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

<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">F</font>,   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> = <font color="Maroon" title="c2">o</font>;<br/>
<b>consider </b><font color="Maroon" title="c5">mF</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">F</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span><b> such that </b><br/><a NAME="E2:11_1_1"/><i><font color="Green" title="E8">A1</font></i>: 
for <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">F</font><br/> for <font color="Olive" title="b2">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> holds  <font color="Maroon" title="c5">mF</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">x</font>)
 <b>from </b><i><a class="ref" href="binop_1.html#S4" title="BINOP_1:sch.4">BINOP_1:sch 4</a>();<br/></i>
<b>consider </b><font color="Maroon" title="c6">mo</font> being   <a href="relset_1.html#NM3" title="RELSET_1:NM.3">Relation</a> of <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span><b> such that </b><br/><a NAME="E4:11_1_1"/><i><font color="Green" title="E9">A2</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">mo</font> iff ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp;  ex <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">b</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b2">a</font>,<font color="Olive" title="b3">b</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> &amp; <font color="Olive" title="b3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">o</font> ) ) )
 <b>by </b><i><a class="txt" href="symsp_1.html#E5"><i><font color="Green" title="E3">Lm2</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">MPS</font> =  <a href="symsp_1.html#G1" title="SYMSP_1:aggr.1">SymStr</a>(# <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c3">md</font>,<font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c5">mF</font>,<font color="Maroon" title="c6">mo</font> #) as  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">Abelian</a> <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">add-associative</a> <a href="rlvect_1.html#V5" title="RLVECT_1:attr.5">right_zeroed</a> <a href="rlvect_1.html#V6" title="RLVECT_1:attr.6">right_complementable</a>  <a href="symsp_1.html#L1" title="SYMSP_1:struct.1">SymStr</a> of <font color="Maroon" title="c4">F</font> <b>by </b><i><a class="txt" href="symsp_1.html#E6"><i><font color="Green" title="E4">Lm3</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c7">MPS</font>
;<br/>

<b>thus </b><a NAME="E6:11_1_1"/>
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font>, <font color="Olive" title="b4">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">MPS</font><br/> for <font color="Olive" title="b5">l</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">F</font> holds <br/> ( ( <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c7">MPS</font> implies  ex <font color="Olive" title="b6">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">MPS</font> st not <font color="Olive" title="b6">y</font> <a href="symsp_1.html#R1" title="SYMSP_1:pred.1">_|_</a> <font color="Olive" title="b1">a</font> ) &amp; ( <font color="Olive" title="b1">a</font> <a href="symsp_1.html#R1" title="SYMSP_1:pred.1">_|_</a> <font color="Olive" title="b2">b</font> implies <font color="Olive" title="b5">l</font> <a href="vectsp_1.html#K5" title="VECTSP_1:func.5">*</a> <font color="Olive" title="b1">a</font> <a href="symsp_1.html#R1" title="SYMSP_1:pred.1">_|_</a> <font color="Olive" title="b2">b</font> ) &amp; ( <font color="Olive" title="b2">b</font> <a href="symsp_1.html#R1" title="SYMSP_1:pred.1">_|_</a> <font color="Olive" title="b1">a</font> &amp; <font color="Olive" title="b3">c</font> <a href="symsp_1.html#R1" title="SYMSP_1:pred.1">_|_</a> <font color="Olive" title="b1">a</font> implies <font color="Olive" title="b2">b</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b3">c</font> <a href="symsp_1.html#R1" title="SYMSP_1:pred.1">_|_</a> <font color="Olive" title="b1">a</font> ) &amp; ( not <font color="Olive" title="b2">b</font> <a href="symsp_1.html#R1" title="SYMSP_1:pred.1">_|_</a> <font color="Olive" title="b1">a</font> implies  ex <font color="Olive" title="b6">k</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">F</font> st <font color="Olive" title="b4">x</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b6">k</font> <a href="vectsp_1.html#K5" title="VECTSP_1:func.5">*</a> <font color="Olive" title="b2">b</font></span>)</span> <a href="symsp_1.html#R1" title="SYMSP_1:pred.1">_|_</a> <font color="Olive" title="b1">a</font> ) &amp; ( <font color="Olive" title="b1">a</font> <a href="symsp_1.html#R1" title="SYMSP_1:pred.1">_|_</a> <font color="Olive" title="b2">b</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b3">c</font> &amp; <font color="Olive" title="b2">b</font> <a href="symsp_1.html#R1" title="SYMSP_1:pred.1">_|_</a> <font color="Olive" title="b3">c</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b1">a</font> implies <font color="Olive" title="b3">c</font> <a href="symsp_1.html#R1" title="SYMSP_1:pred.1">_|_</a> <font color="Olive" title="b1">a</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b2">b</font> ) )
 <b>by </b><i><a class="txt" href="symsp_1.html#E2:11_1_1"><i><font color="Green" title="E8">A1</font></i></a>, <a class="txt" href="symsp_1.html#E4:11_1_1"><i><font color="Green" title="E9">A2</font></i></a>, <a class="txt" href="symsp_1.html#E8"><i><font color="Green" title="E6">Lm5</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="symsp_1.html#D2" target="_self" title="SYMSP_1:def.2">SYMSP_1:def 2</a><br/>

<b>thus </b><a NAME="E7:11_1_1"/>
<font color="Maroon" title="c7">MPS</font> is <a href="vectsp_1.html#V12" title="VECTSP_1:attr.12">VectSp-like</a>
 <b>by </b><i><a class="txt" href="symsp_1.html#E2:11_1_1"><i><font color="Green" title="E8">A1</font></i></a>, <a class="txt" href="symsp_1.html#E7"><i><font color="Green" title="E5">Lm4</font></i></a></i>;<br/>

<b>thus </b><a NAME="E8:11_1_1"/>
<font color="Maroon" title="c7">MPS</font> is <a href="symsp_1.html#V1" title="SYMSP_1:attr.1">strict</a>
 ;<br/>


</div>
