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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="vectsp_1.html#NM2">Field</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be  <a href="matrlin.html#V2" target="_self">finite-dimensional</a> <a href="vectsp_1.html#NM6">VectSp</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be   <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="matrlin.html#M1" target="_self">OrdBasis</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>7</sub></font> be    <a href="matrlin.html#M1" target="_self">OrdBasis</a> of <font color="Maroon">c<sub>3</sub></font>;<br/>













<b>assume </b><b>that </b><br/><a NAME="E1:62"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> is <a href="mod_2.html#V1">linear</a> &amp; <font color="Maroon">c<sub>5</sub></font> is <a href="mod_2.html#V1">linear</a> )
 <b>and </b><br/><a NAME="E2:62"/><i><font color="Green">E51</font></i>: 
 <a href="matrlin.html#K12" target="_self">AutMt</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="matrlin.html#K12" target="_self">AutMt</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 <b>and </b><br/><a NAME="E3:62"/><i><font color="Green">E52</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/>

<a NAME="E4:62"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font> is    <a href="vectsp_7.html#M1">Basis</a> of <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="matrlin.html#D4" target="_self">Def4</a></i>;<br/>
<a NAME="E5:62"/><b>then </b><i><font color="Green">E53</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>
 
;<br/>
<a NAME="E6:62"/><b>then </b><i><font color="Green">E54</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E7:62"/><i><font color="Green">E55</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="matrlin.html#E5:62"><i><font color="Green">E53</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<i><font color="Green">E56</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseqop.html#K5">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> = 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="txt" href="matrlin.html#E6:62"><i><font color="Green">E54</font></i></a>, <a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>
<br/>.= 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseqop.html#K5">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="matrlin.html#E7:62"><i><font color="Green">E55</font></i></a>, <a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>8</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:62_2"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseqop.html#K5">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="ref" href="finseq_3.html#T25">FINSEQ_3:25</a></i>;<br/><a NAME="E3:62_2"/><i><font color="Green">E58</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseqop.html#K5">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="relat_1.html#T44">RELAT_1:44</a></i>;<br/><b>then </b><i><font color="Green">E59</font></i>: <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="matrlin.html#K11" target="_self">|--</a> <font color="Maroon">c<sub>7</sub></font> = 
<span class="p1">(<span class="default"><a href="matrlin.html#K12" target="_self">AutMt</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="matrlin.html#K1" target="_self">/.</a> <font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="txt" href="matrlin.html#E2:62"><i><font color="Green">E51</font></i></a>, <a class="txt" href="matrlin.html#E1:62_2"><i><font color="Green">E57</font></i></a>, <a class="ref" href="matrlin.html#D10" target="_self">Def10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="matrlin.html#K11" target="_self">|--</a> <font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="txt" href="matrlin.html#E1:62_2"><i><font color="Green">E57</font></i></a>, <a class="txt" href="matrlin.html#E3:62_2"><i><font color="Green">E58</font></i></a>, <a class="ref" href="matrlin.html#D10" target="_self">Def10</a></i>
;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseqop.html#K5">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> = 
<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="matrlin.html#E1:62_2"><i><font color="Green">E57</font></i></a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="matrlin.html#E1:62_2"><i><font color="Green">E57</font></i></a>, <a class="txt" href="matrlin.html#E3:62_2"><i><font color="Green">E58</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="matrlin.html#E4:62_2"><i><font color="Green">E59</font></i></a>, <a class="ref" href="matrlin.html#T39" target="_self">Th39</a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="matrlin.html#E1:62_2"><i><font color="Green">E57</font></i></a>, <a class="txt" href="matrlin.html#E3:62_2"><i><font color="Green">E58</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseqop.html#K5">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="txt" href="matrlin.html#E8:62"><i><font color="Green">E56</font></i></a>, <a class="txt" href="matrlin.html#E1:62_2"><i><font color="Green">E57</font></i></a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>
;<br/></div><b>end;</b></div>
<a NAME="E10:62"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="finseqop.html#K5">*</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="finseqop.html#K5">*</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="matrlin.html#E8:62"><i><font color="Green">E56</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>
<b>hence </b><a NAME="E11:62"/>
<font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="matrlin.html#E1:62"><i><font color="Green">E50</font></i></a>, <a class="txt" href="matrlin.html#E3:62"><i><font color="Green">E52</font></i></a>, <a class="ref" href="matrlin.html#T26" target="_self">Th26</a></i>;<br/>


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