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<b>let </b><font color="Maroon">GF</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">GF</font>;<br/><b>let </b><font color="Maroon">A</font> be  <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">V</font>, <font color="Maroon">B</font> be  <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">V</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>assume </b><b>that </b><br/><a NAME="E1:20"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">v</font> <a href="rlvect_1.html#R1">in</a>  <a href="vectsp_7.html#K1">Lin</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font></span>)</span>
 <b>and </b><br/><a NAME="E2:20"/><i><font color="Green">E40</font></i>: 
not <font color="Maroon">v</font> <a href="rlvect_1.html#R1">in</a>  <a href="vectsp_7.html#K1">Lin</a> <font color="Maroon">B</font>
 ;<br/>

<b>consider </b><font color="Maroon">L</font> being    <a href="vectsp_6.html#M2">Linear_Combination</a> of <font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font><b> such that </b><br/><a NAME="E4:20"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="vectsp_6.html#K4">Sum</a> <font color="Maroon">L</font>
 <b>by </b><i>, <a class="ref" href="vectsp_7.html#T12">VECTSP_7:12</a></i>;<br/>
<a NAME="E5:20"/><i><font color="Green">E43</font></i>: 
 <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">L</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font>
 
<b>by </b><i><a class="ref" href="vectsp_6.html#D7">VECTSP_6:def 7</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:20_1"/><i><font color="Green">E44</font></i>: 
for <font color="Olive">w</font> being  <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">V</font>  st <font color="Olive">w</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> holds <br/><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Olive">w</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">GF</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:20_1_1"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">L</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 ;<br/><a NAME="E2:20_1_1"/><b>then </b>
 ex <font color="Olive">u</font> being  <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">V</font> st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">u</font> &amp; <font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Olive">u</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">GF</font> )
 <b>by </b><i><a class="ref" href="vectsp_6.html#T19">VECTSP_6:19</a></i>;<br/><b>hence </b><a NAME="E3:20_1_1"/>
contradiction
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><a NAME="E3:20_1"/><b>then </b>
 <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">L</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><a NAME="E4:20_1"/><b>then </b>
 <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">L</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T73">XBOOLE_1:73</a></i>;<br/><a NAME="E5:20_1"/><b>then </b>
<font color="Maroon">L</font> is    <a href="vectsp_6.html#M2">Linear_Combination</a> of <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="vectsp_6.html#D7">VECTSP_6:def 7</a></i>;<br/><b>hence </b><a NAME="E6:20_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="vectsp_7.html#T12">VECTSP_7:12</a></i>;<br/></div><b>end;</b></div>
<b>then consider </b><font color="Maroon">w</font> being   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E8:20"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">w</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E9:20"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">w</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">GF</font>
 ;<br/>
<b>take </b>
<font color="Maroon">w</font>
;<br/>

<b>set </b><font color="Maroon">a</font> = <font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">w</font>;<br/>
<b>consider </b><font color="Maroon">F</font> being    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E11:20"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">F</font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>and </b><br/><a NAME="E12:20"/><i><font color="Green">E51</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">L</font>
 <b>and </b><br/><a NAME="E13:20"/><i><font color="Green">E63</font></i>: 
 <a href="vectsp_6.html#K4">Sum</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="vectsp_6.html#K3">(#)</a> <font color="Maroon">F</font></span>)</span>
 <b>by </b><i><a class="ref" href="vectsp_6.html#D9">VECTSP_6:def 9</a></i>;<br/>
<a NAME="E14:20"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">w</font> <a href="hidden.html#R2">in</a>  <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">L</font>
 
<b>by </b><i>, <a class="ref" href="vectsp_6.html#T19">VECTSP_6:19</a></i>;<br/>
<a NAME="E15:20"/><b>then </b><i><font color="Green">E69</font></i>: 
<font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="finseq_4.html#K6">-|</a> <font color="Maroon">w</font></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">w</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_4.html#K7">|--</a> <font color="Maroon">w</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_4.html#T66">FINSEQ_4:66</a></i>;<br/>
<b>reconsider </b><font color="Maroon">Fw1</font> = <font color="Maroon">F</font> <a href="finseq_4.html#K6">-|</a> <font color="Maroon">w</font> as    <a href="finseq_1.html#M2">FinSequence</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="finseq_4.html#T53">FINSEQ_4:53</a></i>;<br/>
<b>reconsider </b><font color="Maroon">Fw2</font> = <font color="Maroon">F</font> <a href="finseq_4.html#K7">|--</a> <font color="Maroon">w</font> as    <a href="finseq_1.html#M2">FinSequence</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="finseq_4.html#T65">FINSEQ_4:65</a></i>;<br/>
<a NAME="E18:20"/>
<font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Fw1</font> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">w</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">Fw2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="vectsp_9.html#E5"><i><font color="Green">Lemma30</font></i></a>, <a class="ref" href="finseq_1.html#T45">FINSEQ_1:45</a></i>;<br/>
<b>then </b><font color="Maroon">L</font> <a href="vectsp_6.html#K3">(#)</a> <font color="Maroon">F</font> = 
<span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="vectsp_6.html#K3">(#)</a> <font color="Maroon">Fw1</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="vectsp_6.html#K3">(#)</a> <span class="p2">(<span class="default"><span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">w</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">Fw2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_6.html#T37">VECTSP_6:37</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="vectsp_6.html#K3">(#)</a> <font color="Maroon">Fw1</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="vectsp_6.html#K3">(#)</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">w</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="vectsp_6.html#K3">(#)</a> <font color="Maroon">Fw2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_6.html#T37">VECTSP_6:37</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="vectsp_6.html#K3">(#)</a> <font color="Maroon">Fw1</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="vectsp_6.html#K3">(#)</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">w</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="vectsp_6.html#K3">(#)</a> <font color="Maroon">Fw2</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T45">FINSEQ_1:45</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="vectsp_6.html#K3">(#)</a> <font color="Maroon">Fw1</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">w</font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">w</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="vectsp_6.html#K3">(#)</a> <font color="Maroon">Fw2</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_6.html#T34">VECTSP_6:34</a></i>
;<br/>
<b>then </b><i><font color="Green">E70</font></i>:  <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="vectsp_6.html#K3">(#)</a> <font color="Maroon">F</font></span>)</span> = 
 <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="vectsp_6.html#K3">(#)</a> <font color="Maroon">Fw1</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">w</font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">w</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p3">(<span class="default"><font color="Maroon">L</font> <a href="vectsp_6.html#K3">(#)</a> <font color="Maroon">Fw2</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T45">FINSEQ_1:45</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="vectsp_6.html#K3">(#)</a> <font color="Maroon">Fw1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p2">(<span class="default"><span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">w</font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">w</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p3">(<span class="default"><font color="Maroon">L</font> <a href="vectsp_6.html#K3">(#)</a> <font color="Maroon">Fw2</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T58">RLVECT_1:58</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="vectsp_6.html#K3">(#)</a> <font color="Maroon">Fw1</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="rlvect_1.html#K9">Sum</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">w</font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">w</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p3">(<span class="default"><font color="Maroon">L</font> <a href="vectsp_6.html#K3">(#)</a> <font color="Maroon">Fw2</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T58">RLVECT_1:58</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="vectsp_6.html#K3">(#)</a> <font color="Maroon">Fw1</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="rlvect_1.html#K9">Sum</a> <span class="p3">(<span class="default"><font color="Maroon">L</font> <a href="vectsp_6.html#K3">(#)</a> <font color="Maroon">Fw2</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">w</font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">w</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T61">RLVECT_1:61</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p3">(<span class="default"><font color="Maroon">L</font> <a href="vectsp_6.html#K3">(#)</a> <font color="Maroon">Fw1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p3">(<span class="default"><font color="Maroon">L</font> <a href="vectsp_6.html#K3">(#)</a> <font color="Maroon">Fw2</font></span>)</span></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">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">w</font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">w</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#T7">VECTSP_1:7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">L</font> <a href="vectsp_6.html#K3">(#)</a> <font color="Maroon">Fw1</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p3">(<span class="default"><font color="Maroon">L</font> <a href="vectsp_6.html#K3">(#)</a> <font color="Maroon">Fw2</font></span>)</span></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">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">w</font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">w</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T58">RLVECT_1:58</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="vectsp_6.html#K3">(#)</a> <span class="p3">(<span class="default"><font color="Maroon">Fw1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">Fw2</font></span>)</span></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">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">w</font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">w</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_6.html#T37">VECTSP_6:37</a></i>
;<br/>
<b>set </b><font color="Maroon">Fw</font> = <font color="Maroon">Fw1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">Fw2</font>;<br/>
<b>consider </b><font color="Maroon">K</font> being    <a href="vectsp_6.html#M1">Linear_Combination</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E22:20"/><i><font color="Green">E74</font></i>: 
(  <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">K</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">Fw1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">Fw2</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">L</font></span>)</span> &amp; <font color="Maroon">L</font> <a href="vectsp_6.html#K3">(#)</a> <span class="p1">(<span class="default"><font color="Maroon">Fw1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">Fw2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">K</font> <a href="vectsp_6.html#K3">(#)</a> <span class="p1">(<span class="default"><font color="Maroon">Fw1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">Fw2</font></span>)</span> )
 <b>by </b><i/>;<br/>
<a NAME="E23:20"/>
<font color="Maroon">F</font> <a href="finseq_4.html#R2">just_once_values</a> <font color="Maroon">w</font>
 
<b>by </b><i>, , , <a class="ref" href="finseq_4.html#T10">FINSEQ_4:10</a></i>;<br/>
<a NAME="E24:20"/><b>then </b><i><font color="Green">E75</font></i>: 
<font color="Maroon">Fw1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">Fw2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="finseq_3.html#K1">-</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">w</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="finseq_4.html#T69">FINSEQ_4:69</a></i>;<br/>
<a NAME="E25:20"/><b>then </b><i><font color="Green">E76</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">Fw1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">Fw2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">w</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T72">FINSEQ_3:72</a></i>;<br/>
<a NAME="E26:20"/><i><font color="Green">E77</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">Fw1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">Fw2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">L</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">w</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i>, <a class="ref" href="vectsp_9.html#T5" target="_self">Th5</a>, <a class="ref" href="finseq_3.html#T72">FINSEQ_3:72</a></i>;<br/>
<a NAME="E27:20"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">Fw1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">Fw2</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">L</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E28:20"/><b>then </b><i><font color="Green">E78</font></i>: 
 <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">K</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">Fw1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">Fw2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="vectsp_9.html#E1:20"><i><font color="Green">E32</font></i></a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/>
<a NAME="E29:20"/><i><font color="Green">E79</font></i>: 
<span class="p1">(<span class="default"><a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">L</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">w</font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">w</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T33">XBOOLE_1:33</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">K</font> = <font color="Maroon">K</font> as    <a href="vectsp_6.html#M2">Linear_Combination</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">w</font></span><a href="domain_1.html#K6">}</a></span> <b>by </b><i>, , <a class="ref" href="vectsp_6.html#D7">VECTSP_6:def 7</a></i>;<br/>
<a NAME="E31:20"/>
( <font color="Maroon">Fw1</font> is <a href="funct_1.html#V2">one-to-one</a> &amp; <font color="Maroon">Fw2</font> is <a href="funct_1.html#V2">one-to-one</a> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">Fw1</font> <a href="xboole_0.html#R1">misses</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">Fw2</font> )
 
<b>by </b><i>, , , <a class="ref" href="finseq_4.html#T67">FINSEQ_4:67</a>, <a class="ref" href="finseq_4.html#T68">FINSEQ_4:68</a>, <a class="ref" href="finseq_4.html#T72">FINSEQ_4:72</a></i>;<br/>
<a NAME="E32:20"/><b>then </b>
<font color="Maroon">Fw1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">Fw2</font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T98">FINSEQ_3:98</a></i>;<br/>
<a NAME="E33:20"/><b>then </b>
 <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="vectsp_6.html#K3">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">Fw1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">Fw2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="vectsp_6.html#K4">Sum</a> <font color="Maroon">K</font>
 
<b>by </b><i>, <a class="ref" href="vectsp_6.html#D9">VECTSP_6:def 9</a></i>;<br/>
<b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">w</font></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">w</font></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span> <a href="vectsp_1.html#K6">*</a> <span class="p2">(<span class="default"><a href="vectsp_6.html#K4">Sum</a> <font color="Maroon">K</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">w</font></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span> <a href="vectsp_1.html#K6">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">w</font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">w</font></span>)</span></span>)</span>
<b>by </b><i>, , , <a class="txt" href="vectsp_9.html#E1:20"><i><font color="Green">E32</font></i></a>, <a class="ref" href="vectsp_1.html#D26">VECTSP_1:def 26</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">w</font></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span> <a href="vectsp_1.html#K6">*</a> <span class="p2">(<span class="default"><a href="vectsp_6.html#K4">Sum</a> <font color="Maroon">K</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">w</font>
<b>by </b><i>, <a class="ref" href="vectsp_1.html#T67">VECTSP_1:67</a></i>
;<br/>
<b>then </b><i><font color="Green">E80</font></i>: <font color="Maroon">w</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">w</font></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">w</font></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span> <a href="vectsp_1.html#K6">*</a> <span class="p2">(<span class="default"><a href="vectsp_6.html#K4">Sum</a> <font color="Maroon">K</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_2.html#T22">VECTSP_2:22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">w</font></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span> <a href="vectsp_1.html#K6">*</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><a href="vectsp_6.html#K4">Sum</a> <font color="Maroon">K</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#T70">VECTSP_1:70</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">w</font></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span> <a href="vectsp_1.html#K6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="vectsp_6.html#K4">Sum</a> <font color="Maroon">K</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">v</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D11">RLVECT_1:def 11</a></i>
;<br/>
<a NAME="E36:20"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E37:20"/><b>then </b>
<font color="Maroon">v</font> <a href="rlvect_1.html#R1">in</a>  <a href="vectsp_7.html#K1">Lin</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="vectsp_7.html#T13">VECTSP_7:13</a></i>;<br/>
<b>then consider </b><font color="Maroon">Lv</font> being    <a href="vectsp_6.html#M2">Linear_Combination</a> of <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span><b> such that </b><br/><a NAME="E39:20"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="vectsp_6.html#K4">Sum</a> <font color="Maroon">Lv</font>
 <b>by </b><i><a class="ref" href="vectsp_7.html#T12">VECTSP_7:12</a></i>;<br/>
<i><font color="Green">E82</font></i>: <font color="Maroon">w</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">w</font></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span> <a href="vectsp_1.html#K6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="vectsp_6.html#K4">Sum</a> <span class="p3">(<span class="default"><a href="vectsp_6.html#K7">-</a> <font color="Maroon">K</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><a href="vectsp_6.html#K4">Sum</a> <font color="Maroon">Lv</font></span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="vectsp_6.html#T79">VECTSP_6:79</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">w</font></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span> <a href="vectsp_1.html#K6">*</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K4">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="vectsp_6.html#K7">-</a> <font color="Maroon">K</font></span>)</span> <a href="vectsp_6.html#K5">+</a> <font color="Maroon">Lv</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_6.html#T77">VECTSP_6:77</a></i>
<br/>.= 
 <a href="vectsp_6.html#K4">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">w</font></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span> <a href="vectsp_6.html#K6">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="vectsp_6.html#K7">-</a> <font color="Maroon">K</font></span>)</span> <a href="vectsp_6.html#K5">+</a> <font color="Maroon">Lv</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_6.html#T78">VECTSP_6:78</a></i>
;<br/>
<b>set </b><font color="Maroon">LC</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">w</font></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span> <a href="vectsp_6.html#K6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="vectsp_6.html#K7">-</a> <font color="Maroon">K</font></span>)</span> <a href="vectsp_6.html#K5">+</a> <font color="Maroon">Lv</font></span>)</span>;<br/>
<a NAME="E41:20"/><i><font color="Green">E83</font></i>: 
 <a href="vectsp_6.html#K1">Carrier</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">w</font></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span> <a href="vectsp_6.html#K6">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="vectsp_6.html#K7">-</a> <font color="Maroon">K</font></span>)</span> <a href="vectsp_6.html#K5">+</a> <font color="Maroon">Lv</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="vectsp_6.html#K1">Carrier</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="vectsp_6.html#K7">-</a> <font color="Maroon">K</font></span>)</span> <a href="vectsp_6.html#K5">+</a> <font color="Maroon">Lv</font></span>)</span>
 
<b>by </b><i><a class="ref" href="vectsp_6.html#T58">VECTSP_6:58</a></i>;<br/>
<a NAME="E42:20"/><i><font color="Green">E85</font></i>: 
 <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">Lv</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="vectsp_6.html#D7">VECTSP_6:def 7</a></i>;<br/>
<a NAME="E43:20"/>
 <a href="vectsp_6.html#K1">Carrier</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="vectsp_6.html#K7">-</a> <font color="Maroon">K</font></span>)</span> <a href="vectsp_6.html#K5">+</a> <font color="Maroon">Lv</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K1">Carrier</a> <span class="p2">(<span class="default"><a href="vectsp_6.html#K7">-</a> <font color="Maroon">K</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">Lv</font></span>)</span>
 
<b>by </b><i><a class="ref" href="vectsp_6.html#T51">VECTSP_6:51</a></i>;<br/>
<a NAME="E44:20"/><b>then </b><i><font color="Green">E86</font></i>: 
 <a href="vectsp_6.html#K1">Carrier</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="vectsp_6.html#K7">-</a> <font color="Maroon">K</font></span>)</span> <a href="vectsp_6.html#K5">+</a> <font color="Maroon">Lv</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">K</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">Lv</font></span>)</span>
 
<b>by </b><i><a class="ref" href="vectsp_6.html#T69">VECTSP_6:69</a></i>;<br/>
<a NAME="E45:20"/>
<span class="p1">(<span class="default"><a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">K</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">Lv</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">w</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i>, , , <a class="txt" href="vectsp_9.html#E4:20"><i><font color="Green">E42</font></i></a>, <a class="ref" href="xboole_1.html#T13">XBOOLE_1:13</a></i>;<br/>
<a NAME="E46:20"/><b>then </b>
 <a href="vectsp_6.html#K1">Carrier</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="vectsp_6.html#K7">-</a> <font color="Maroon">K</font></span>)</span> <a href="vectsp_6.html#K5">+</a> <font color="Maroon">Lv</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">w</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="txt" href="vectsp_9.html#E5:20"><i><font color="Green">E43</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E47:20"/><b>then </b>
 <a href="vectsp_6.html#K1">Carrier</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">w</font></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span> <a href="vectsp_6.html#K6">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="vectsp_6.html#K7">-</a> <font color="Maroon">K</font></span>)</span> <a href="vectsp_6.html#K5">+</a> <font color="Maroon">Lv</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">w</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E48:20"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">w</font></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span> <a href="vectsp_6.html#K6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="vectsp_6.html#K7">-</a> <font color="Maroon">K</font></span>)</span> <a href="vectsp_6.html#K5">+</a> <font color="Maroon">Lv</font></span>)</span> is    <a href="vectsp_6.html#M2">Linear_Combination</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">w</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="vectsp_6.html#D7">VECTSP_6:def 7</a></i>;<br/>
<b>hence </b><a NAME="E49:20"/>
( <font color="Maroon">w</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">w</font> <a href="rlvect_1.html#R1">in</a>  <a href="vectsp_7.html#K1">Lin</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">w</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="subset_1.html#K4">\/</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="txt" href="vectsp_9.html#E2:20"><i><font color="Green">E40</font></i></a>, <a class="ref" href="vectsp_7.html#T12">VECTSP_7:12</a></i>;<br/>


</div>
