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<b>let </b><font color="Maroon" title="c1">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<a NAME="E1:33"/>
 <a href="grcat_1.html#K7" title="GRCAT_1:func.7">id</a> <span class="p1">(<span class="default"><a href="vectmetr.html#K4" title="VECTMETR:func.4">RLMSpace</a> <font color="Maroon" title="c1">n</font></span>)</span> is <a href="vectmetr.html#V3" title="VECTMETR:attr.3">isometric</a>
 
<b>by </b><i><a class="ref" href="vectmetr.html#T5" target="_self" title="VECTMETR:th.5">Th5</a></i>;<br/>
<a NAME="E2:33"/><b>then </b><i><font color="Green" title="E14">A1</font></i>: 
 <a href="grcat_1.html#K7" title="GRCAT_1:func.7">id</a> <span class="p1">(<span class="default"><a href="vectmetr.html#K4" title="VECTMETR:func.4">RLMSpace</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="vectmetr.html#K2" title="VECTMETR:func.2">ISOM</a> <span class="p1">(<span class="default"><a href="vectmetr.html#K4" title="VECTMETR:func.4">RLMSpace</a> <font color="Maroon" title="c1">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="vectmetr.html#D4" target="_self" title="VECTMETR:def.4">Def4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c2">e</font> =  <a href="grcat_1.html#K7" title="GRCAT_1:func.7">id</a> <span class="p1">(<span class="default"><a href="vectmetr.html#K4" title="VECTMETR:func.4">RLMSpace</a> <font color="Maroon" title="c1">n</font></span>)</span> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="vectmetr.html#K5" title="VECTMETR:func.5">IsomGroup</a> <font color="Maroon" title="c1">n</font></span>)</span> <b>by </b><i><a class="ref" href="vectmetr.html#D9" target="_self" title="VECTMETR:def.9">Def9</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">h</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="vectmetr.html#K5" title="VECTMETR:func.5">IsomGroup</a> <font color="Maroon" title="c1">n</font></span>)</span>;<br/><a NAME="E1:33_1"/>
<font color="Maroon" title="c3">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="vectmetr.html#K5" title="VECTMETR:func.5">IsomGroup</a> <font color="Maroon" title="c1">n</font></span>)</span>
 ;<br/><a NAME="E2:33_1"/><b>then </b><i><font color="Green" title="E15">A2</font></i>: 
<font color="Maroon" title="c3">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="vectmetr.html#K2" title="VECTMETR:func.2">ISOM</a> <span class="p1">(<span class="default"><a href="vectmetr.html#K4" title="VECTMETR:func.4">RLMSpace</a> <font color="Maroon" title="c1">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="vectmetr.html#D9" target="_self" title="VECTMETR:def.9">Def9</a></i>;<br/><b>then consider </b><font color="Maroon" title="c4">h1</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="vectmetr.html#K4" title="VECTMETR:func.4">RLMSpace</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="vectmetr.html#K4" title="VECTMETR:func.4">RLMSpace</a> <font color="Maroon" title="c1">n</font></span>)</span><b> such that </b><br/><a NAME="E4:33_1"/><i><font color="Green" title="E16">A3</font></i>: 
<font color="Maroon" title="c4">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">h</font>
 <b>and </b><br/><a NAME="E5:33_1"/>
<font color="Maroon" title="c4">h1</font> is <a href="vectmetr.html#V3" title="VECTMETR:attr.3">isometric</a>
 <b>by </b><i><a class="ref" href="vectmetr.html#D4" target="_self" title="VECTMETR:def.4">Def4</a></i>;<br/><b>thus </b><font color="Maroon" title="c3">h</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">e</font> = 
the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <span class="p1">(<span class="default"><a href="vectmetr.html#K5" title="VECTMETR:func.5">IsomGroup</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c3">h</font>,<font color="Maroon" title="c2">e</font>
<b>by </b><i><a class="ref" href="group_1.html#D1" title="GROUP_1:def.1">GROUP_1:def 1</a></i>
<br/>.= 
<font color="Maroon" title="c4">h1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="vectmetr.html#K4" title="VECTMETR:func.4">RLMSpace</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="vectmetr.html#E2:33"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="vectmetr.html#E2:33_1"><i><font color="Green" title="E15">A2</font></i></a>, <a class="txt" href="vectmetr.html#E4:33_1"><i><font color="Green" title="E16">A3</font></i></a>, <a class="ref" href="vectmetr.html#D9" target="_self" title="VECTMETR:def.9">Def9</a></i>
<br/>.= 
<font color="Maroon" title="c3">h</font>
<b>by </b><i><a class="txt" href="vectmetr.html#E4:33_1"><i><font color="Green" title="E16">A3</font></i></a>, <a class="ref" href="funct_2.html#T23" title="FUNCT_2:th.23">FUNCT_2:23</a></i>
;<br/><b>thus </b><font color="Maroon" title="c2">e</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c3">h</font> = 
the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <span class="p1">(<span class="default"><a href="vectmetr.html#K5" title="VECTMETR:func.5">IsomGroup</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c2">e</font>,<font color="Maroon" title="c3">h</font>
<b>by </b><i><a class="ref" href="group_1.html#D1" title="GROUP_1:def.1">GROUP_1:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="vectmetr.html#K4" title="VECTMETR:func.4">RLMSpace</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">h1</font>
<b>by </b><i><a class="txt" href="vectmetr.html#E2:33"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="vectmetr.html#E2:33_1"><i><font color="Green" title="E15">A2</font></i></a>, <a class="txt" href="vectmetr.html#E4:33_1"><i><font color="Green" title="E16">A3</font></i></a>, <a class="ref" href="vectmetr.html#D9" target="_self" title="VECTMETR:def.9">Def9</a></i>
<br/>.= 
<font color="Maroon" title="c3">h</font>
<b>by </b><i><a class="txt" href="vectmetr.html#E4:33_1"><i><font color="Green" title="E16">A3</font></i></a>, <a class="ref" href="funct_2.html#T23" title="FUNCT_2:th.23">FUNCT_2:23</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:33"/>
 <a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <span class="p1">(<span class="default"><a href="vectmetr.html#K5" title="VECTMETR:func.5">IsomGroup</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="grcat_1.html#K7" title="GRCAT_1:func.7">id</a> <span class="p1">(<span class="default"><a href="vectmetr.html#K4" title="VECTMETR:func.4">RLMSpace</a> <font color="Maroon" title="c1">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="group_1.html#T10" title="GROUP_1:th.10">GROUP_1:10</a></i>;<br/>


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