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

<b>let </b><font color="Maroon">F</font> be   <a href="vectsp_1.html#NM2">Field</a>;<br/><b>let </b><font color="Maroon">A</font> be  <a href="vectsp_1.html#V11">strict</a> <a href="vectsp_1.html#NM6">VectSp</a> of <font color="Maroon">F</font>, <font color="Maroon">B</font> be  <a href="vectsp_1.html#V11">strict</a> <a href="vectsp_1.html#NM6">VectSp</a> of <font color="Maroon">F</font>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font>;<br/>





<b>assume </b><a NAME="E1:22"/><i><font color="Green">E24</font></i>: 
( <font color="Maroon">f</font> is <a href="mod_2.html#V1">linear</a> &amp; <font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a> )
 ;<br/>

<a NAME="E2:22"/>
for <font color="Olive">x1</font>, <font color="Olive">x2</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="vectsp_8.html#K6" target="_self">FuncLatt</a> <font color="Maroon">f</font></span>)</span> &amp; <font color="Olive">x2</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="vectsp_8.html#K6" target="_self">FuncLatt</a> <font color="Maroon">f</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="vectsp_8.html#K6" target="_self">FuncLatt</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x1</font> <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">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x2</font> holds <br/><font color="Olive">x1</font> <a href="hidden.html#R1">=</a> <font color="Olive">x2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">x2</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:22_1"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="vectsp_8.html#K6" target="_self">FuncLatt</a> <font color="Maroon">f</font></span>)</span> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="vectsp_8.html#K6" target="_self">FuncLatt</a> <font color="Maroon">f</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="vectsp_8.html#K6" target="_self">FuncLatt</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> <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">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font> )
 ;<br/>

<a NAME="E2:22_1"/><b>then </b>
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="vectsp_8.html#K1" target="_self">lattice</a> <font color="Maroon">A</font></span>)</span> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="vectsp_8.html#K1" target="_self">lattice</a> <font color="Maroon">A</font></span>)</span> )
 
;<br/>
<a NAME="E3:22_1"/><b>then </b><i><font color="Green">E34</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="vectsp_8.html#K2" target="_self">Subspaces</a> <font color="Maroon">A</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a>  <a href="vectsp_8.html#K2" target="_self">Subspaces</a> <font color="Maroon">A</font> )
 
;<br/>
<b>then consider </b><font color="Maroon">X1</font> being  <a href="vectsp_1.html#V11">strict</a>  <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">A</font><b> such that </b><br/><a NAME="E5:22_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">X1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</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">X2</font> being  <a href="vectsp_1.html#V11">strict</a>  <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">A</font><b> such that </b><br/><a NAME="E7:22_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">X2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</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">A1</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">B</font><b> such that </b><br/><a NAME="E9:22_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">A1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X1</font>
 ;<br/>
<b>consider </b><font color="Maroon">A2</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">B</font><b> such that </b><br/><a NAME="E11:22_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">A2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X2</font>
 ;<br/>
<a NAME="E12:22_1"/><i><font color="Green">E47</font></i>: 
( <span class="p1">(<span class="default"><a href="vectsp_8.html#K6" target="_self">FuncLatt</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">X1</font> <a href="hidden.html#R1">=</a>  <a href="vectsp_7.html#K1">Lin</a> <font color="Maroon">A1</font> &amp; <span class="p1">(<span class="default"><a href="vectsp_8.html#K6" target="_self">FuncLatt</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">X2</font> <a href="hidden.html#R1">=</a>  <a href="vectsp_7.html#K1">Lin</a> <font color="Maroon">A2</font> )
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E13:22_1"/><i><font color="Green">E48</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E14:22_1"/>
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">A</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">X1</font>
 
<b>by </b><i><a class="ref" href="vectsp_4.html#T25">VECTSP_4:25</a></i>;<br/>
<a NAME="E15:22_1"/><b>then </b><i><font color="Green">E56</font></i>: 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X1</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">y</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">B</font><b> such that </b><br/><a NAME="E17:22_1"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</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">A</font></span>)</span>
 ;<br/>
<a NAME="E18:22_1"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X1</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#E1:22"><i><font color="Green">E24</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E19:22_1"/>
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X1</font> is <a href="vectsp_4.html#V1">lineary-closed</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_1"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">BB</font> = <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X1</font>;<br/>
<a NAME="E1:22_1_1"/><i><font color="Green">E60</font></i>: 
for <font color="Olive">v</font>, <font color="Olive">u</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">B</font>  st <font color="Olive">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X1</font> &amp; <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X1</font> holds <br/><font color="Olive">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">v</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">B</font>, <font color="Maroon">u</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">B</font>;<br/>

<b>assume </b><a NAME="E1:22_1_1_1"/><i><font color="Green">E62</font></i>: 
( <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X1</font> &amp; <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X1</font> )
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font><b> such that </b><br/><a NAME="E3:22_1_1_1"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X1</font> &amp; <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="funct_2.html#T116">FUNCT_2:116</a></i>;<br/>
<a NAME="E4:22_1_1_1"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">x</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">X1</font>
 
<b>by </b><i><a class="ref" href="vectsp_8.html#D4" target="_self">Def4</a>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon">y</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font><b> such that </b><br/><a NAME="E6:22_1_1_1"/><i><font color="Green">E67</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X1</font> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> )
 <b>by </b><i>, <a class="ref" href="funct_2.html#T116">FUNCT_2:116</a></i>;<br/>
<a NAME="E7:22_1_1_1"/>
<font color="Maroon">y</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">X1</font>
 
<b>by </b><i>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E8:22_1_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">X1</font>
 
<b>by </b><i>, <a class="ref" href="vectsp_4.html#T28">VECTSP_4:28</a></i>;<br/>
<a NAME="E9:22_1_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X1</font>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E10:22_1_1_1"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X1</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/>
<b>hence </b><a NAME="E11:22_1_1_1"/>
<font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X1</font>
 <b>by </b><i>, <a class="ref" href="vectsp_8.html#D4" target="_self">Def4</a>, , <a class="ref" href="mod_2.html#D5">MOD_2:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:22_1_1"/>
for <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">F</font><br/> for <font color="Olive">v</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">B</font>  st <font color="Olive">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X1</font> holds <br/><font color="Olive">a</font> <a href="vectsp_1.html#K6">*</a> <font color="Olive">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">F</font>;<br/><b>let </b><font color="Maroon">v</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">B</font>;<br/>



<b>assume </b><a NAME="E1:22_1_1_2"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X1</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font><b> such that </b><br/><a NAME="E3:22_1_1_2"/><i><font color="Green">E68</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X1</font> &amp; <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="funct_2.html#T116">FUNCT_2:116</a></i>;<br/>
<a NAME="E4:22_1_1_2"/>
<font color="Maroon">x</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">X1</font>
 
<b>by </b><i>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E5:22_1_1_2"/><b>then </b>
<font color="Maroon">a</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">x</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">X1</font>
 
<b>by </b><i><a class="ref" href="vectsp_4.html#T29">VECTSP_4:29</a></i>;<br/>
<a NAME="E6:22_1_1_2"/><b>then </b>
<font color="Maroon">a</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X1</font>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E7:22_1_1_2"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X1</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/>
<b>hence </b><a NAME="E8:22_1_1_2"/>
<font color="Maroon">a</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X1</font>
 <b>by </b><i>, , <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="E3:22_1_1"/>
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X1</font> is <a href="vectsp_4.html#V1">lineary-closed</a>
 <b>by </b><i>, <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">B1</font> being  <a href="vectsp_1.html#V11">strict</a>  <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">B</font><b> such that </b><br/><a NAME="E21:22_1"/><i><font color="Green">E69</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X1</font>
 <b>by </b><i><a class="ref" href="vectsp_8.html#D3" target="_self">Def3</a>, <a class="ref" href="vectsp_4.html#T42">VECTSP_4:42</a></i>;<br/>
<a NAME="E22:22_1"/>
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">A</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">X2</font>
 
<b>by </b><i><a class="ref" href="vectsp_4.html#T25">VECTSP_4:25</a></i>;<br/>
<a NAME="E23:22_1"/><b>then </b><i><font color="Green">E70</font></i>: 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X2</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">y</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">B</font><b> such that </b><br/><a NAME="E25:22_1"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</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">A</font></span>)</span>
 ;<br/>
<a NAME="E26:22_1"/><i><font color="Green">E104</font></i>: 
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, , <a class="ref" href="vectsp_8.html#D4" target="_self">Def4</a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E27:22_1"/>
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X2</font> is <a href="vectsp_4.html#V1">lineary-closed</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_2"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">BB</font> = <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X2</font>;<br/>
<a NAME="E1:22_1_2"/><i><font color="Green">E105</font></i>: 
for <font color="Olive">v</font>, <font color="Olive">u</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">B</font>  st <font color="Olive">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X2</font> &amp; <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X2</font> holds <br/><font color="Olive">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">v</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">B</font>, <font color="Maroon">u</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">B</font>;<br/>

<b>assume </b><a NAME="E1:22_1_2_1"/><i><font color="Green">E106</font></i>: 
( <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X2</font> &amp; <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X2</font> )
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font><b> such that </b><br/><a NAME="E3:22_1_2_1"/><i><font color="Green">E107</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X2</font> &amp; <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="funct_2.html#T116">FUNCT_2:116</a></i>;<br/>
<a NAME="E4:22_1_2_1"/><i><font color="Green">E108</font></i>: 
<font color="Maroon">x</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">X2</font>
 
<b>by </b><i><a class="txt" href="vectsp_8.html#E1:22_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon">y</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font><b> such that </b><br/><a NAME="E6:22_1_2_1"/><i><font color="Green">E109</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X2</font> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> )
 <b>by </b><i>, <a class="ref" href="funct_2.html#T116">FUNCT_2:116</a></i>;<br/>
<a NAME="E7:22_1_2_1"/>
<font color="Maroon">y</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">X2</font>
 
<b>by </b><i><a class="txt" href="vectsp_8.html#E3:22_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E8:22_1_2_1"/><b>then </b>
<font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">X2</font>
 
<b>by </b><i>, <a class="ref" href="vectsp_4.html#T28">VECTSP_4:28</a></i>;<br/>
<a NAME="E9:22_1_2_1"/><b>then </b>
<font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X2</font>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E10:22_1_2_1"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X2</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/>
<b>hence </b><a NAME="E11:22_1_2_1"/>
<font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X2</font>
 <b>by </b><i>, <a class="txt" href="vectsp_8.html#E1:22_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="vectsp_8.html#E3:22_1"><i><font color="Green">E34</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="E2:22_1_2"/>
for <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">F</font><br/> for <font color="Olive">v</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">B</font>  st <font color="Olive">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X2</font> holds <br/><font color="Olive">a</font> <a href="vectsp_1.html#K6">*</a> <font color="Olive">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">F</font>;<br/><b>let </b><font color="Maroon">v</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">B</font>;<br/>



<b>assume </b><a NAME="E1:22_1_2_2"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X2</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font><b> such that </b><br/><a NAME="E3:22_1_2_2"/><i><font color="Green">E110</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X2</font> &amp; <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="funct_2.html#T116">FUNCT_2:116</a></i>;<br/>
<a NAME="E4:22_1_2_2"/>
<font color="Maroon">x</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">X2</font>
 
<b>by </b><i>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E5:22_1_2_2"/><b>then </b>
<font color="Maroon">a</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">x</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">X2</font>
 
<b>by </b><i><a class="ref" href="vectsp_4.html#T29">VECTSP_4:29</a></i>;<br/>
<a NAME="E6:22_1_2_2"/><b>then </b>
<font color="Maroon">a</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X2</font>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E7:22_1_2_2"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X2</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/>
<b>hence </b><a NAME="E8:22_1_2_2"/>
<font color="Maroon">a</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X2</font>
 <b>by </b><i>, , <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="E3:22_1_2"/>
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X2</font> is <a href="vectsp_4.html#V1">lineary-closed</a>
 <b>by </b><i>, <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">B2</font> being  <a href="vectsp_1.html#V11">strict</a>  <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">B</font><b> such that </b><br/><a NAME="E29:22_1"/><i><font color="Green">E111</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X2</font>
 <b>by </b><i>, <a class="ref" href="vectsp_4.html#T42">VECTSP_4:42</a></i>;<br/>
<a NAME="E30:22_1"/>
 <a href="vectsp_7.html#K1">Lin</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">B2</font>
 
<b>by </b><i>, <a class="ref" href="vectsp_7.html#T16">VECTSP_7:16</a></i>;<br/>
<a NAME="E31:22_1"/><b>then </b><i><font color="Green">E112</font></i>: 
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X2</font>
 
<b>by </b><i>, , <a class="ref" href="vectsp_8.html#D2" target="_self">Def2</a>, , , , , , <a class="ref" href="vectsp_7.html#T16">VECTSP_7:16</a></i>;<br/>
<a NAME="E32:22_1"/>
( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X1</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X2</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> )
 
<b>by </b><i>, <a class="ref" href="vectsp_4.html#D2">VECTSP_4:def 2</a></i>;<br/>
<a NAME="E33:22_1"/><b>then </b>
( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X1</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X2</font> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X2</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X1</font> )
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#T157">FUNCT_1:157</a></i>;<br/>
<a NAME="E34:22_1"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X1</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X2</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<b>hence </b><a NAME="E35:22_1"/>
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font>
 <b>by </b><i>, <a class="ref" href="vectsp_8.html#D2" target="_self">Def2</a>, <a class="ref" href="vectsp_4.html#T37">VECTSP_4:37</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:22"/>
 <a href="vectsp_8.html#K6" target="_self">FuncLatt</a> <font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i><a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/>


</div>
