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<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">f</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="vectmetr.html#K5" target="_self">IsomGroup</a> <font color="Maroon">n</font></span>)</span>;<br/><b>let </b><font color="Maroon">g</font> be   <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>;<br/>





<b>assume </b><a NAME="E1:34"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font>
 ;<br/>

<a NAME="E2:34"/>
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="vectmetr.html#K5" target="_self">IsomGroup</a> <font color="Maroon">n</font></span>)</span>
 
;<br/>
<a NAME="E3:34"/><b>then </b><i><font color="Green">E44</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>by </b><i/>;<br/>
<b>then 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="E5:34"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E6:34"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">f1</font> is <a href="vectmetr.html#V3" target="_self">isometric</a>
 <b>by </b><i/>;<br/>
<a NAME="E7:34"/>
<font color="Maroon">g</font> <a href="tops_2.html#NK3" target="_self">"</a>  is <a href="vectmetr.html#V3" target="_self">isometric</a>
 
<b>by </b><i>, , , </i>;<br/>
<a NAME="E8:34"/><b>then </b><i><font color="Green">E57</font></i>: 
<font color="Maroon">g</font> <a href="tops_2.html#NK3" target="_self">"</a>  <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>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">g1</font> = <font color="Maroon">g</font> <a href="tops_2.html#NK3" target="_self">"</a>  as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="vectmetr.html#K5" target="_self">IsomGroup</a> <font color="Maroon">n</font></span>)</span> <b>by </b><i/>;<br/>
<i><font color="Green">E58</font></i>:  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font> = 
the <a href="struct_0.html#U1">carrier</a> of <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><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
 <a href="pre_topc.html#K2">[#]</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><a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a></i>
;<br/>
<i><font color="Green">E59</font></i>:  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f1</font> = 
the <a href="struct_0.html#U1">carrier</a> of <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>, </i>
<br/>.= 
 <a href="pre_topc.html#K2">[#]</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><a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a></i>
;<br/>
<a NAME="E12:34"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">f1</font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i>, </i>;<br/>
<i><font color="Green">E61</font></i>: <font color="Maroon">f</font> <a href="group_1.html#K1">*</a> <font color="Maroon">g1</font> = 
the <a href="group_1.html#U1">mult</a> of <span class="p1">(<span class="default"><a href="vectmetr.html#K5" target="_self">IsomGroup</a> <font color="Maroon">n</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">f</font>,<font color="Maroon">g1</font>
<b>by </b><i><a class="ref" href="group_1.html#D1">GROUP_1:def 1</a></i>
<br/>.= 
<font color="Maroon">f1</font> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span>
<b>by </b><i>, , , </i>
<br/>.= 
 <a href="partfun1.html#K6">id</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f1</font></span>)</span>
<b>by </b><i>, , , , <a class="ref" href="tops_2.html#T65">TOPS_2:65</a></i>
<br/>.= 
 <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <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>, , <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
 <a href="grcat_1.html#K7">id</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><a class="ref" href="grcat_1.html#D11">GRCAT_1:def 11</a></i>
<br/>.= 
 <a href="group_1.html#K2">1.</a> <span class="p1">(<span class="default"><a href="vectmetr.html#K5" target="_self">IsomGroup</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i/>
;<br/>
<font color="Maroon">g1</font> <a href="group_1.html#K1">*</a> <font color="Maroon">f</font> = 
the <a href="group_1.html#U1">mult</a> of <span class="p1">(<span class="default"><a href="vectmetr.html#K5" target="_self">IsomGroup</a> <font color="Maroon">n</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">g1</font>,<font color="Maroon">f</font>
<b>by </b><i><a class="ref" href="group_1.html#D1">GROUP_1:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">f1</font>
<b>by </b><i>, , , </i>
<br/>.= 
 <a href="partfun1.html#K6">id</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font></span>)</span>
<b>by </b><i>, , , , <a class="ref" href="tops_2.html#T65">TOPS_2:65</a></i>
<br/>.= 
 <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <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><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
 <a href="grcat_1.html#K7">id</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><a class="ref" href="grcat_1.html#D11">GRCAT_1:def 11</a></i>
<br/>.= 
 <a href="group_1.html#K2">1.</a> <span class="p1">(<span class="default"><a href="vectmetr.html#K5" target="_self">IsomGroup</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i/>
;<br/>
<b>hence </b><a NAME="E15:34"/>
<font color="Maroon">f</font> <a href="group_1.html#K3">"</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="tops_2.html#NK3" target="_self">"</a> 
 <b>by </b><i>, <a class="ref" href="group_1.html#T12">GROUP_1:12</a></i>;<br/>


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