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

<b>set </b><font color="Maroon">V</font> =  <a href="vectmetr.html#K4" target="_self">RLMSpace</a> <font color="Maroon">n</font>;<br/>
<b>let </b><font color="Maroon">G1</font> be  <a href="group_1.html#V1">strict</a>  <a href="group_1.html#L1">HGrStr</a> , <font color="Maroon">G2</font> be  <a href="group_1.html#V1">strict</a>  <a href="group_1.html#L1">HGrStr</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:30_1_2"/><i><font color="Green">E65</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G1</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>
 <b>and </b><br/><a NAME="E2:30_1_2"/><i><font color="Green">E66</font></i>: 
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">G1</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>
 <b>and </b><br/><a NAME="E3:30_1_2"/><i><font color="Green">E70</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G2</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>
 <b>and </b><br/><a NAME="E4:30_1_2"/><i><font color="Green">E72</font></i>: 
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">G2</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><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">f</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G1</font>, <font color="Maroon">g</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G1</font>;<br/><b>consider </b><font color="Maroon">f1</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_2_1"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E3:30_1_2_1"/>
<font color="Maroon">f1</font> is <a href="vectmetr.html#V3" target="_self">isometric</a>
 <b>by </b><i>, </i>;<br/><b>consider </b><font color="Maroon">g1</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_2_1"/><i><font color="Green">E77</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_2_1"/>
<font color="Maroon">g1</font> is <a href="vectmetr.html#V3" target="_self">isometric</a>
 <b>by </b><i>, </i>;<br/><b>thus </b>the <a href="group_1.html#U1">mult</a> of <font color="Maroon">G1</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">f</font>,<font color="Maroon">g</font> = 
<font color="Maroon">f1</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">g1</font>
<b>by </b><i>, , , </i>
<br/>.= 
the <a href="group_1.html#U1">mult</a> of <font color="Maroon">G2</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">f</font>,<font color="Maroon">g</font>
<b>by </b><i>, , , </i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:30_1_2"/>
<font color="Maroon">G1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G2</font>
 <b>by </b><i>, , <a class="ref" href="binop_1.html#T2">BINOP_1:2</a></i>;<br/>


</div>
