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

<b>let </b><font color="Maroon">V</font> be   <a href="rlvect_1.html#NM2">RealLinearSpace</a>;<br/><b>let </b><font color="Maroon">V1</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">V</font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:37"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">V1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>and </b><br/><a NAME="E2:37"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">V1</font> is <a href="rlsub_1.html#V1" target="_self">lineary-closed</a>
 ;<br/>

<b>reconsider </b><font color="Maroon">D</font> = <font color="Maroon">V1</font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">d1</font> =  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> <b>by </b><i>, </i>;<br/>
<b>set </b><font color="Maroon">A</font> = the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">V</font> <a href="realset1.html#K1">||</a> <font color="Maroon">V1</font>;<br/>
<b>set </b><font color="Maroon">M</font> = the <a href="rlvect_1.html#U2">Mult</a> of <font color="Maroon">V</font> <a href="relat_1.html#K7">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K1">REAL</a> ,<font color="Maroon">V1</font></span><a href="zfmisc_1.html#K2">:]</a></span>;<br/>
<b>set </b><font color="Maroon">VV</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>;<br/>
<a NAME="E5:37"/>
 <a href="relat_1.html#K1">dom</a> the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">V</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E6:37"/><b>then </b>
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">V</font> <a href="realset1.html#K1">||</a> <font color="Maroon">V1</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="subset_1.html#K9">/\</a> <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">V1</font>,<font color="Maroon">V1</font></span><a href="mcart_1.html#K12">:]</a></span> &amp; <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">V1</font>,<font color="Maroon">V1</font></span><a href="mcart_1.html#K12">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font></span><a href="zfmisc_1.html#K2">:]</a></span> )
 
<b>by </b><i><a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>;<br/>
<a NAME="E7:37"/><b>then </b><i><font color="Green">E35</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">V</font> <a href="realset1.html#K1">||</a> <font color="Maroon">V1</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">D</font>,<font color="Maroon">D</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/>
<a NAME="E8:37"/>
 <a href="relat_1.html#K1">dom</a> the <a href="rlvect_1.html#U2">Mult</a> of <font color="Maroon">V</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K1">REAL</a> ,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E9:37"/><b>then </b>
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="rlvect_1.html#U2">Mult</a> of <font color="Maroon">V</font> <a href="relat_1.html#K7">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K1">REAL</a> ,<font color="Maroon">V1</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K1">REAL</a> ,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#K3">/\</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K1">REAL</a> ,<font color="Maroon">V1</font></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K1">REAL</a> ,<font color="Maroon">V1</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K1">REAL</a> ,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font></span><a href="zfmisc_1.html#K2">:]</a></span> )
 
<b>by </b><i><a class="ref" href="relat_1.html#T90">RELAT_1:90</a>, <a class="ref" href="zfmisc_1.html#T118">ZFMISC_1:118</a></i>;<br/>
<a NAME="E10:37"/><b>then </b><i><font color="Green">E37</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="rlvect_1.html#U2">Mult</a> of <font color="Maroon">V</font> <a href="relat_1.html#K7">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K1">REAL</a> ,<font color="Maroon">V1</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K1">REAL</a> ,<font color="Maroon">D</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/>
<a NAME="E11:37"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">D</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default">the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">V</font> <a href="realset1.html#K1">||</a> <font color="Maroon">V1</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>thus </b><a NAME="E1:37_1_1"/>
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> implies  ex <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">V</font> <a href="realset1.html#K1">||</a> <font color="Maroon">V1</font></span>)</span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">V</font> <a href="realset1.html#K1">||</a> <font color="Maroon">V1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:37_1_1_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">v1</font> = <font color="Maroon">y</font>, <font color="Maroon">v0</font> = <font color="Maroon">d1</font> as    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font> ;<br/>
<a NAME="E3:37_1_1_1"/><i><font color="Green">E40</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">d1</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">D</font>,<font color="Maroon">D</font></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">d1</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font></span><a href="zfmisc_1.html#K2">:]</a></span> )
 
<b>by </b><i><a class="txt" href="rlsub_1.html#E1:37"><i><font color="Green">E32</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>then </b><span class="p1">(<span class="default">the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">V</font> <a href="realset1.html#K1">||</a> <font color="Maroon">V1</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">d1</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> = 
<font color="Maroon">v0</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">v1</font>
<b>by </b><i><a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon">y</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>
;<br/>
<b>hence </b><a NAME="E5:37_1_1_1"/>
 ex <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">V</font> <a href="realset1.html#K1">||</a> <font color="Maroon">V1</font></span>)</span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">V</font> <a href="realset1.html#K1">||</a> <font color="Maroon">V1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> )
 <b>by </b><i>, <a class="txt" href="rlsub_1.html#E2:37"><i><font color="Green">E33</font></i></a></i>;<br/>


</div><b>end;</b></div><b>given </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><a NAME="E2:37_1_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">V</font> <a href="realset1.html#K1">||</a> <font color="Maroon">V1</font></span>)</span>
 <b>and </b><br/><a NAME="E3:37_1_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">V</font> <a href="realset1.html#K1">||</a> <font color="Maroon">V1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 ;<br/><b>consider </b><font color="Maroon">x1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">x2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:37_1_1"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> )
 <b>and </b><br/><a NAME="E6:37_1_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i>, <a class="txt" href="rlsub_1.html#E1:37"><i><font color="Green">E32</font></i></a>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/><a NAME="E7:37_1_1"/><i><font color="Green">E45</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">V1</font>,<font color="Maroon">V1</font></span><a href="mcart_1.html#K12">:]</a></span> )
 <b>by </b><i><a class="ref" href="rlsub_1.html#T5" target="_self">Th5</a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>reconsider </b><font color="Maroon">v1</font> = <font color="Maroon">x1</font>, <font color="Maroon">v2</font> = <font color="Maroon">x2</font> as    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font> <b>by </b><i><a class="ref" href="rlsub_1.html#T5" target="_self">Th5</a></i>;<br/><a NAME="E9:37_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">v2</font>
 <b>by </b><i><a class="txt" href="rlsub_1.html#E2:37"><i><font color="Green">E33</font></i></a>, <a class="txt" href="rlsub_1.html#E7:37"><i><font color="Green">E35</font></i></a>, <a class="ref" href="rlsub_1.html#T7" target="_self">Th7</a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/><b>hence </b><a NAME="E10:37_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 <b>by </b><i>, <a class="ref" href="rlsub_1.html#T5" target="_self">Th5</a>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:37_1"/>
<font color="Maroon">D</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default">the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">V</font> <a href="realset1.html#K1">||</a> <font color="Maroon">V1</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E12:37"/><i><font color="Green">E102</font></i>: 
<font color="Maroon">D</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default">the <a href="rlvect_1.html#U2">Mult</a> of <font color="Maroon">V</font> <a href="relat_1.html#K7">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K1">REAL</a> ,<font color="Maroon">V1</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>thus </b><a NAME="E1:37_2_1"/>
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> implies  ex <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="rlvect_1.html#U2">Mult</a> of <font color="Maroon">V</font> <a href="relat_1.html#K7">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K1">REAL</a> ,<font color="Maroon">V1</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">the <a href="rlvect_1.html#U2">Mult</a> of <font color="Maroon">V</font> <a href="relat_1.html#K7">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K1">REAL</a> ,<font color="Maroon">V1</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:37_2_1_1"/><i><font color="Green">E103</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">v1</font> = <font color="Maroon">y</font> as    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font> ;<br/>
<a NAME="E3:37_2_1_1"/><i><font color="Green">E104</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K1">REAL</a> ,<font color="Maroon">D</font></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K1">REAL</a> ,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font></span><a href="zfmisc_1.html#K2">:]</a></span> )
 
<b>by </b><i><a class="txt" href="rlsub_1.html#E2:37"><i><font color="Green">E33</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>then </b><span class="p1">(<span class="default">the <a href="rlvect_1.html#U2">Mult</a> of <font color="Maroon">V</font> <a href="relat_1.html#K7">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K1">REAL</a> ,<font color="Maroon">V1</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> = 
1 <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v1</font>
<b>by </b><i><a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon">y</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>
;<br/>
<b>hence </b><a NAME="E5:37_2_1_1"/>
 ex <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="rlvect_1.html#U2">Mult</a> of <font color="Maroon">V</font> <a href="relat_1.html#K7">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K1">REAL</a> ,<font color="Maroon">V1</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">the <a href="rlvect_1.html#U2">Mult</a> of <font color="Maroon">V</font> <a href="relat_1.html#K7">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K1">REAL</a> ,<font color="Maroon">V1</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> )
 <b>by </b><i><a class="ref" href="rlsub_1.html#D1" target="_self">Def1</a>, <a class="ref" href="rlsub_1.html#T5" target="_self">Th5</a></i>;<br/>


</div><b>end;</b></div><b>given </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><a NAME="E2:37_2_1"/><i><font color="Green">E105</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="rlvect_1.html#U2">Mult</a> of <font color="Maroon">V</font> <a href="relat_1.html#K7">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K1">REAL</a> ,<font color="Maroon">V1</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span>
 <b>and </b><br/><a NAME="E3:37_2_1"/><i><font color="Green">E106</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">the <a href="rlvect_1.html#U2">Mult</a> of <font color="Maroon">V</font> <a href="relat_1.html#K7">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K1">REAL</a> ,<font color="Maroon">V1</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 ;<br/><b>consider </b><font color="Maroon">x1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">x2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:37_2_1"/><i><font color="Green">E107</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a> 
 <b>and </b><br/><a NAME="E6:37_2_1"/><i><font color="Green">E108</font></i>: 
<font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 <b>and </b><br/><a NAME="E7:37_2_1"/><i><font color="Green">E109</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="rlsub_1.html#D1" target="_self">Def1</a>, <a class="txt" href="rlsub_1.html#E2:37"><i><font color="Green">E33</font></i></a>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/><a NAME="E8:37_2_1"/><i><font color="Green">E110</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K1">REAL</a> ,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K1">REAL</a> ,<font color="Maroon">V1</font></span><a href="zfmisc_1.html#K2">:]</a></span> )
 <b>by </b><i><a class="txt" href="rlsub_1.html#E7:37"><i><font color="Green">E35</font></i></a>, <a class="ref" href="rlsub_1.html#T7" target="_self">Th7</a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>reconsider </b><font color="Maroon">v2</font> = <font color="Maroon">x2</font> as    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font> <b>by </b><i><a class="ref" href="rlsub_1.html#T7" target="_self">Th7</a></i>;<br/><b>reconsider </b><font color="Maroon">xx1</font> = <font color="Maroon">x1</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="txt" href="rlsub_1.html#E7:37"><i><font color="Green">E35</font></i></a></i>;<br/><a NAME="E11:37_2_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">xx1</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v2</font>
 <b>by </b><i><a class="ref" href="rlsub_1.html#T5" target="_self">Th5</a>, <a class="txt" href="rlsub_1.html#E10:37"><i><font color="Green">E37</font></i></a>, <a class="txt" href="rlsub_1.html#E11:37"><i><font color="Green">E38</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/><b>hence </b><a NAME="E12:37_2_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 <b>by </b><i>, <a class="ref" href="rlsub_1.html#T7" target="_self">Th7</a>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:37_2"/>
<font color="Maroon">D</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default">the <a href="rlvect_1.html#U2">Mult</a> of <font color="Maroon">V</font> <a href="relat_1.html#K7">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K1">REAL</a> ,<font color="Maroon">V1</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">A</font> = the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">V</font> <a href="realset1.html#K1">||</a> <font color="Maroon">V1</font> as   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">D</font>,<font color="Maroon">D</font></span><a href="zfmisc_1.html#K2">:]</a></span>,<font color="Maroon">D</font> <b>by </b><i>, <a class="ref" href="rlsub_1.html#T4" target="_self">Th4</a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>
<b>reconsider </b><font color="Maroon">M</font> = the <a href="rlvect_1.html#U2">Mult</a> of <font color="Maroon">V</font> <a href="relat_1.html#K7">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K1">REAL</a> ,<font color="Maroon">V1</font></span><a href="zfmisc_1.html#K2">:]</a></span> as   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K1">REAL</a> ,<font color="Maroon">D</font></span><a href="zfmisc_1.html#K2">:]</a></span>,<font color="Maroon">D</font> <b>by </b><i><a class="ref" href="rlsub_1.html#D1" target="_self">Def1</a>, <a class="txt" href="rlsub_1.html#E1:37"><i><font color="Green">E32</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>
<b>set </b><font color="Maroon">W</font> =  <a href="rlvect_1.html#G2">RLSStruct</a>(# <font color="Maroon">D</font>,<font color="Maroon">d1</font>,<font color="Maroon">A</font>,<font color="Maroon">M</font> #);<br/>
<a NAME="E15:37"/>
(  <a href="rlvect_1.html#G2">RLSStruct</a>(# <font color="Maroon">D</font>,<font color="Maroon">d1</font>,<font color="Maroon">A</font>,<font color="Maroon">M</font> #) is    <a href="rlsub_1.html#M1" target="_self">Subspace</a> of <font color="Maroon">V</font> &amp; the <a href="struct_0.html#U1">carrier</a> of <a href="rlvect_1.html#G2">RLSStruct</a>(# <font color="Maroon">D</font>,<font color="Maroon">d1</font>,<font color="Maroon">A</font>,<font color="Maroon">M</font> #) <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E16:37"/>
 ex <font color="Olive">W</font> being <a href="rlvect_1.html#V2">strict</a>  <a href="rlsub_1.html#M1" target="_self">Subspace</a> of <font color="Maroon">V</font> st <font color="Maroon">V1</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">W</font>
 ;<br/>


</div>
