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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="vectsp_1.html#NM2">Field</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be  <a href="vectsp_1.html#V11">strict</a> <a href="vectsp_1.html#NM6">VectSp</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>;<br/>





<b>assume </b><a NAME="E1:20"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>4</sub></font> is <a href="mod_2.html#V1">linear</a>
 ;<br/>

<a NAME="E2:20"/>
 <a href="vectsp_8.html#K6" target="_self">FuncLatt</a> <font color="Maroon">c<sub>4</sub></font> is    <a href="vectsp_8.html#M4" target="_self">sup-Semilattice-Homomorphism</a> of  <a href="vectsp_8.html#K1" target="_self">lattice</a> <font color="Maroon">c<sub>2</sub></font>, <a href="vectsp_8.html#K1" target="_self">lattice</a> <font color="Maroon">c<sub>3</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="vectsp_8.html#K1" target="_self">lattice</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="vectsp_8.html#D9" target="_self">VECTSP_8:def 9</a><br/>

<a NAME="E1:20_1"/>
<span class="p1">(<span class="default"><a href="vectsp_8.html#K6" target="_self">FuncLatt</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="vectsp_8.html#K6" target="_self">FuncLatt</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="vectsp_8.html#K6" target="_self">FuncLatt</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">c<sub>7</sub></font> being  <a href="vectsp_1.html#V11">strict</a>  <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E2:20_1_1"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="vectsp_5.html#D3">VECTSP_5:def 3</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>8</sub></font> being  <a href="vectsp_1.html#V11">strict</a>  <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E4:20_1_1"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="vectsp_5.html#D3">VECTSP_5:def 3</a></i>;<br/>
<a NAME="E5:20_1_1"/><i><font color="Green">E17</font></i>: 
<span class="p1">(<span class="default"><a href="vectsp_8.html#K6" target="_self">FuncLatt</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="lattices.html#K3">"\/"</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="vectsp_8.html#K6" target="_self">FuncLatt</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="vectsp_5.html#K1">+</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="vectsp_8.html#E2:20_1_1"><i><font color="Green">E15</font></i></a>, <a class="txt" href="vectsp_8.html#E4:20_1_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="vectsp_8.html#T7" target="_self">Th7</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <span class="p1">(<span class="default"><a href="vectsp_8.html#K6" target="_self">FuncLatt</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="vectsp_8.html#K1" target="_self">lattice</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> ;<br/>
<a NAME="E7:20_1_1"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="vectsp_7.html#K1">Lin</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>7</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="vectsp_8.html#E2:20_1_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="vectsp_8.html#D7" target="_self">Def7</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <span class="p1">(<span class="default"><a href="vectsp_8.html#K6" target="_self">FuncLatt</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="vectsp_8.html#K1" target="_self">lattice</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> ;<br/>
<a NAME="E9:20_1_1"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="vectsp_7.html#K1">Lin</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>8</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="vectsp_8.html#E4:20_1_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="vectsp_8.html#D7" target="_self">Def7</a></i>;<br/>
<a NAME="E10:20_1_1"/><i><font color="Green">E20</font></i>: 
<span class="p1">(<span class="default"><a href="vectsp_8.html#K6" target="_self">FuncLatt</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="vectsp_5.html#K1">+</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="vectsp_7.html#K1">Lin</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="vectsp_5.html#K1">+</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="vectsp_8.html#D7" target="_self">Def7</a></i>;<br/>
<a NAME="E11:20_1_1"/><i><font color="Green">E21</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</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="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E12:20_1_1"/>
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>2</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="ref" href="vectsp_4.html#T25">VECTSP_4:25</a></i>;<br/>
<a NAME="E13:20_1_1"/><b>then </b><i><font color="Green">E22</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> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>11</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E15:20_1_1"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/>
<a NAME="E16:20_1_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>7</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_8.html#E11:20_1_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="vectsp_8.html#E13:20_1_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="vectsp_8.html#E15:20_1_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E17:20_1_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>7</sub></font> is <a href="vectsp_4.html#V1">lineary-closed</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_1"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>7</sub></font>;<br/>
<a NAME="E1:20_1_1_1"/><i><font color="Green">E26</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>3</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>7</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>4</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>7</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>13</sub></font>, <font color="Maroon">c<sub>14</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>3</sub></font>;<br/>

<b>assume </b><a NAME="E1:20_1_1_1_1"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>7</sub></font> )
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>15</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E3:20_1_1_1_1"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>15</sub></font> )
 <b>by </b><i><a class="ref" href="funct_2.html#T116">FUNCT_2:116</a></i>;<br/>
<a NAME="E4:20_1_1_1_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="vectsp_8.html#E3:20_1_1_1_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>16</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="E6:20_1_1_1_1"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>16</sub></font> )
 <b>by </b><i><a class="txt" href="vectsp_8.html#E1:20_1_1_1_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="funct_2.html#T116">FUNCT_2:116</a></i>;<br/>
<a NAME="E7:20_1_1_1_1"/>
<font color="Maroon">c<sub>16</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="vectsp_8.html#E6:20_1_1_1_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E8:20_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>15</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>16</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="vectsp_8.html#E4:20_1_1_1_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="vectsp_4.html#T28">VECTSP_4:28</a></i>;<br/>
<a NAME="E9:20_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>15</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E10:20_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/>
<b>hence </b><a NAME="E11:20_1_1_1_1"/>
<font color="Maroon">c<sub>13</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="vectsp_8.html#E1:20"><i><font color="Green">E14</font></i></a>, <a class="txt" href="vectsp_8.html#E3:20_1_1_1_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="vectsp_8.html#E6:20_1_1_1_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="mod_2.html#D5">MOD_2:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:20_1_1_1"/>
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>3</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>7</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>4</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>7</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>13</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>14</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>3</sub></font>;<br/>



<b>assume </b><a NAME="E1:20_1_1_1_2"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>7</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>15</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E3:20_1_1_1_2"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>15</sub></font> )
 <b>by </b><i><a class="ref" href="funct_2.html#T116">FUNCT_2:116</a></i>;<br/>
<a NAME="E4:20_1_1_1_2"/>
<font color="Maroon">c<sub>15</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="vectsp_8.html#E3:20_1_1_1_2"><i><font color="Green">E27</font></i></a>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E5:20_1_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c<sub>15</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="ref" href="vectsp_4.html#T29">VECTSP_4:29</a></i>;<br/>
<a NAME="E6:20_1_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E7:20_1_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/>
<b>hence </b><a NAME="E8:20_1_1_1_2"/>
<font color="Maroon">c<sub>13</sub></font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="vectsp_8.html#E1:20"><i><font color="Green">E14</font></i></a>, <a class="txt" href="vectsp_8.html#E3:20_1_1_1_2"><i><font color="Green">E27</font></i></a>, <a class="ref" href="mod_2.html#D5">MOD_2:def 5</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:20_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>7</sub></font> is <a href="vectsp_4.html#V1">lineary-closed</a>
 <b>by </b><i><a class="txt" href="vectsp_8.html#E1:20_1_1_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="vectsp_4.html#D1">VECTSP_4:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">c<sub>12</sub></font> being  <a href="vectsp_1.html#V11">strict</a>  <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E19:20_1_1"/><i><font color="Green">E26</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="vectsp_8.html#E16:20_1_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="vectsp_4.html#T42">VECTSP_4:42</a></i>;<br/>
<a NAME="E20:20_1_1"/><i><font color="Green">E27</font></i>: 
 <a href="vectsp_7.html#K1">Lin</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="txt" href="vectsp_8.html#E19:20_1_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="vectsp_7.html#T16">VECTSP_7:16</a></i>;<br/>
<a NAME="E21:20_1_1"/>
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>2</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="ref" href="vectsp_4.html#T25">VECTSP_4:25</a></i>;<br/>
<a NAME="E22:20_1_1"/><b>then </b><i><font color="Green">E28</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> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<b>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>3</sub></font><b> such that </b><br/><a NAME="E24:20_1_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/>
<a NAME="E25:20_1_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>8</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_8.html#E11:20_1_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="vectsp_8.html#E22:20_1_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="vectsp_8.html#E24:20_1_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E26:20_1_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>8</sub></font> is <a href="vectsp_4.html#V1">lineary-closed</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_2"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>8</sub></font>;<br/>
<a NAME="E1:20_1_1_2"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>3</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>8</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>4</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>8</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>15</sub></font>, <font color="Maroon">c<sub>16</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>3</sub></font>;<br/>

<b>assume </b><a NAME="E1:20_1_1_2_1"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>8</sub></font> )
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>17</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="E3:20_1_1_2_1"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>17</sub></font> )
 <b>by </b><i><a class="ref" href="funct_2.html#T116">FUNCT_2:116</a></i>;<br/>
<a NAME="E4:20_1_1_2_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="vectsp_8.html#E3:20_1_1_2_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>18</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="E6:20_1_1_2_1"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>18</sub></font> )
 <b>by </b><i><a class="txt" href="vectsp_8.html#E1:20_1_1_2_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="funct_2.html#T116">FUNCT_2:116</a></i>;<br/>
<a NAME="E7:20_1_1_2_1"/>
<font color="Maroon">c<sub>18</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="vectsp_8.html#E6:20_1_1_2_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E8:20_1_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>17</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>18</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="vectsp_8.html#E4:20_1_1_2_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="vectsp_4.html#T28">VECTSP_4:28</a></i>;<br/>
<a NAME="E9:20_1_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>17</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E10:20_1_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/>
<b>hence </b><a NAME="E11:20_1_1_2_1"/>
<font color="Maroon">c<sub>15</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="vectsp_8.html#E1:20"><i><font color="Green">E14</font></i></a>, <a class="txt" href="vectsp_8.html#E3:20_1_1_2_1"><i><font color="Green">E34</font></i></a>, <a class="txt" href="vectsp_8.html#E6:20_1_1_2_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="mod_2.html#D5">MOD_2:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:20_1_1_2"/>
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>3</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>8</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>4</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>8</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>15</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>16</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>3</sub></font>;<br/>



<b>assume </b><a NAME="E1:20_1_1_2_2"/>
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>8</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>17</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="E3:20_1_1_2_2"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>17</sub></font> )
 <b>by </b><i><a class="ref" href="funct_2.html#T116">FUNCT_2:116</a></i>;<br/>
<a NAME="E4:20_1_1_2_2"/>
<font color="Maroon">c<sub>17</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="vectsp_8.html#E3:20_1_1_2_2"><i><font color="Green">E33</font></i></a>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E5:20_1_1_2_2"/><b>then </b>
<font color="Maroon">c<sub>15</sub></font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c<sub>17</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="ref" href="vectsp_4.html#T29">VECTSP_4:29</a></i>;<br/>
<a NAME="E6:20_1_1_2_2"/><b>then </b>
<font color="Maroon">c<sub>15</sub></font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E7:20_1_1_2_2"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/>
<b>hence </b><a NAME="E8:20_1_1_2_2"/>
<font color="Maroon">c<sub>15</sub></font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="vectsp_8.html#E1:20"><i><font color="Green">E14</font></i></a>, <a class="txt" href="vectsp_8.html#E3:20_1_1_2_2"><i><font color="Green">E33</font></i></a>, <a class="ref" href="mod_2.html#D5">MOD_2:def 5</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:20_1_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>8</sub></font> is <a href="vectsp_4.html#V1">lineary-closed</a>
 <b>by </b><i><a class="txt" href="vectsp_8.html#E1:20_1_1_2"><i><font color="Green">E32</font></i></a>, <a class="ref" href="vectsp_4.html#D1">VECTSP_4:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">c<sub>14</sub></font> being  <a href="vectsp_1.html#V11">strict</a>  <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E28:20_1_1"/><i><font color="Green">E32</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="vectsp_8.html#E25:20_1_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="vectsp_4.html#T42">VECTSP_4:42</a></i>;<br/>
<a NAME="E29:20_1_1"/><i><font color="Green">E33</font></i>: 
 <a href="vectsp_7.html#K1">Lin</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="txt" href="vectsp_8.html#E28:20_1_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="vectsp_7.html#T16">VECTSP_7:16</a></i>;<br/>
<a NAME="E30:20_1_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="vectsp_5.html#K1">+</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> is <a href="vectsp_4.html#V1">lineary-closed</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_3"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="vectsp_5.html#K1">+</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>;<br/>
<a NAME="E1:20_1_1_3"/><i><font color="Green">E35</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>3</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="vectsp_5.html#K1">+</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="vectsp_5.html#K1">+</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> 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>4</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="vectsp_5.html#K1">+</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>16</sub></font>, <font color="Maroon">c<sub>17</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>3</sub></font>;<br/>

<b>assume </b><a NAME="E1:20_1_1_3_1"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="vectsp_5.html#K1">+</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="vectsp_5.html#K1">+</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> )
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>18</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="E3:20_1_1_3_1"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="vectsp_5.html#K1">+</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>18</sub></font> )
 <b>by </b><i><a class="ref" href="funct_2.html#T116">FUNCT_2:116</a></i>;<br/>
<a NAME="E4:20_1_1_3_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>7</sub></font> <a href="vectsp_5.html#K1">+</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="vectsp_8.html#E3:20_1_1_3_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>19</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="E6:20_1_1_3_1"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="vectsp_5.html#K1">+</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font> )
 <b>by </b><i><a class="txt" href="vectsp_8.html#E1:20_1_1_3_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="funct_2.html#T116">FUNCT_2:116</a></i>;<br/>
<a NAME="E7:20_1_1_3_1"/>
<font color="Maroon">c<sub>19</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>7</sub></font> <a href="vectsp_5.html#K1">+</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="vectsp_8.html#E6:20_1_1_3_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E8:20_1_1_3_1"/><b>then </b>
<font color="Maroon">c<sub>18</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>19</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>7</sub></font> <a href="vectsp_5.html#K1">+</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="vectsp_8.html#E4:20_1_1_3_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="vectsp_4.html#T28">VECTSP_4:28</a></i>;<br/>
<a NAME="E9:20_1_1_3_1"/><b>then </b>
<font color="Maroon">c<sub>18</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="vectsp_5.html#K1">+</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E10:20_1_1_3_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="vectsp_5.html#K1">+</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/>
<b>hence </b><a NAME="E11:20_1_1_3_1"/>
<font color="Maroon">c<sub>16</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="vectsp_5.html#K1">+</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="vectsp_8.html#E1:20"><i><font color="Green">E14</font></i></a>, <a class="txt" href="vectsp_8.html#E3:20_1_1_3_1"><i><font color="Green">E37</font></i></a>, <a class="txt" href="vectsp_8.html#E6:20_1_1_3_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="mod_2.html#D5">MOD_2:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:20_1_1_3"/>
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>3</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="vectsp_5.html#K1">+</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> 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>4</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="vectsp_5.html#K1">+</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_3_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>16</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>17</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>3</sub></font>;<br/>



<b>assume </b><a NAME="E1:20_1_1_3_2"/>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="vectsp_5.html#K1">+</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>18</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="E3:20_1_1_3_2"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="vectsp_5.html#K1">+</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>18</sub></font> )
 <b>by </b><i><a class="ref" href="funct_2.html#T116">FUNCT_2:116</a></i>;<br/>
<a NAME="E4:20_1_1_3_2"/>
<font color="Maroon">c<sub>18</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>7</sub></font> <a href="vectsp_5.html#K1">+</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="vectsp_8.html#E3:20_1_1_3_2"><i><font color="Green">E36</font></i></a>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E5:20_1_1_3_2"/><b>then </b>
<font color="Maroon">c<sub>16</sub></font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c<sub>18</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>7</sub></font> <a href="vectsp_5.html#K1">+</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="ref" href="vectsp_4.html#T29">VECTSP_4:29</a></i>;<br/>
<a NAME="E6:20_1_1_3_2"/><b>then </b>
<font color="Maroon">c<sub>16</sub></font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="vectsp_5.html#K1">+</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E7:20_1_1_3_2"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="vectsp_5.html#K1">+</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/>
<b>hence </b><a NAME="E8:20_1_1_3_2"/>
<font color="Maroon">c<sub>16</sub></font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="vectsp_5.html#K1">+</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="vectsp_8.html#E1:20"><i><font color="Green">E14</font></i></a>, <a class="txt" href="vectsp_8.html#E3:20_1_1_3_2"><i><font color="Green">E36</font></i></a>, <a class="ref" href="mod_2.html#D5">MOD_2:def 5</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:20_1_1_3"/>
<font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="vectsp_5.html#K1">+</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> is <a href="vectsp_4.html#V1">lineary-closed</a>
 <b>by </b><i><a class="txt" href="vectsp_8.html#E1:20_1_1_3"><i><font color="Green">E35</font></i></a>, <a class="ref" href="vectsp_4.html#D1">VECTSP_4:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">c<sub>15</sub></font> =  <a href="vectsp_7.html#K1">Lin</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="vectsp_5.html#K1">+</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> as    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">c<sub>3</sub></font> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>12</sub></font> <a href="vectsp_5.html#K1">+</a> <font color="Maroon">c<sub>14</sub></font> as    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">c<sub>3</sub></font> ;<br/>
<a NAME="E33:20_1_1"/>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_4"><b>proof </b></a><div class="add">

<a NAME="E1:20_1_1_4"/><i><font color="Green">E35</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>15</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>16</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>17</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="E1:20_1_1_4_1"/>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>15</sub></font>
 ;<br/>

<a NAME="E2:20_1_1_4_1"/><b>then </b><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>15</sub></font>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E3:20_1_1_4_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="vectsp_5.html#K1">+</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="vectsp_5.html#K1">+</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_4_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:20_1_1_4_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="vectsp_5.html#K1">+</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="vectsp_5.html#K1">+</a> <font color="Maroon">c<sub>14</sub></font></span>)</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="20_1_1_4_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>18</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="E1:20_1_1_4_1_1_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="vectsp_5.html#K1">+</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>18</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>3</sub></font> ;<br/>
<b>consider </b><font color="Maroon">c<sub>20</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="E4:20_1_1_4_1_1_1"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="vectsp_5.html#K1">+</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>20</sub></font> )
 <b>by </b><i><a class="txt" href="vectsp_8.html#E1:20_1_1_4_1_1_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="funct_2.html#T116">FUNCT_2:116</a></i>;<br/>
<a NAME="E5:20_1_1_4_1_1_1"/>
<font color="Maroon">c<sub>20</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>7</sub></font> <a href="vectsp_5.html#K1">+</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="vectsp_8.html#E4:20_1_1_4_1_1_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>21</sub></font>, <font color="Maroon">c<sub>22</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="E7:20_1_1_4_1_1_1"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">c<sub>21</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>22</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>22</sub></font> )
 <b>by </b><i><a class="ref" href="vectsp_5.html#T5">VECTSP_5:5</a></i>;<br/>
<a NAME="E8:20_1_1_4_1_1_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="vectsp_8.html#E7:20_1_1_4_1_1_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E9:20_1_1_4_1_1_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>22</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="vectsp_8.html#E1:20"><i><font color="Green">E14</font></i></a>, <a class="txt" href="vectsp_8.html#E4:20_1_1_4_1_1_1"><i><font color="Green">E38</font></i></a>, <a class="txt" href="vectsp_8.html#E7:20_1_1_4_1_1_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="mod_2.html#D5">MOD_2:def 5</a></i>;<br/>
<a NAME="E10:20_1_1_4_1_1_1"/>
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="vectsp_8.html#E7:20_1_1_4_1_1_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E11:20_1_1_4_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/>
<a NAME="E12:20_1_1_4_1_1_1"/><b>then </b><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>21</sub></font> <a href="rlvect_1.html#R1">in</a>  <a href="vectsp_7.html#K1">Lin</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>7</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="vectsp_7.html#T13">VECTSP_7:13</a></i>;<br/>
<a NAME="E13:20_1_1_4_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="vectsp_8.html#E8:20_1_1_4_1_1_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/>
<a NAME="E14:20_1_1_4_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>22</sub></font> <a href="rlvect_1.html#R1">in</a>  <a href="vectsp_7.html#K1">Lin</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>8</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="vectsp_7.html#T13">VECTSP_7:13</a></i>;<br/>
<a NAME="E15:20_1_1_4_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>19</sub></font> <a href="rlvect_1.html#R1">in</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="vectsp_5.html#K1">+</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="vectsp_8.html#E9:20_1_1_4_1_1_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="vectsp_8.html#E12:20_1_1_4_1_1_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="vectsp_5.html#T5">VECTSP_5:5</a></i>;<br/>
<b>hence </b><a NAME="E16:20_1_1_4_1_1_1"/>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="vectsp_5.html#K1">+</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="vectsp_8.html#E20:20_1_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="vectsp_8.html#E29:20_1_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>


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

<b>thus </b><a NAME="E2:20_1_1_4_1_1"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="vectsp_5.html#K1">+</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="vectsp_5.html#K1">+</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_4_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>18</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="E1:20_1_1_4_1_1_2"/>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="vectsp_5.html#K1">+</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
 ;<br/>

<a NAME="E2:20_1_1_4_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>18</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>12</sub></font> <a href="vectsp_5.html#K1">+</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>19</sub></font>, <font color="Maroon">c<sub>20</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E4:20_1_1_4_1_1_2"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">c<sub>19</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>20</sub></font> )
 <b>by </b><i><a class="ref" href="vectsp_5.html#T5">VECTSP_5:5</a></i>;<br/>
<a NAME="E5:20_1_1_4_1_1_2"/>
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="vectsp_8.html#E16:20_1_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="vectsp_8.html#E17:20_1_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="vectsp_8.html#E20:20_1_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="vectsp_8.html#E4:20_1_1_4_1_1_2"><i><font color="Green">E37</font></i></a>, <a class="ref" href="vectsp_8.html#T10" target="_self">Th10</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>21</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="E7:20_1_1_4_1_1_2"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>21</sub></font> )
 <b>by </b><i><a class="ref" href="funct_2.html#T116">FUNCT_2:116</a></i>;<br/>
<a NAME="E8:20_1_1_4_1_1_2"/>
<font color="Maroon">c<sub>21</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="vectsp_8.html#E7:20_1_1_4_1_1_2"><i><font color="Green">E38</font></i></a>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E9:20_1_1_4_1_1_2"/><b>then </b><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>21</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>7</sub></font> <a href="vectsp_5.html#K1">+</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="ref" href="vectsp_5.html#T6">VECTSP_5:6</a></i>;<br/>
<a NAME="E10:20_1_1_4_1_1_2"/>
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="vectsp_8.html#E25:20_1_1"><i><font color="Green">E30</font></i></a>, <a class="txt" href="vectsp_8.html#E26:20_1_1"><i><font color="Green">E31</font></i></a>, <a class="txt" href="vectsp_8.html#E29:20_1_1"><i><font color="Green">E33</font></i></a>, <a class="txt" href="vectsp_8.html#E4:20_1_1_4_1_1_2"><i><font color="Green">E37</font></i></a>, <a class="ref" href="vectsp_8.html#T10" target="_self">Th10</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>22</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="E12:20_1_1_4_1_1_2"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>22</sub></font> )
 <b>by </b><i><a class="ref" href="funct_2.html#T116">FUNCT_2:116</a></i>;<br/>
<a NAME="E13:20_1_1_4_1_1_2"/>
<font color="Maroon">c<sub>22</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="vectsp_8.html#E12:20_1_1_4_1_1_2"><i><font color="Green">E40</font></i></a>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E14:20_1_1_4_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>22</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>7</sub></font> <a href="vectsp_5.html#K1">+</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="ref" href="vectsp_5.html#T6">VECTSP_5:6</a></i>;<br/>
<a NAME="E15:20_1_1_4_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>22</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>21</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>7</sub></font> <a href="vectsp_5.html#K1">+</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="vectsp_8.html#E9:20_1_1_4_1_1_2"><i><font color="Green">E39</font></i></a>, <a class="ref" href="vectsp_4.html#T28">VECTSP_4:28</a></i>;<br/>
<a NAME="E16:20_1_1_4_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>22</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="vectsp_5.html#K1">+</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E17:20_1_1_4_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="vectsp_5.html#K1">+</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/>
<b>hence </b><a NAME="E18:20_1_1_4_1_1_2"/>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="vectsp_5.html#K1">+</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="vectsp_8.html#E1:20"><i><font color="Green">E14</font></i></a>, <a class="txt" href="vectsp_8.html#E4:20_1_1_4_1_1_2"><i><font color="Green">E37</font></i></a>, <a class="txt" href="vectsp_8.html#E7:20_1_1_4_1_1_2"><i><font color="Green">E38</font></i></a>, <a class="txt" href="vectsp_8.html#E12:20_1_1_4_1_1_2"><i><font color="Green">E40</font></i></a>, <a class="ref" href="mod_2.html#D5">MOD_2:def 5</a></i>;<br/>


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


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:20_1_1_4_1"/>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="txt" href="vectsp_8.html#E30:20_1_1"><i><font color="Green">E34</font></i></a>, <a class="txt" href="vectsp_8.html#E2:20_1_1_4_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="vectsp_8.html#T10" target="_self">Th10</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:20_1_1_4"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>16</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>15</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_4_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>17</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="E1:20_1_1_4_2"/>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>16</sub></font>
 ;<br/>

<a NAME="E2:20_1_1_4_2"/><b>then </b>
<font color="Maroon">c<sub>17</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>12</sub></font> <a href="vectsp_5.html#K1">+</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>18</sub></font>, <font color="Maroon">c<sub>19</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E4:20_1_1_4_2"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">c<sub>18</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>19</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>19</sub></font> )
 <b>by </b><i><a class="ref" href="vectsp_5.html#T5">VECTSP_5:5</a></i>;<br/>
<a NAME="E5:20_1_1_4_2"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="vectsp_8.html#E19:20_1_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="vectsp_8.html#E4:20_1_1_4_2"><i><font color="Green">E36</font></i></a>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E6:20_1_1_4_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="vectsp_5.html#K1">+</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_4_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>20</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="E1:20_1_1_4_2_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>7</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>21</sub></font> = <font color="Maroon">c<sub>20</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>3</sub></font> ;<br/>
<b>consider </b><font color="Maroon">c<sub>22</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="E4:20_1_1_4_2_1"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>22</sub></font> )
 <b>by </b><i><a class="txt" href="vectsp_8.html#E1:20_1_1_4_2_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="funct_2.html#T116">FUNCT_2:116</a></i>;<br/>
<a NAME="E5:20_1_1_4_2_1"/>
the <a href="struct_0.html#U1">carrier</a> of <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 <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="vectsp_5.html#K1">+</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_4_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>23</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="E1:20_1_1_4_2_1_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>7</sub></font>
 ;<br/>

<a NAME="E2:20_1_1_4_2_1_1"/><b>then </b><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>23</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E3:20_1_1_4_2_1_1"/>
the <a href="struct_0.html#U1">carrier</a> of <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="vectsp_4.html#D2">VECTSP_4:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>24</sub></font> = <font color="Maroon">c<sub>23</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="vectsp_8.html#E1:20_1_1_4_2_1_1"><i><font color="Green">E40</font></i></a></i>;<br/>
<a NAME="E5:20_1_1_4_2_1_1"/>
<font color="Maroon">c<sub>24</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>7</sub></font> <a href="vectsp_5.html#K1">+</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="vectsp_8.html#E2:20_1_1_4_2_1_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="vectsp_5.html#T6">VECTSP_5:6</a></i>;<br/>
<b>hence </b><a NAME="E6:20_1_1_4_2_1_1"/>
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="vectsp_5.html#K1">+</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:20_1_1_4_2_1"/>
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="vectsp_5.html#K1">+</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="vectsp_8.html#E4:20_1_1_4_2_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:20_1_1_4_2"/><b>then </b><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>15</sub></font>
 
<b>by </b><i><a class="txt" href="vectsp_8.html#E5:20_1_1_4_2"><i><font color="Green">E37</font></i></a>, <a class="ref" href="vectsp_7.html#T13">VECTSP_7:13</a></i>;<br/>
<a NAME="E8:20_1_1_4_2"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="vectsp_8.html#E28:20_1_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="vectsp_8.html#E4:20_1_1_4_2"><i><font color="Green">E36</font></i></a>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E9:20_1_1_4_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="vectsp_5.html#K1">+</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_4_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>20</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="E1:20_1_1_4_2_2"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>8</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>21</sub></font> = <font color="Maroon">c<sub>20</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>3</sub></font> ;<br/>
<b>consider </b><font color="Maroon">c<sub>22</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="E4:20_1_1_4_2_2"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>22</sub></font> )
 <b>by </b><i><a class="txt" href="vectsp_8.html#E1:20_1_1_4_2_2"><i><font color="Green">E40</font></i></a>, <a class="ref" href="funct_2.html#T116">FUNCT_2:116</a></i>;<br/>
<a NAME="E5:20_1_1_4_2_2"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="vectsp_5.html#K1">+</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_4_2_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>23</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="E1:20_1_1_4_2_2_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>8</sub></font>
 ;<br/>

<a NAME="E2:20_1_1_4_2_2_1"/><b>then </b><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>23</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E3:20_1_1_4_2_2_1"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>8</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="vectsp_4.html#D2">VECTSP_4:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>24</sub></font> = <font color="Maroon">c<sub>23</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="vectsp_8.html#E1:20_1_1_4_2_2_1"><i><font color="Green">E42</font></i></a></i>;<br/>
<a NAME="E5:20_1_1_4_2_2_1"/>
<font color="Maroon">c<sub>24</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>7</sub></font> <a href="vectsp_5.html#K1">+</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="vectsp_8.html#E2:20_1_1_4_2_2_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="vectsp_5.html#T6">VECTSP_5:6</a></i>;<br/>
<b>hence </b><a NAME="E6:20_1_1_4_2_2_1"/>
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="vectsp_5.html#K1">+</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:20_1_1_4_2_2"/>
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="vectsp_5.html#K1">+</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="vectsp_8.html#E4:20_1_1_4_2_2"><i><font color="Green">E41</font></i></a>, <a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:20_1_1_4_2"/><b>then </b>
<font color="Maroon">c<sub>19</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>15</sub></font>
 
<b>by </b><i><a class="txt" href="vectsp_8.html#E8:20_1_1_4_2"><i><font color="Green">E39</font></i></a>, <a class="ref" href="vectsp_7.html#T13">VECTSP_7:13</a></i>;<br/>
<a NAME="E11:20_1_1_4_2"/><b>then </b>
<font color="Maroon">c<sub>17</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>15</sub></font>
 
<b>by </b><i><a class="txt" href="vectsp_8.html#E4:20_1_1_4_2"><i><font color="Green">E36</font></i></a>, <a class="txt" href="vectsp_8.html#E7:20_1_1_4_2"><i><font color="Green">E38</font></i></a>, <a class="ref" href="vectsp_4.html#T28">VECTSP_4:28</a></i>;<br/>
<b>hence </b><a NAME="E12:20_1_1_4_2"/>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:20_1_1_4"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>15</sub></font>
 
<b>by </b><i><a class="txt" href="vectsp_8.html#E1:20_1_1_4"><i><font color="Green">E35</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<b>hence </b><a NAME="E4:20_1_1_4"/>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="ref" href="vectsp_4.html#T37">VECTSP_4:37</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E34:20_1_1"/>
<span class="p1">(<span class="default"><a href="vectsp_8.html#K6" target="_self">FuncLatt</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="vectsp_8.html#K6" target="_self">FuncLatt</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="vectsp_8.html#K6" target="_self">FuncLatt</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="vectsp_8.html#E5:20_1_1"><i><font color="Green">E17</font></i></a>, <a class="txt" href="vectsp_8.html#E7:20_1_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="vectsp_8.html#E9:20_1_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="vectsp_8.html#E10:20_1_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="vectsp_8.html#E20:20_1_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="vectsp_8.html#E29:20_1_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="vectsp_8.html#T7" target="_self">Th7</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:20_1"/>
<span class="p1">(<span class="default"><a href="vectsp_8.html#K6" target="_self">FuncLatt</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="vectsp_8.html#K6" target="_self">FuncLatt</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="vectsp_8.html#K6" target="_self">FuncLatt</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:20"/>
 <a href="vectsp_8.html#K6" target="_self">FuncLatt</a> <font color="Maroon">c<sub>4</sub></font> is    <a href="vectsp_8.html#M4" target="_self">sup-Semilattice-Homomorphism</a> of  <a href="vectsp_8.html#K1" target="_self">lattice</a> <font color="Maroon">c<sub>2</sub></font>, <a href="vectsp_8.html#K1" target="_self">lattice</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>


</div>
