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

<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>4</sub></font>,   <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> = <font color="Maroon">c<sub>2</sub></font>;<br/>
<b>consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="funct_2.html#NM1">Function</a> of <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">c<sub>4</sub></font>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>,<span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span><b> such that </b><br/><a NAME="E2:11_1_1"/><i><font color="Green">E8</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>4</sub></font><br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> holds <font color="Maroon">c<sub>5</sub></font> <a href="binop_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>)
 <b>from </b><i><a class="ref" href="binop_1.html#S4">BINOP_1:sch 4</a>();<br/></i>
<b>consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="relset_1.html#NM3">Relation</a> of <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span><b> such that </b><br/><a NAME="E4:11_1_1"/><i><font color="Green">E9</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> iff ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; ex <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> ) ) )
 <b>by </b><i><a class="txt" href="symsp_1.html#E5"><i><font color="Green">Lemma3</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> =  <a href="symsp_1.html#G1" target="_self">SymStr</a>(# <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> #) as  non <a href="struct_0.html#V3">empty</a> <a href="rlvect_1.html#V3">Abelian</a> <a href="rlvect_1.html#V4">add-associative</a> <a href="rlvect_1.html#V5">right_zeroed</a> <a href="rlvect_1.html#V6">right_complementable</a>  <a href="symsp_1.html#L1" target="_self">SymStr</a> of <font color="Maroon">c<sub>4</sub></font> <b>by </b><i><a class="txt" href="symsp_1.html#E6"><i><font color="Green">Lemma4</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>7</sub></font>
;<br/>

<b>thus </b><a NAME="E6:11_1_1"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>7</sub></font><br/>for <font color="Olive">b<sub>5</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>4</sub></font> holds <br/>( ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>7</sub></font> implies ex <font color="Olive">b<sub>6</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>7</sub></font> st not <font color="Olive">b<sub>6</sub></font> <a href="symsp_1.html#R1" target="_self">_|_</a> <font color="Olive">b<sub>1</sub></font> ) &amp; ( <font color="Olive">b<sub>1</sub></font> <a href="symsp_1.html#R1" target="_self">_|_</a> <font color="Olive">b<sub>2</sub></font> implies <font color="Olive">b<sub>5</sub></font> <a href="vectsp_1.html#K6">*</a> <font color="Olive">b<sub>1</sub></font> <a href="symsp_1.html#R1" target="_self">_|_</a> <font color="Olive">b<sub>2</sub></font> ) &amp; ( <font color="Olive">b<sub>2</sub></font> <a href="symsp_1.html#R1" target="_self">_|_</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="symsp_1.html#R1" target="_self">_|_</a> <font color="Olive">b<sub>1</sub></font> implies <font color="Olive">b<sub>2</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">b<sub>3</sub></font> <a href="symsp_1.html#R1" target="_self">_|_</a> <font color="Olive">b<sub>1</sub></font> ) &amp; ( not <font color="Olive">b<sub>2</sub></font> <a href="symsp_1.html#R1" target="_self">_|_</a> <font color="Olive">b<sub>1</sub></font> implies ex <font color="Olive">b<sub>6</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>4</sub></font> st <font color="Olive">b<sub>4</sub></font> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>6</sub></font> <a href="vectsp_1.html#K6">*</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="symsp_1.html#R1" target="_self">_|_</a> <font color="Olive">b<sub>1</sub></font> ) &amp; ( <font color="Olive">b<sub>1</sub></font> <a href="symsp_1.html#R1" target="_self">_|_</a> <font color="Olive">b<sub>2</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="symsp_1.html#R1" target="_self">_|_</a> <font color="Olive">b<sub>3</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">b<sub>1</sub></font> implies <font color="Olive">b<sub>3</sub></font> <a href="symsp_1.html#R1" target="_self">_|_</a> <font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">b<sub>2</sub></font> ) )
 <b>by </b><i><a class="txt" href="symsp_1.html#E2:11_1_1"><i><font color="Green">E8</font></i></a>, <a class="txt" href="symsp_1.html#E4:11_1_1"><i><font color="Green">E9</font></i></a>, <a class="txt" href="symsp_1.html#E8"><i><font color="Green">Lemma6</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="symsp_1.html#D2" target="_self">SYMSP_1:def 2</a><br/>

<b>thus </b><a NAME="E7:11_1_1"/>
<font color="Maroon">c<sub>7</sub></font> is <a href="vectsp_1.html#V12">VectSp-like</a>
 <b>by </b><i><a class="txt" href="symsp_1.html#E2:11_1_1"><i><font color="Green">E8</font></i></a>, <a class="txt" href="symsp_1.html#E7"><i><font color="Green">Lemma5</font></i></a></i>;<br/>

<b>thus </b><a NAME="E8:11_1_1"/>
<font color="Maroon">c<sub>7</sub></font> is <a href="symsp_1.html#V1" target="_self">strict</a>
 ;<br/>


</div>
