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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">f</font>, <font color="Olive">g</font> being  <a href="funct_1.html#NM1">Function</a> st <br/>( <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>2</sub></font> &amp; <font color="Maroon">a<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">f</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Olive">g</font> );<br/>
<a NAME="E1:30_1_1"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="vectmetr.html#K2" target="_self">ISOM</a> <span class="p1">(<span class="default"><a href="vectmetr.html#K4" target="_self">RLMSpace</a> <font color="Maroon">n</font></span>)</span>  ex <font color="Olive">z</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="vectmetr.html#K2" target="_self">ISOM</a> <span class="p1">(<span class="default"><a href="vectmetr.html#K4" target="_self">RLMSpace</a> <font color="Maroon">n</font></span>)</span> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>,<font color="Olive">z</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="vectmetr.html#K2" target="_self">ISOM</a> <span class="p1">(<span class="default"><a href="vectmetr.html#K4" target="_self">RLMSpace</a> <font color="Maroon">n</font></span>)</span>, <font color="Maroon">y</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="vectmetr.html#K2" target="_self">ISOM</a> <span class="p1">(<span class="default"><a href="vectmetr.html#K4" target="_self">RLMSpace</a> <font color="Maroon">n</font></span>)</span>;<br/>

<b>consider </b><font color="Maroon">x1</font> being   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="vectmetr.html#K4" target="_self">RLMSpace</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="vectmetr.html#K4" target="_self">RLMSpace</a> <font color="Maroon">n</font></span>)</span><b> such that </b><br/><a NAME="E2:30_1_1_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>and </b><br/><a NAME="E3:30_1_1_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">x1</font> is <a href="vectmetr.html#V3" target="_self">isometric</a>
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">y1</font> being   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="vectmetr.html#K4" target="_self">RLMSpace</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="vectmetr.html#K4" target="_self">RLMSpace</a> <font color="Maroon">n</font></span>)</span><b> such that </b><br/><a NAME="E5:30_1_1_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 <b>and </b><br/><a NAME="E6:30_1_1_1"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">y1</font> is <a href="vectmetr.html#V3" target="_self">isometric</a>
 <b>by </b><i/>;<br/>
<a NAME="E7:30_1_1_1"/>
<font color="Maroon">x1</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">y1</font> is <a href="vectmetr.html#V3" target="_self">isometric</a>
 
<b>by </b><i>, , </i>;<br/>
<b>then reconsider </b><font color="Maroon">z</font> = <font color="Maroon">x1</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">y1</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="vectmetr.html#K2" target="_self">ISOM</a> <span class="p1">(<span class="default"><a href="vectmetr.html#K4" target="_self">RLMSpace</a> <font color="Maroon">n</font></span>)</span> <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">z</font>
;<br/>

<b>take </b>
<font color="Maroon">x1</font>
;<br/>

<b>take </b>
<font color="Maroon">y1</font>
;<br/>

<b>thus </b><a NAME="E9:30_1_1_1"/>
( <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">y1</font> )
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">o</font> being   <a href="binop_1.html#NM2">BinOp</a> of  <a href="vectmetr.html#K2" target="_self">ISOM</a> <span class="p1">(<span class="default"><a href="vectmetr.html#K4" target="_self">RLMSpace</a> <font color="Maroon">n</font></span>)</span><b> such that </b><br/><a NAME="E3:30_1_1"/><i><font color="Green">E58</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="vectmetr.html#K2" target="_self">ISOM</a> <span class="p1">(<span class="default"><a href="vectmetr.html#K4" target="_self">RLMSpace</a> <font color="Maroon">n</font></span>)</span> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">a</font>,<font color="Olive">b</font>,<font color="Maroon">o</font> <a href="binop_1.html#K2">.</a> <font color="Olive">a</font>,<font color="Olive">b</font>]
 <b>from </b><i><a class="ref" href="binop_1.html#S3">BINOP_1:sch 3</a>();<br/></i>
<b>take </b><font color="Maroon">G</font> =  <a href="group_1.html#G1">HGrStr</a>(# <span class="p1">(<span class="default"><a href="vectmetr.html#K2" target="_self">ISOM</a> <span class="p2">(<span class="default"><a href="vectmetr.html#K4" target="_self">RLMSpace</a> <font color="Maroon">n</font></span>)</span></span>)</span>,<font color="Maroon">o</font> #);<br/>

<a NAME="E4:30_1_1"/>
for <font color="Olive">f</font>, <font color="Olive">g</font> being  <a href="funct_1.html#NM1">Function</a>  st <font color="Olive">f</font> <a href="hidden.html#R2">in</a>  <a href="vectmetr.html#K2" target="_self">ISOM</a> <span class="p1">(<span class="default"><a href="vectmetr.html#K4" target="_self">RLMSpace</a> <font color="Maroon">n</font></span>)</span> &amp; <font color="Olive">g</font> <a href="hidden.html#R2">in</a>  <a href="vectmetr.html#K2" target="_self">ISOM</a> <span class="p1">(<span class="default"><a href="vectmetr.html#K4" target="_self">RLMSpace</a> <font color="Maroon">n</font></span>)</span> holds <br/>the <a href="group_1.html#U1">mult</a> of <font color="Maroon">G</font> <a href="binop_1.html#K1">.</a> <font color="Olive">f</font>,<font color="Olive">g</font> <a href="hidden.html#R1">=</a> <font color="Olive">f</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Olive">g</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">f</font> be   <a href="funct_1.html#NM1">Function</a>, <font color="Maroon">g</font> be   <a href="funct_1.html#NM1">Function</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:30_1_1_2"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <a href="vectmetr.html#K2" target="_self">ISOM</a> <span class="p1">(<span class="default"><a href="vectmetr.html#K4" target="_self">RLMSpace</a> <font color="Maroon">n</font></span>)</span>
 <b>and </b><br/><a NAME="E2:30_1_1_2"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">g</font> <a href="hidden.html#R2">in</a>  <a href="vectmetr.html#K2" target="_self">ISOM</a> <span class="p1">(<span class="default"><a href="vectmetr.html#K4" target="_self">RLMSpace</a> <font color="Maroon">n</font></span>)</span>
 ;<br/>

<b>consider </b><font color="Maroon">f1</font> being   <a href="funct_1.html#NM1">Function</a>, <font color="Maroon">g1</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E4:30_1_1_2"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E5:30_1_1_2"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">g1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font>
 <b>and </b><br/><a NAME="E6:30_1_1_2"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">o</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">f</font>,<font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">g1</font>
 <b>by </b><i>, , </i>;<br/>
<b>thus </b><a NAME="E7:30_1_1_2"/>
the <a href="group_1.html#U1">mult</a> of <font color="Maroon">G</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">f</font>,<font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">g</font>
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:30_1_1"/>
( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="vectmetr.html#K2" target="_self">ISOM</a> <span class="p1">(<span class="default"><a href="vectmetr.html#K4" target="_self">RLMSpace</a> <font color="Maroon">n</font></span>)</span> &amp; ( for <font color="Olive">f</font>, <font color="Olive">g</font> being  <a href="funct_1.html#NM1">Function</a>  st <font color="Olive">f</font> <a href="hidden.html#R2">in</a>  <a href="vectmetr.html#K2" target="_self">ISOM</a> <span class="p1">(<span class="default"><a href="vectmetr.html#K4" target="_self">RLMSpace</a> <font color="Maroon">n</font></span>)</span> &amp; <font color="Olive">g</font> <a href="hidden.html#R2">in</a>  <a href="vectmetr.html#K2" target="_self">ISOM</a> <span class="p1">(<span class="default"><a href="vectmetr.html#K4" target="_self">RLMSpace</a> <font color="Maroon">n</font></span>)</span> holds <br/>the <a href="group_1.html#U1">mult</a> of <font color="Maroon">G</font> <a href="binop_1.html#K1">.</a> <font color="Olive">f</font>,<font color="Olive">g</font> <a href="hidden.html#R1">=</a> <font color="Olive">f</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Olive">g</font> ) )
 ;<br/>


</div>
