</a>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means for <font color="Olive">b<sub>1</sub></font> being   <a href="vectsp_6.html#M2" target="_self">Linear_Combination</a> of <font color="Maroon">c<sub>3</sub></font> st  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K1" target="_self">Carrier</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> holds <br/> <a href="vectsp_6.html#K4" target="_self">Sum</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>;<br/>
<a NAME="E3:19_1"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="vectsp_6.html#M2" target="_self">Linear_Combination</a> of <font color="Maroon">c<sub>3</sub></font>;<br/>

<b>assume </b><a NAME="E1:19_1_1"/>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K1" target="_self">Carrier</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<a NAME="E2:19_1_1"/><b>then </b>
 <a href="vectsp_6.html#K1" target="_self">Carrier</a> <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="card_2.html#T59">CARD_2:59</a></i>;<br/>
<a NAME="E3:19_1_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#R2">=</a>  <a href="vectsp_6.html#K2" target="_self">ZeroLC</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="vectsp_6.html#D6" target="_self">Def6</a></i>;<br/>
<a NAME="E4:19_1_1"/><b>then </b>
 <a href="vectsp_6.html#K4" target="_self">Sum</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="vectsp_6.html#E47"><i><font color="Green">Lemma13</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:19_1_1"/>
 <a href="vectsp_6.html#K4" target="_self">Sum</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="vectsp_6.html#E1:19_1"><i><font color="Green">E15</font></i></a>, <a class="txt" href="vectsp_6.html#E2:19_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="vectsp_4.html#T4">VECTSP_4:4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:19_1"/><i><font color="Green">E18</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:19_1_2"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>4</sub></font>]
 ;<br/>

<b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="vectsp_6.html#M2" target="_self">Linear_Combination</a> of <font color="Maroon">c<sub>3</sub></font>;<br/>

<b>assume </b><a NAME="E2:19_1_2"/><i><font color="Green">E20</font></i>: 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K1" target="_self">Carrier</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>6</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>2</sub></font><b> such that </b><br/><a NAME="E4:19_1_2"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>6</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>and </b><br/><a NAME="E5:19_1_2"/><i><font color="Green">E22</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="vectsp_6.html#K1" target="_self">Carrier</a> <font color="Maroon">c<sub>5</sub></font>
 <b>and </b><br/><a NAME="E6:19_1_2"/><i><font color="Green">E23</font></i>: 
 <a href="vectsp_6.html#K4" target="_self">Sum</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="vectsp_6.html#K3" target="_self">(#)</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="vectsp_6.html#D9" target="_self">Def9</a></i>;<br/>
<a NAME="E7:19_1_2"/><i><font color="Green">E24</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="txt" href="vectsp_6.html#E2:19_1_2"><i><font color="Green">E20</font></i></a>, <a class="txt" href="vectsp_6.html#E4:19_1_2"><i><font color="Green">E21</font></i></a>, <a class="txt" href="vectsp_6.html#E5:19_1_2"><i><font color="Green">E22</font></i></a>, <a class="ref" href="finseq_4.html#T77">FINSEQ_4:77</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> as    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="finseq_1.html#T23">FINSEQ_1:23</a></i>;<br/>
<a NAME="E9:19_1_2"/><i><font color="Green">E25</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="vectsp_6.html#E7:19_1_2"><i><font color="Green">E24</font></i></a>, <a class="ref" href="finseq_3.html#T59">FINSEQ_3:59</a></i>;<br/>
<a NAME="E10:19_1_2"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>4</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"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T6">FINSEQ_1:6</a></i>;<br/>
<a NAME="E11:19_1_2"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="vectsp_6.html#E7:19_1_2"><i><font color="Green">E24</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="finseq_2.html#T13">FINSEQ_2:13</a></i>;<br/>
<a NAME="E13:19_1_2"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="vectsp_6.html#E7:19_1_2"><i><font color="Green">E24</font></i></a>, <a class="txt" href="vectsp_6.html#E10:19_1_2"><i><font color="Green">E26</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E14:19_1_2"/><b>then </b><i><font color="Green">E28</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="vectsp_6.html#K1" target="_self">Carrier</a> <font color="Maroon">c<sub>5</sub></font> &amp;  <a href="vectsp_6.html#K1" target="_self">Carrier</a> <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="txt" href="vectsp_6.html#E5:19_1_2"><i><font color="Green">E22</font></i></a>, <a class="ref" href="vectsp_6.html#D7" target="_self">Def7</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E15:19_1_2"/><b>then </b><i><font color="Green">E29</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="vectsp_6.html#E2:19_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="vectsp_4.html#D1">VECTSP_4:def 1</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> = <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<b>consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>,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="E17:19_1_2"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E18:19_1_2"/><i><font color="Green">E31</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>8</sub></font> holds <br/><font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S6">FUNCT_2:sch 6</a>();<br/></i>
<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>9</sub></font> as    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>,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#T11">FUNCT_2:11</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>11</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><b>assume </b><a NAME="E1:19_1_2_1"/><i><font color="Green">E32</font></i>: 
not <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="vectsp_6.html#K1" target="_self">Carrier</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><b>hence </b><font color="Maroon">c<sub>10</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>11</sub></font> = 
<font color="Maroon">c<sub>5</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="vectsp_6.html#E14:19_1_2"><i><font color="Green">E28</font></i></a>, <a class="txt" href="vectsp_6.html#E18:19_1_2"><i><font color="Green">E31</font></i></a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="txt" href="vectsp_6.html#E1:19_1_2_1"><i><font color="Green">E32</font></i></a></i>
;<br/><br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>10</sub></font> as    <a href="vectsp_6.html#M1" target="_self">Linear_Combination</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="vectsp_6.html#D4" target="_self">Def4</a></i>;<br/>
<a NAME="E22:19_1_2"/><i><font color="Green">E32</font></i>: 
 <a href="vectsp_6.html#K1" target="_self">Carrier</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K1" target="_self">Carrier</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>8</sub></font></span><a href="domain_1.html#K6">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_2_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:19_1_2_2"/>
 <a href="vectsp_6.html#K1" target="_self">Carrier</a> <font color="Maroon">c<sub>11</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K1" target="_self">Carrier</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>8</sub></font></span><a href="domain_1.html#K6">}</a></span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_2_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>12</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:19_1_2_2_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="vectsp_6.html#K1" target="_self">Carrier</a> <font color="Maroon">c<sub>11</sub></font>
 ;<br/>

<a NAME="E2:19_1_2_2_1"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font> : <font color="Maroon">c<sub>11</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</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>}</span> 
 
;<br/>
<b>then consider </b><font color="Maroon">c<sub>13</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E4:19_1_2_2_1"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</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/>
<a NAME="E5:19_1_2_2_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font>
 
<b>by </b><i><a class="txt" href="vectsp_6.html#E17:19_1_2"><i><font color="Green">E30</font></i></a>, <a class="txt" href="vectsp_6.html#E18:19_1_2"><i><font color="Green">E31</font></i></a>, <a class="txt" href="vectsp_6.html#E4:19_1_2_2_1"><i><font color="Green">E33</font></i></a></i>;<br/>
<a NAME="E6:19_1_2_2_1"/><b>then </b>
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</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> <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>}</span>  &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>8</sub></font> )
 
<b>by </b><i><a class="txt" href="vectsp_6.html#E17:19_1_2"><i><font color="Green">E30</font></i></a>, <a class="txt" href="vectsp_6.html#E4:19_1_2_2_1"><i><font color="Green">E33</font></i></a></i>;<br/>
<a NAME="E7:19_1_2_2_1"/><b>then </b>
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="vectsp_6.html#K1" target="_self">Carrier</a> <font color="Maroon">c<sub>5</sub></font> &amp; not <font color="Maroon">c<sub>12</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>8</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>hence </b><a NAME="E8:19_1_2_2_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K1" target="_self">Carrier</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>8</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon">c<sub>12</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:19_1_2_2"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K1" target="_self">Carrier</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>8</sub></font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/>

<a NAME="E3:19_1_2_2"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="vectsp_6.html#K1" target="_self">Carrier</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E4:19_1_2_2"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</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> <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>}</span> 
 
;<br/>
<b>then consider </b><font color="Maroon">c<sub>13</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E6:19_1_2_2"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</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/>
<a NAME="E7:19_1_2_2"/>
not <font color="Maroon">c<sub>12</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>8</sub></font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="txt" href="vectsp_6.html#E2:19_1_2_2"><i><font color="Green">E33</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E8:19_1_2_2"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E9:19_1_2_2"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font>
 
<b>by </b><i><a class="txt" href="vectsp_6.html#E18:19_1_2"><i><font color="Green">E31</font></i></a>, <a class="txt" href="vectsp_6.html#E6:19_1_2_2"><i><font color="Green">E34</font></i></a></i>;<br/>
<a NAME="E10:19_1_2_2"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font> : <font color="Maroon">c<sub>11</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</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>}</span> 
 
<b>by </b><i><a class="txt" href="vectsp_6.html#E6:19_1_2_2"><i><font color="Green">E34</font></i></a></i>;<br/>
<b>hence </b><a NAME="E11:19_1_2_2"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="vectsp_6.html#K1" target="_self">Carrier</a> <font color="Maroon">c<sub>11</sub></font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E23:19_1_2"/><b>then </b>
(  <a href="vectsp_6.html#K1" target="_self">Carrier</a> <font color="Maroon">c<sub>11</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>8</sub></font></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>8</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="txt" href="vectsp_6.html#E14:19_1_2"><i><font color="Green">E28</font></i></a>, <a class="ref" href="xboole_1.html#T33">XBOOLE_1:33</a>, <a class="ref" href="xboole_1.html#T36">XBOOLE_1:36</a></i>;<br/>
<a NAME="E24:19_1_2"/><b>then </b>
 <a href="vectsp_6.html#K1" target="_self">Carrier</a> <font color="Maroon">c<sub>11</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</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="vectsp_6.html#M2" target="_self">Linear_Combination</a> of <font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="ref" href="vectsp_6.html#D7" target="_self">Def7</a></i>;<br/>
<a NAME="E26:19_1_2"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>7</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="vectsp_6.html#E4:19_1_2"><i><font color="Green">E21</font></i></a>, <a class="ref" href="funct_1.html#T84">FUNCT_1:84</a></i>;<br/>
<a NAME="E27:19_1_2"/><i><font color="Green">E34</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="vectsp_6.html#K1" target="_self">Carrier</a> <font color="Maroon">c<sub>12</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_2_3"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:19_1_2_3"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a>  <a href="vectsp_6.html#K1" target="_self">Carrier</a> <font color="Maroon">c<sub>12</sub></font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_2_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>13</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:19_1_2_3_1"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>14</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:19_1_2_3_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>7</sub></font>
 <b>and </b><br/><a NAME="E4:19_1_2_3_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>14</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="vectsp_6.html#E3:19_1_2_3_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="finseq_3.html#T25">FINSEQ_3:25</a></i>;<br/>
<a NAME="E6:19_1_2_3_1"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T89">RELAT_1:89</a></i>;<br/>
<a NAME="E7:19_1_2_3_1"/><b>then </b><i><font color="Green">E37</font></i>: 
( <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</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>15</sub></font> )
 
<b>by </b><i><a class="txt" href="vectsp_6.html#E3:19_1_2_3_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/>
<a NAME="E8:19_1_2_3_1"/><b>then </b><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>13</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="vectsp_6.html#E4:19_1_2_3_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_2_3_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:19_1_2_3_1_1"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/><a NAME="E2:19_1_2_3_1_1"/><b>then </b>
( <font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="txt" href="vectsp_6.html#E4:19_1_2"><i><font color="Green">E21</font></i></a>, <a class="txt" href="vectsp_6.html#E9:19_1_2"><i><font color="Green">E25</font></i></a>, <a class="txt" href="vectsp_6.html#E13:19_1_2"><i><font color="Green">E27</font></i></a>, <a class="txt" href="vectsp_6.html#E3:19_1_2_3_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="vectsp_6.html#E4:19_1_2_3_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="vectsp_6.html#E7:19_1_2_3_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a>, <a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/><b>hence </b><a NAME="E3:19_1_2_3_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div>
<a NAME="E10:19_1_2_3_1"/><b>then </b>
not <font color="Maroon">c<sub>13</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>8</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>hence </b><a NAME="E11:19_1_2_3_1"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="vectsp_6.html#K1" target="_self">Carrier</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="vectsp_6.html#E5:19_1_2"><i><font color="Green">E22</font></i></a>, <a class="txt" href="vectsp_6.html#E22:19_1_2"><i><font color="Green">E32</font></i></a>, <a class="txt" href="vectsp_6.html#E8:19_1_2_3_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon">c<sub>13</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:19_1_2_3"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="vectsp_6.html#K1" target="_self">Carrier</a> <font color="Maroon">c<sub>12</sub></font>
 ;<br/>

<a NAME="E3:19_1_2_3"/><b>then </b>
<font color="Maroon">c<sub>13</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="vectsp_6.html#E5:19_1_2"><i><font color="Green">E22</font></i></a>, <a class="txt" href="vectsp_6.html#E22:19_1_2"><i><font color="Green">E32</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>14</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:19_1_2_3"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>6</sub></font>
 <b>and </b><br/><a NAME="E6:19_1_2_3"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>14</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="vectsp_6.html#E5:19_1_2_3"><i><font color="Green">E36</font></i></a>, <a class="ref" href="finseq_3.html#T25">FINSEQ_3:25</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_2_3_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:19_1_2_3_2"/>
not <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><a NAME="E2:19_1_2_3_2"/><b>then </b>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="vectsp_6.html#E5:19_1_2_3"><i><font color="Green">E36</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E3:19_1_2_3_2"/><b>then </b>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="vectsp_6.html#E7:19_1_2"><i><font color="Green">E24</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E4:19_1_2_3_2"/><b>then </b>
<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"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="finseq_3.html#T16">FINSEQ_3:16</a></i>;<br/><a NAME="E5:19_1_2_3_2"/><b>then </b>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E6:19_1_2_3_2"/><b>then </b>
not <font color="Maroon">c<sub>8</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>8</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="txt" href="vectsp_6.html#E22:19_1_2"><i><font color="Green">E32</font></i></a>, <a class="txt" href="vectsp_6.html#E2:19_1_2_3"><i><font color="Green">E35</font></i></a>, <a class="txt" href="vectsp_6.html#E6:19_1_2_3"><i><font color="Green">E37</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>hence </b><a NAME="E7:19_1_2_3_2"/>
contradiction
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div>
<a NAME="E9:19_1_2_3"/><b>then </b>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="vectsp_6.html#E5:19_1_2_3"><i><font color="Green">E36</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E10:19_1_2_3"/><b>then </b><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>;<br/>
<a NAME="E11:19_1_2_3"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</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>15</sub></font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/>
<b>hence </b><a NAME="E12:19_1_2_3"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="vectsp_6.html#E6:19_1_2_3"><i><font color="Green">E37</font></i></a>, <a class="txt" href="vectsp_6.html#E10:19_1_2_3"><i><font color="Green">E38</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E28:19_1_2"/><b>then </b><i><font color="Green">E35</font></i>: 
 <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="vectsp_6.html#K3" target="_self">(#)</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="vectsp_6.html#K4" target="_self">Sum</a> <font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="txt" href="vectsp_6.html#E26:19_1_2"><i><font color="Green">E33</font></i></a>, <a class="ref" href="vectsp_6.html#D9" target="_self">Def9</a></i>;<br/>
<a NAME="E29:19_1_2"/><i><font color="Green">E36</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="vectsp_6.html#K3" target="_self">(#)</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="vectsp_6.html#K3" target="_self">(#)</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="txt" href="vectsp_6.html#E7:19_1_2"><i><font color="Green">E24</font></i></a>, <a class="txt" href="vectsp_6.html#E9:19_1_2"><i><font color="Green">E25</font></i></a>, <a class="ref" href="vectsp_6.html#D8" target="_self">Def8</a></i>;<br/>
<a NAME="E30:19_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/>
<b>then </b><span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> = 
 <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="finseq_1.html#T9">FINSEQ_1:9</a></i>
<br/>.= 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="vectsp_6.html#K3" target="_self">(#)</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="vectsp_6.html#E29:19_1_2"><i><font color="Green">E36</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
;<br/>
<a NAME="E32:19_1_2"/><b>then </b><i><font color="Green">E37</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="vectsp_6.html#K3" target="_self">(#)</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="vectsp_6.html#K3" target="_self">(#)</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="vectsp_6.html#E29:19_1_2"><i><font color="Green">E36</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_2_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>13</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:19_1_2_5"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="vectsp_6.html#K3" target="_self">(#)</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>13</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="ref" href="finseq_3.html#T25">FINSEQ_3:25</a></i>;<br/><a NAME="E3:19_1_2_5"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="vectsp_6.html#E9:19_1_2"><i><font color="Green">E25</font></i></a>, <a class="txt" href="vectsp_6.html#E29:19_1_2"><i><font color="Green">E36</font></i></a>, <a class="txt" href="vectsp_6.html#E1:19_1_2_5"><i><font color="Green">E38</font></i></a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/><a NAME="E4:19_1_2_5"/><b>then </b><i><font color="Green">E40</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>7</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font> ;<br/><a NAME="E6:19_1_2_5"/>
not <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"><font color="Maroon">c<sub>8</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="txt" href="vectsp_6.html#E22:19_1_2"><i><font color="Green">E32</font></i></a>, <a class="txt" href="vectsp_6.html#E27:19_1_2"><i><font color="Green">E34</font></i></a>, <a class="txt" href="vectsp_6.html#E4:19_1_2_5"><i><font color="Green">E40</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E7:19_1_2_5"/><b>then </b><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><i><font color="Green">E42</font></i>: <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="vectsp_6.html#K3" target="_self">(#)</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c<sub>15</sub></font>
<b>by </b><i><a class="txt" href="vectsp_6.html#E3:19_1_2_5"><i><font color="Green">E39</font></i></a>, <a class="ref" href="vectsp_6.html#T32" target="_self">Th32</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c<sub>15</sub></font>
<b>by </b><i><a class="txt" href="vectsp_6.html#E18:19_1_2"><i><font color="Green">E31</font></i></a>, <a class="txt" href="vectsp_6.html#E7:19_1_2_5"><i><font color="Green">E41</font></i></a></i>
;<br/><a NAME="E9:19_1_2_5"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="vectsp_6.html#K3" target="_self">(#)</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="vectsp_6.html#E32:19_1_2"><i><font color="Green">E37</font></i></a>, <a class="txt" href="vectsp_6.html#E1:19_1_2_5"><i><font color="Green">E38</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E10:19_1_2_5"/><b>then </b><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="vectsp_6.html#E7:19_1_2"><i><font color="Green">E24</font></i></a>, <a class="txt" href="vectsp_6.html#E29:19_1_2"><i><font color="Green">E36</font></i></a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/><a NAME="E11:19_1_2_5"/><b>then </b>
( <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</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> &amp;  <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>2</sub></font> )
 <b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font> ;<br/><a NAME="E13:19_1_2_5"/>
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="vectsp_6.html#E3:19_1_2_5"><i><font color="Green">E39</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/><b>hence </b><a NAME="E14:19_1_2_5"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="vectsp_6.html#K3" target="_self">(#)</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</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>5</sub></font> <a href="vectsp_6.html#K3" target="_self">(#)</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="vectsp_6.html#E8:19_1_2_5"><i><font color="Green">E42</font></i></a>, <a class="txt" href="vectsp_6.html#E10:19_1_2_5"><i><font color="Green">E43</font></i></a>, <a class="ref" href="vectsp_6.html#T32" target="_self">Th32</a></i>;<br/></div><b>end;</b></div>
<a NAME="E34:19_1_2"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="vectsp_6.html#K3" target="_self">(#)</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="vectsp_6.html#K3" target="_self">(#)</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="vectsp_6.html#E32:19_1_2"><i><font color="Green">E37</font></i></a>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>
<a NAME="E35:19_1_2"/><b>then </b><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="vectsp_6.html#K3" target="_self">(#)</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="vectsp_6.html#K3" target="_self">(#)</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="vectsp_6.html#K3" target="_self">(#)</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="vectsp_6.html#E29:19_1_2"><i><font color="Green">E36</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E36:19_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="vectsp_6.html#K3" target="_self">(#)</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="vectsp_6.html#E7:19_1_2"><i><font color="Green">E24</font></i></a>, <a class="txt" href="vectsp_6.html#E13:19_1_2"><i><font color="Green">E27</font></i></a>, <a class="ref" href="vectsp_6.html#T32" target="_self">Th32</a></i>;<br/>
<a NAME="E37:19_1_2"/><b>then </b><i><font color="Green">E39</font></i>: 
 <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="vectsp_6.html#K3" target="_self">(#)</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="vectsp_6.html#K3" target="_self">(#)</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="vectsp_6.html#E7:19_1_2"><i><font color="Green">E24</font></i></a>, <a class="txt" href="vectsp_6.html#E29:19_1_2"><i><font color="Green">E36</font></i></a>, <a class="txt" href="vectsp_6.html#E35:19_1_2"><i><font color="Green">E38</font></i></a>, <a class="ref" href="rlvect_1.html#T55">RLVECT_1:55</a></i>;<br/>
<a NAME="E38:19_1_2"/>
<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="vectsp_6.html#E13:19_1_2"><i><font color="Green">E27</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E39:19_1_2"/><b>then </b>
(  <a href="vectsp_6.html#K1" target="_self">Carrier</a> <font color="Maroon">c<sub>5</sub></font> is <a href="finset_1.html#V1">finite</a> &amp; <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>8</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="vectsp_6.html#K1" target="_self">Carrier</a> <font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="txt" href="vectsp_6.html#E5:19_1_2"><i><font color="Green">E22</font></i></a>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<b>then </b> <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K1" target="_self">Carrier</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>8</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
<b>by </b><i><a class="txt" href="vectsp_6.html#E2:19_1_2"><i><font color="Green">E20</font></i></a>, <a class="txt" href="vectsp_6.html#E22:19_1_2"><i><font color="Green">E32</font></i></a>, <a class="ref" href="card_2.html#T63">CARD_2:63</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
<b>by </b><i><a class="ref" href="card_1.html#T79">CARD_1:79</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T26">XCMPLX_1:26</a></i>
;<br/>
<a NAME="E41:19_1_2"/><b>then </b>
 <a href="vectsp_6.html#K4" target="_self">Sum</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="vectsp_6.html#E1:19_1_2"><i><font color="Green">E19</font></i></a></i>;<br/>
<b>hence </b><a NAME="E42:19_1_2"/>
 <a href="vectsp_6.html#K4" target="_self">Sum</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="vectsp_6.html#E2:19_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="vectsp_6.html#E6:19_1_2"><i><font color="Green">E23</font></i></a>, <a class="txt" href="vectsp_6.html#E15:19_1_2"><i><font color="Green">E29</font></i></a>, <a class="txt" href="vectsp_6.html#E28:19_1_2"><i><font color="Green">E35</font></i></a>, <a class="txt" href="vectsp_6.html#E37:19_1_2"><i><font color="Green">E39</font></i></a>, <a class="ref" href="vectsp_4.html#D1">VECTSP_4:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:19_1"/><i><font color="Green">E19</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="vectsp_6.html#E3:19_1"><i><font color="Green">E17</font></i></a>, <a class="txt" href="vectsp_6.html#E4:19_1"><i><font color="Green">E18</font></i></a>);<br/></i>
<b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="vectsp_6.html#M2" target="_self">Linear_Combination</a> of <font color="Maroon">c<sub>3</sub></font>;<br/>

<a NAME="E6:19_1"/>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K1" target="_self">Carrier</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K1" target="_self">Carrier</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
;<br/>
<b>hence </b><a NAME="E7:19_1"/>
 <a href="vectsp_6.html#K4" target="_self">Sum</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="vectsp_6.html#E5:19_1"><i><font color="Green">E19</font></i></a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E3:19"/><i><font color="Green">E15</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="vectsp_6.html#M2" target="_self">Linear_Combination</a> of <font color="Maroon">c<sub>3</sub></font> holds  <a href="vectsp_6.html#K4" target="_self">Sum</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E4:19"/>
(  <a href="vectsp_6.html#K2" target="_self">ZeroLC</a> <font color="Maroon">c<sub>2</sub></font> is    <a href="vectsp_6.html#M2" target="_self">Linear_Combination</a> of <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="vectsp_6.html#K4" target="_self">Sum</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K2" target="_self">ZeroLC</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="vectsp_6.html#E47"><i><font color="Green">Lemma13</font></i></a>, <a class="ref" href="vectsp_6.html#T26" target="_self">Th26</a></i>;<br/>
<a NAME="E5:19"/><b>then </b><i><font color="Green">E16</font></i>: 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="vectsp_6.html#E3:19"><i><font color="Green">E15</font></i></a></i>;<br/>
<b>thus </b><a NAME="E6:19"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="vectsp_6.html#E3:19"><i><font color="Green">E15</font></i></a></i>;<br/>

<a NAME="E7:19"/><i><font color="Green">E17</font></i>: 
for <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><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>2</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="vectsp_1.html#K6">*</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>



<b>assume </b><a NAME="E1:19_2"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:19_2_1_1"/>
( <font color="Maroon">c<sub>4</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>4</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="19_2_1_1_1"><b>suppose </b></a><a NAME="E1:19_2_1_1_1"/>
<font color="Maroon">c<sub>4</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"><b>hence </b><a NAME="E2:19_2_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="vectsp_6.html#E5:19"><i><font color="Green">E16</font></i></a>, <a class="ref" href="vectsp_1.html#T59">VECTSP_1:59</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_2_1_1_2"><b>suppose </b></a><a NAME="E1:19_2_1_1_2"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>4</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"><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> =  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font>;<br/><b>consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>,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="E3:19_2_1_1_2"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E4:19_2_1_1_2"/><i><font color="Green">E21</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>5</sub></font> holds <br/><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S6">FUNCT_2:sch 6</a>();<br/></i><b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> as    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>,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#T11">FUNCT_2:11</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><b>assume </b><a NAME="E1:19_2_1_1_2_1"/>
not <font color="Maroon">c<sub>8</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>5</sub></font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/><a NAME="E2:19_2_1_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:19_2_1_1_2_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="vectsp_6.html#E4:19_2_1_1_2"><i><font color="Green">E21</font></i></a></i>;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> as    <a href="vectsp_6.html#M1" target="_self">Linear_Combination</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="vectsp_6.html#D4" target="_self">Def4</a></i>;<br/><a NAME="E8:19_2_1_1_2"/><i><font color="Green">E22</font></i>: 
 <a href="vectsp_6.html#K1" target="_self">Carrier</a> <font color="Maroon">c<sub>8</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>5</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2_1_1_2_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:19_2_1_1_2_2"/>
 <a href="vectsp_6.html#K1" target="_self">Carrier</a> <font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2_1_1_2_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:19_2_1_1_2_2_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="vectsp_6.html#K1" target="_self">Carrier</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/>

<a NAME="E2:19_2_1_1_2_2_1"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font> : <font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</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>}</span> 
 
;<br/>
<b>then consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E4:19_2_1_1_2_2_1"/><i><font color="Green">E23</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>10</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/>
<a NAME="E5:19_2_1_1_2_2_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="vectsp_6.html#E4:19_2_1_1_2"><i><font color="Green">E21</font></i></a>, <a class="txt" href="vectsp_6.html#E4:19_2_1_1_2_2_1"><i><font color="Green">E23</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:19_2_1_1_2_2_1"/>
<font color="Maroon">c<sub>9</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>5</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="txt" href="vectsp_6.html#E4:19_2_1_1_2_2_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon">c<sub>9</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:19_2_1_1_2_2"/>
<font color="Maroon">c<sub>9</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>5</sub></font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/>

<a NAME="E3:19_2_1_1_2_2"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E4:19_2_1_1_2_2"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font> : <font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</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>}</span> 
 
<b>by </b><i><a class="txt" href="vectsp_6.html#E1:19_2_1_1_2"><i><font color="Green">E19</font></i></a>, <a class="txt" href="vectsp_6.html#E3:19_2_1_1_2"><i><font color="Green">E20</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:19_2_1_1_2_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="vectsp_6.html#K1" target="_self">Carrier</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/>


</div><b>end;</b></div><a NAME="E9:19_2_1_1_2"/>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="vectsp_6.html#E1:19_2"><i><font color="Green">E18</font></i></a>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font> as    <a href="vectsp_6.html#M2" target="_self">Linear_Combination</a> of <font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="txt" href="vectsp_6.html#E8:19_2_1_1_2"><i><font color="Green">E22</font></i></a>, <a class="ref" href="vectsp_6.html#D7" target="_self">Def7</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>10</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>2</sub></font><b> such that </b><br/><a NAME="E12:19_2_1_1_2"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>10</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>and </b><br/><a NAME="E13:19_2_1_1_2"/><i><font color="Green">E24</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="vectsp_6.html#K1" target="_self">Carrier</a> <font color="Maroon">c<sub>9</sub></font>
 <b>and </b><br/><a NAME="E14:19_2_1_1_2"/><i><font color="Green">E25</font></i>: 
 <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="vectsp_6.html#K3" target="_self">(#)</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="vectsp_6.html#K4" target="_self">Sum</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="ref" href="vectsp_6.html#D9" target="_self">Def9</a></i>;<br/><a NAME="E15:19_2_1_1_2"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="vectsp_6.html#E8:19_2_1_1_2"><i><font color="Green">E22</font></i></a>, <a class="txt" href="vectsp_6.html#E12:19_2_1_1_2"><i><font color="Green">E23</font></i></a>, <a class="txt" href="vectsp_6.html#E13:19_2_1_1_2"><i><font color="Green">E24</font></i></a>, <a class="ref" href="finseq_3.html#T106">FINSEQ_3:106</a></i>;<br/><a NAME="E16:19_2_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="vectsp_6.html#K3" target="_self">(#)</a> <font color="Maroon">c<sub>10</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="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="vectsp_6.html#T34" target="_self">Th34</a></i>;<br/><a NAME="E17:19_2_1_1_2"/><b>then </b>
 <a href="vectsp_6.html#K4" target="_self">Sum</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="vectsp_6.html#E3:19_2_1_1_2"><i><font color="Green">E20</font></i></a>, <a class="txt" href="vectsp_6.html#E14:19_2_1_1_2"><i><font color="Green">E25</font></i></a>, <a class="ref" href="rlvect_1.html#T61">RLVECT_1:61</a></i>;<br/><b>hence </b><a NAME="E18:19_2_1_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="vectsp_6.html#E3:19"><i><font color="Green">E15</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:19_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E8:19"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="vectsp_4.html#D1">VECTSP_4:def 1</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:19_3"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E2:19_3"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:19_3_1_1"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>5</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_3_1_1_1"><b>suppose </b></a><a NAME="E1:19_3_1_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><div class="add"><b>then </b><font color="Maroon">c<sub>4</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>5</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="vectsp_1.html#D26">VECTSP_1:def 26</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D26">VECTSP_1:def 26</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</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="vectsp_1.html#K6">*</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="vectsp_1.html#D26">VECTSP_1:def 26</a></i>
;<br/><b>hence </b><a NAME="E3:19_3_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="vectsp_6.html#E7:19"><i><font color="Green">E17</font></i></a>, <a class="txt" href="vectsp_6.html#E1:19_3"><i><font color="Green">E18</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_3_1_1_2"><b>suppose </b></a><a NAME="E1:19_3_1_1_2"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><div class="add"><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> =  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font>;<br/><b>consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>,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="E3:19_3_1_1_2"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E4:19_3_1_1_2"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E5:19_3_1_1_2"/><i><font color="Green">E23</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>5</sub></font> holds <br/><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S7">FUNCT_2:sch 7</a>(<a class="txt" href="vectsp_6.html#E1:19_3_1_1_2"><i><font color="Green">E20</font></i></a>);<br/></i><b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> as    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>,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#T11">FUNCT_2:11</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_3_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><b>assume </b><a NAME="E1:19_3_1_1_2_1"/>
not <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K7">}</a></span>
 ;<br/><a NAME="E2:19_3_1_1_2_1"/><b>then </b>
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><b>hence </b><a NAME="E3:19_3_1_1_2_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="vectsp_6.html#E5:19_3_1_1_2"><i><font color="Green">E23</font></i></a></i>;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> as    <a href="vectsp_6.html#M1" target="_self">Linear_Combination</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="vectsp_6.html#D4" target="_self">Def4</a></i>;<br/><a NAME="E9:19_3_1_1_2"/><i><font color="Green">E24</font></i>: 
 <a href="vectsp_6.html#K1" target="_self">Carrier</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K7">}</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_3_1_1_2_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:19_3_1_1_2_2"/>
 <a href="vectsp_6.html#K1" target="_self">Carrier</a> <font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K7">}</a></span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_3_1_1_2_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:19_3_1_1_2_2_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="vectsp_6.html#K1" target="_self">Carrier</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/>

<a NAME="E2:19_3_1_1_2_2_1"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font> : <font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</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>}</span> 
 
;<br/>
<a NAME="E3:19_3_1_1_2_2_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>2</sub></font> st <br/>( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</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/>
<a NAME="E4:19_3_1_1_2_2_1"/><b>then </b>
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> or <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="txt" href="vectsp_6.html#E5:19_3_1_1_2"><i><font color="Green">E23</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:19_3_1_1_2_2_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K7">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon">c<sub>9</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:19_3_1_1_2_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K7">}</a></span>
 ;<br/>

<a NAME="E3:19_3_1_1_2_2"/><b>then </b>
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> or <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<a NAME="E4:19_3_1_1_2_2"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font> : <font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</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>}</span> 
 
<b>by </b><i><a class="txt" href="vectsp_6.html#E1:19"><i><font color="Green">E14</font></i></a>, <a class="txt" href="vectsp_6.html#E3:19_3_1_1_2"><i><font color="Green">E21</font></i></a>, <a class="txt" href="vectsp_6.html#E4:19_3_1_1_2"><i><font color="Green">E22</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:19_3_1_1_2_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="vectsp_6.html#K1" target="_self">Carrier</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/>


</div><b>end;</b></div><a NAME="E10:19_3_1_1_2"/><b>then </b>
 <a href="vectsp_6.html#K1" target="_self">Carrier</a> <font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="vectsp_6.html#E1:19_3"><i><font color="Green">E18</font></i></a>, <a class="txt" href="vectsp_6.html#E2:19_3"><i><font color="Green">E19</font></i></a>, <a class="ref" href="zfmisc_1.html#T38">ZFMISC_1:38</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font> as    <a href="vectsp_6.html#M2" target="_self">Linear_Combination</a> of <font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="ref" href="vectsp_6.html#D7" target="_self">Def7</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>10</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>2</sub></font><b> such that </b><br/><a NAME="E13:19_3_1_1_2"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>10</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>and </b><br/><a NAME="E14:19_3_1_1_2"/><i><font color="Green">E26</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="vectsp_6.html#K1" target="_self">Carrier</a> <font color="Maroon">c<sub>9</sub></font>
 <b>and </b><br/><a NAME="E15:19_3_1_1_2"/><i><font color="Green">E27</font></i>: 
 <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="vectsp_6.html#K3" target="_self">(#)</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="vectsp_6.html#K4" target="_self">Sum</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="ref" href="vectsp_6.html#D9" target="_self">Def9</a></i>;<br/><a NAME="E16:19_3_1_1_2"/>
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="finseq_4.html#K2">*&gt;</a></span> or <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="finseq_4.html#K2">*&gt;</a></span> )
 <b>by </b><i><a class="txt" href="vectsp_6.html#E1:19_3_1_1_2"><i><font color="Green">E20</font></i></a>, <a class="txt" href="vectsp_6.html#E9:19_3_1_1_2"><i><font color="Green">E24</font></i></a>, <a class="txt" href="vectsp_6.html#E13:19_3_1_1_2"><i><font color="Green">E25</font></i></a>, <a class="txt" href="vectsp_6.html#E14:19_3_1_1_2"><i><font color="Green">E26</font></i></a>, <a class="ref" href="finseq_3.html#T108">FINSEQ_3:108</a></i>;<br/><a NAME="E17:19_3_1_1_2"/><b>then </b>
( ( <font color="Maroon">c<sub>9</sub></font> <a href="vectsp_6.html#K3" target="_self">(#)</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> or <font color="Maroon">c<sub>9</sub></font> <a href="vectsp_6.html#K3" target="_self">(#)</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> ) &amp; <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="vectsp_6.html#E3:19_3_1_1_2"><i><font color="Green">E21</font></i></a>, <a class="txt" href="vectsp_6.html#E4:19_3_1_1_2"><i><font color="Green">E22</font></i></a>, <a class="ref" href="vectsp_6.html#T35" target="_self">Th35</a>, <a class="ref" href="vectsp_1.html#D26">VECTSP_1:def 26</a></i>;<br/><a NAME="E18:19_3_1_1_2"/><b>then </b>
 <a href="vectsp_6.html#K4" target="_self">Sum</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="vectsp_6.html#E15:19_3_1_1_2"><i><font color="Green">E27</font></i></a>, <a class="ref" href="rlvect_1.html#T62">RLVECT_1:62</a></i>;<br/><b>hence </b><a NAME="E19:19_3_1_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="vectsp_6.html#E3:19"><i><font color="Green">E15</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:19_3"/>
<font color="Maroon">c<sub>4</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E9:19"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font><br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> holds <br/>(  not <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> or <font color="Olive">b<sub>1</sub></font> <a href="vectsp_1.html#K6">*</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="vectsp_6.html#E7:19"><i><font color="Green">E17</font></i></a></i>;<br/>


</div>
