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<div class="add">

<b>reconsider </b><font color="Maroon">c<sub>1</sub></font> =  <a href="vectsp_1.html#K7">comp</a> <a href="algstr_1.html#K2">L_Trivial</a>  as   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>,<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span> <b>by </b><i><a class="ref" href="algstr_1.html#D4">ALGSTR_1:def 4</a></i>;<br/>
<a NAME="E2:20"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span> holds <br/><a href="vectsp_2.html#K7">op1</a>  <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:20_1"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>2</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <a href="algstr_1.html#K2">L_Trivial</a>  <b>by </b><i><a class="ref" href="algstr_1.html#D4">ALGSTR_1:def 4</a></i>;<br/>
<a NAME="E3:20_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="grcat_1.html#E1:20_1"><i><font color="Green">E12</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>then </b><a href="vectsp_2.html#K7">op1</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> = 
 <a href="rlvect_1.html#K5">-</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="grcat_1.html#T5" target="_self">Th5</a>, <a class="ref" href="algstr_1.html#D4">ALGSTR_1:def 4</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="vectsp_1.html#D25">VECTSP_1:def 25</a></i>
;<br/>
<b>hence </b><a NAME="E5:20_1"/>
<a href="vectsp_2.html#K7">op1</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:20"/>
 <a href="vectsp_1.html#K7">comp</a> <a href="algstr_1.html#K2">L_Trivial</a>  <a href="funct_2.html#R1">=</a>  <a href="vectsp_2.html#K7">op1</a> 
 <b>by </b><i><a class="ref" href="funct_2.html#T18">FUNCT_2:18</a></i>;<br/>


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