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

<b>let </b><font color="Maroon">K</font> be   <a href="vectsp_1.html#NM2">Field</a>;<br/><b>let </b><font color="Maroon">V</font> be   <a href="vectsp_1.html#NM6">VectSp</a> of <font color="Maroon">K</font>;<br/><b>let </b><font color="Maroon">X</font> be    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">fi</font> be   <a href="hahnban1.html#NM2">linear-Functional</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">v</font> be   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">y</font> be   <a href="vectsp_1.html#NM5">Vector</a> of <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="vectsp_5.html#K1">+</a> <span class="p2">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span>;<br/>











<b>assume </b><b>that </b><br/><a NAME="E1:44"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 <b>and </b><br/><a NAME="E2:44"/><i><font color="Green">E35</font></i>: 
not <font color="Maroon">v</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">X</font>
 ;<br/>

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

<b>reconsider </b><font color="Maroon">W1</font> = <font color="Maroon">X</font> as    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">X</font> <a href="vectsp_5.html#K1">+</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <b>by </b><i><a class="ref" href="vectsp_5.html#T11">VECTSP_5:11</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="vectsp_5.html#K1">+</a> <span class="p2">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span>,  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">K</font>] <b>means </b>for <font color="Olive">x</font> being  <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">X</font><br/> for <font color="Olive">s</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">K</font>  st <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="vectsp_5.html#K4">|--</a> <font color="Maroon">W1</font>,<span class="p2">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> <a href="domain_1.html#K2">`1</a>  <a href="hidden.html#R1">=</a> <font color="Olive">x</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="vectsp_5.html#K4">|--</a> <font color="Maroon">W1</font>,<span class="p2">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> <a href="domain_1.html#K3">`2</a>  <a href="hidden.html#R1">=</a> <font color="Olive">s</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v</font> holds <br/><font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">fi</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Olive">s</font> <a href="group_1.html#K10">*</a> <font color="Maroon">r</font></span>)</span>;<br/>
<a NAME="E4:44"/><i><font color="Green">E42</font></i>: 
for <font color="Olive">e</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="vectsp_5.html#K1">+</a> <span class="p2">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span>  ex <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">K</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">e</font>,<font color="Olive">a</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">e</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="vectsp_5.html#K1">+</a> <span class="p2">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span>;<br/>

<b>consider </b><font color="Maroon">x</font> being   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">X</font>, <font color="Maroon">s</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">K</font><b> such that </b><br/><a NAME="E2:44_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">e</font> <a href="vectsp_5.html#K4">|--</a> <font color="Maroon">W1</font>,<span class="p1">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v</font></span>)</span></span><a href="domain_1.html#K1">]</a></span>
 <b>by </b><i>, , </i>;<br/>
<b>take </b><font color="Maroon">u</font> = <span class="p1">(<span class="default"><font color="Maroon">fi</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="group_1.html#K10">*</a> <font color="Maroon">r</font></span>)</span>;<br/>

<b>let </b><font color="Maroon">x'</font> be   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">t</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">K</font>;<br/>



<b>assume </b><a NAME="E3:44_1"/><i><font color="Green">E47</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="vectsp_5.html#K4">|--</a> <font color="Maroon">W1</font>,<span class="p2">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> <a href="domain_1.html#K2">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">x'</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="vectsp_5.html#K4">|--</a> <font color="Maroon">W1</font>,<span class="p2">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> <a href="domain_1.html#K3">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v</font> )
 ;<br/>

<a NAME="E4:44_1"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font>
 
<b>by </b><i>, <a class="ref" href="vectsp_4.html#T25">VECTSP_4:25</a></i>;<br/>
<a NAME="E5:44_1"/>
<font color="Maroon">s</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v</font>
 
<b>by </b><i>, , <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>
<a NAME="E6:44_1"/><b>then </b>
<font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E7:44_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">fi</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x'</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="group_1.html#K10">*</a> <font color="Maroon">r</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">f</font> being   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="vectsp_5.html#K1">+</a> <span class="p2">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">K</font><b> such that </b><br/><a NAME="E6:44"/><i><font color="Green">E69</font></i>: 
for <font color="Olive">e</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="vectsp_5.html#K1">+</a> <span class="p2">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">e</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">e</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>(<a class="txt" href="vectsp10.html#E1:44"><i><font color="Green">E29</font></i></a>);<br/></i>
<a NAME="E7:44"/><i><font color="Green">E74</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">fi</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<div><i><font color="Green">E75</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="44_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:44_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">fi</font>
 ;<br/><b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">a</font> as   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">X</font> ;<br/><a NAME="E3:44_2"/>
<font color="Maroon">X</font> is    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">X</font> <a href="vectsp_5.html#K1">+</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="vectsp_5.html#T11">VECTSP_5:11</a></i>;<br/><a NAME="E4:44_2"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</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">X</font> <a href="vectsp_5.html#K1">+</a> <span class="p2">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span>
 <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">v1</font> = <font color="Maroon">x</font> as   <a href="vectsp_1.html#NM5">Vector</a> of <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="vectsp_5.html#K1">+</a> <span class="p2">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><a NAME="E6:44_2"/>
<font color="Maroon">v1</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/><b>then </b><font color="Maroon">v1</font> <a href="vectsp_5.html#K4">|--</a> <font color="Maroon">W1</font>,<span class="p1">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> = 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">v1</font>,<span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font></span>)</span></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i>, , </i>
<br/>.= 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">v1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v</font></span>)</span></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i/>
;<br/><a NAME="E8:44_2"/><b>then </b><i><font color="Green">E76</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">v1</font> <a href="vectsp_5.html#K4">|--</a> <font color="Maroon">W1</font>,<span class="p2">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> <a href="domain_1.html#K2">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">v1</font> <a href="vectsp_5.html#K4">|--</a> <font color="Maroon">W1</font>,<span class="p2">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> <a href="domain_1.html#K3">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v</font> )
 <b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/><b>thus </b><font color="Maroon">fi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> = 
<span class="p1">(<span class="default"><font color="Maroon">fi</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">fi</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">r</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#T39">VECTSP_1:39</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>
<b>by </b><i>, </i>
;<br/></div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as   <a href="hahnban1.html#NM1">Functional</a> of <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="vectsp_5.html#K1">+</a> <span class="p2">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> ;<br/>
<a NAME="E10:44"/><i><font color="Green">E100</font></i>: 
<font color="Maroon">f</font> is <a href="hahnban1.html#V1">additive</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">v1</font> be   <a href="vectsp_1.html#NM5">Vector</a> of <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="vectsp_5.html#K1">+</a> <span class="p2">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span>, <font color="Maroon">v2</font> be   <a href="vectsp_1.html#NM5">Vector</a> of <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="vectsp_5.html#K1">+</a> <span class="p2">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="hahnban1.html#D11">HAHNBAN1:def 11</a><br/>

<b>consider </b><font color="Maroon">x1</font> being   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">X</font>, <font color="Maroon">s1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">K</font><b> such that </b><br/><a NAME="E2:44_3"/><i><font color="Green">E101</font></i>: 
<font color="Maroon">v1</font> <a href="vectsp_5.html#K4">|--</a> <font color="Maroon">W1</font>,<span class="p1">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x1</font>,<span class="p2">(<span class="default"><font color="Maroon">s1</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v</font></span>)</span></span><a href="domain_1.html#K1">]</a></span>
 <b>by </b><i>, , </i>;<br/>
<b>consider </b><font color="Maroon">x2</font> being   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">X</font>, <font color="Maroon">s2</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">K</font><b> such that </b><br/><a NAME="E4:44_3"/><i><font color="Green">E104</font></i>: 
<font color="Maroon">v2</font> <a href="vectsp_5.html#K4">|--</a> <font color="Maroon">W1</font>,<span class="p1">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x2</font>,<span class="p2">(<span class="default"><font color="Maroon">s2</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v</font></span>)</span></span><a href="domain_1.html#K1">]</a></span>
 <b>by </b><i>, , </i>;<br/>
<a NAME="E5:44_3"/><i><font color="Green">E105</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">v1</font> <a href="vectsp_5.html#K4">|--</a> <font color="Maroon">W1</font>,<span class="p2">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> <a href="domain_1.html#K2">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">v1</font> <a href="vectsp_5.html#K4">|--</a> <font color="Maroon">W1</font>,<span class="p2">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> <a href="domain_1.html#K3">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">s1</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v</font> )
 
<b>by </b><i>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>
<a NAME="E6:44_3"/><i><font color="Green">E106</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">v2</font> <a href="vectsp_5.html#K4">|--</a> <font color="Maroon">W1</font>,<span class="p2">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> <a href="domain_1.html#K2">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">v2</font> <a href="vectsp_5.html#K4">|--</a> <font color="Maroon">W1</font>,<span class="p2">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> <a href="domain_1.html#K3">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">s2</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v</font> )
 
<b>by </b><i><a class="txt" href="vectsp10.html#E2:44"><i><font color="Green">E35</font></i></a>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>
<a NAME="E7:44_3"/>
<span class="p1">(<span class="default"><font color="Maroon">v1</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">v2</font></span>)</span> <a href="vectsp_5.html#K4">|--</a> <font color="Maroon">W1</font>,<span class="p1">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x1</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">x2</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">s1</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">s2</font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v</font></span>)</span></span><a href="domain_1.html#K1">]</a></span>
 
<b>by </b><i>, , , <a class="txt" href="vectsp10.html#E2:44"><i><font color="Green">E35</font></i></a>, </i>;<br/>
<a NAME="E8:44_3"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">v1</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">v2</font></span>)</span> <a href="vectsp_5.html#K4">|--</a> <font color="Maroon">W1</font>,<span class="p2">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> <a href="domain_1.html#K2">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">x2</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">v1</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">v2</font></span>)</span> <a href="vectsp_5.html#K4">|--</a> <font color="Maroon">W1</font>,<span class="p2">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> <a href="domain_1.html#K3">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">s2</font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v</font> )
 
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>
<b>hence </b><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">v1</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">v2</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">fi</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">x1</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">x2</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s1</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">s2</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">r</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">fi</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">x1</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">x2</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s1</font> <a href="group_1.html#K10">*</a> <font color="Maroon">r</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">s2</font> <a href="group_1.html#K10">*</a> <font color="Maroon">r</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D12">VECTSP_1:def 12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">fi</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">fi</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x2</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s1</font> <a href="group_1.html#K10">*</a> <font color="Maroon">r</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">s2</font> <a href="group_1.html#K10">*</a> <font color="Maroon">r</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="hahnban1.html#D11">HAHNBAN1:def 11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">fi</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><font color="Maroon">fi</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x2</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">s1</font> <a href="group_1.html#K10">*</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">s2</font> <a href="group_1.html#K10">*</a> <font color="Maroon">r</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">fi</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><font color="Maroon">s1</font> <a href="group_1.html#K10">*</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">fi</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x2</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">s2</font> <a href="group_1.html#K10">*</a> <font color="Maroon">r</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">fi</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">s1</font> <a href="group_1.html#K10">*</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">fi</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x2</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">s2</font> <a href="group_1.html#K10">*</a> <font color="Maroon">r</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">fi</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x2</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">s2</font> <a href="group_1.html#K10">*</a> <font color="Maroon">r</font></span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v2</font></span>)</span>
<b>by </b><i>, <a class="ref" href="vectsp10.html#T2" target="_self">Th2</a></i>
;<br/><br/>


</div><b>end;</b></div>
<a NAME="E11:44"/>
<font color="Maroon">f</font> is <a href="hahnban1.html#V2">homogeneous</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">v0</font> be   <a href="vectsp_1.html#NM5">Vector</a> of <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="vectsp_5.html#K1">+</a> <span class="p2">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="hahnban1.html#D12">HAHNBAN1:def 12</a><br/><b>let </b><font color="Maroon">t</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">K</font>;<br/>



<b>consider </b><font color="Maroon">x0</font> being   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">X</font>, <font color="Maroon">s0</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">K</font><b> such that </b><br/><a NAME="E2:44_4"/><i><font color="Green">E107</font></i>: 
<font color="Maroon">v0</font> <a href="vectsp_5.html#K4">|--</a> <font color="Maroon">W1</font>,<span class="p1">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x0</font>,<span class="p2">(<span class="default"><font color="Maroon">s0</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v</font></span>)</span></span><a href="domain_1.html#K1">]</a></span>
 <b>by </b><i>, , </i>;<br/>
<a NAME="E3:44_4"/><i><font color="Green">E108</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">v0</font> <a href="vectsp_5.html#K4">|--</a> <font color="Maroon">W1</font>,<span class="p2">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> <a href="domain_1.html#K2">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">x0</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">v0</font> <a href="vectsp_5.html#K4">|--</a> <font color="Maroon">W1</font>,<span class="p2">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> <a href="domain_1.html#K3">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">s0</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v</font> )
 
<b>by </b><i>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>
<a NAME="E4:44_4"/>
<span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v0</font></span>)</span> <a href="vectsp_5.html#K4">|--</a> <font color="Maroon">W1</font>,<span class="p1">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">x0</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="group_1.html#K10">*</a> <font color="Maroon">s0</font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v</font></span>)</span></span><a href="domain_1.html#K1">]</a></span>
 
<b>by </b><i>, , , </i>;<br/>
<a NAME="E5:44_4"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v0</font></span>)</span> <a href="vectsp_5.html#K4">|--</a> <font color="Maroon">W1</font>,<span class="p2">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> <a href="domain_1.html#K2">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">x0</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v0</font></span>)</span> <a href="vectsp_5.html#K4">|--</a> <font color="Maroon">W1</font>,<span class="p2">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> <a href="domain_1.html#K3">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="group_1.html#K10">*</a> <font color="Maroon">s0</font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v</font> )
 
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>
<b>hence </b><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v0</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">fi</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="group_1.html#K10">*</a> <font color="Maroon">s0</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">r</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><font color="Maroon">fi</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="group_1.html#K10">*</a> <font color="Maroon">s0</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">r</font></span>)</span>
<b>by </b><i><a class="ref" href="hahnban1.html#D12">HAHNBAN1:def 12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><font color="Maroon">fi</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><font color="Maroon">s0</font> <a href="group_1.html#K10">*</a> <font color="Maroon">r</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>
<br/>.= 
<font color="Maroon">t</font> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">fi</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x0</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">s0</font> <a href="group_1.html#K10">*</a> <font color="Maroon">r</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D11">VECTSP_1:def 11</a></i>
<br/>.= 
<font color="Maroon">t</font> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v0</font></span>)</span>
<b>by </b><i>, <a class="txt" href="vectsp10.html#E2:44"><i><font color="Green">E35</font></i></a></i>
;<br/><br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as   <a href="hahnban1.html#NM2">linear-Functional</a> of <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="vectsp_5.html#K1">+</a> <span class="p2">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">f</font>
;<br/>

<a NAME="E13:44"/><i><font color="Green">E109</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 <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="vectsp_5.html#K1">+</a> <span class="p2">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E14:44"/>
<font color="Maroon">X</font> is    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">X</font> <a href="vectsp_5.html#K1">+</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="vectsp_5.html#T11">VECTSP_5:11</a></i>;<br/>
<a NAME="E15:44"/><b>then </b>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">fi</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="E16:44"/><b>then </b>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">fi</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K8">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/>
<b>hence </b><a NAME="E17:44"/>
<font color="Maroon">f</font> <a href="relset_1.html#K8">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fi</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>

<a NAME="E18:44"/><i><font color="Green">E120</font></i>: 
<font color="Maroon">y</font> <a href="vectsp_5.html#K4">|--</a> <font color="Maroon">W1</font>,<span class="p1">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">W1</font></span>)</span>,<font color="Maroon">y</font></span><a href="domain_1.html#K1">]</a></span>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E19:44"/><b>then </b><i><font color="Green">E121</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="vectsp_5.html#K4">|--</a> <font color="Maroon">W1</font>,<span class="p2">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> <a href="domain_1.html#K2">`1</a>  <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>
<span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="vectsp_5.html#K4">|--</a> <font color="Maroon">W1</font>,<span class="p2">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> <a href="domain_1.html#K3">`2</a>  = 
<font color="Maroon">y</font>
<b>by </b><i><a class="txt" href="vectsp10.html#E2:44"><i><font color="Green">E35</font></i></a>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">K</font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v</font>
<b>by </b><i>, <a class="ref" href="vectsp_1.html#D26">VECTSP_1:def 26</a></i>
;<br/>
<b>hence </b><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> = 
<span class="p1">(<span class="default"><font color="Maroon">fi</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">K</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">r</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">K</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">r</font></span>)</span>
<b>by </b><i><a class="ref" href="hahnban1.html#D13">HAHNBAN1:def 13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">r</font>
<b>by </b><i><a class="ref" href="vectsp_1.html#D19">VECTSP_1:def 19</a></i>
<br/>.= 
<font color="Maroon">r</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>
;<br/><br/>


</div>
