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

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">y</font>, <font color="Maroon" title="c4">z</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="E1:32_1_1_1"/>
<font color="Maroon" title="c2">x</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="E2:32_1_1_1"/><b>then </b><i><font color="Green" title="E13">A1</font></i>: 
<font color="Maroon" title="c2">x</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="c5">x1</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="E4:32_1_1_1"/><i><font color="Green" title="E14">A2</font></i>: 
<font color="Maroon" title="c5">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font>
 <b>and </b><br/><a NAME="E5:32_1_1_1"/><i><font color="Green" title="E15">A3</font></i>: 
<font color="Maroon" title="c5">x1</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/><a NAME="E6:32_1_1_1"/>
<font color="Maroon" title="c3">y</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="E7:32_1_1_1"/><b>then </b><i><font color="Green" title="E16">A4</font></i>: 
<font color="Maroon" title="c3">y</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="c6">y1</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="E9:32_1_1_1"/><i><font color="Green" title="E17">A5</font></i>: 
<font color="Maroon" title="c6">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">y</font>
 <b>and </b><br/><a NAME="E10:32_1_1_1"/><i><font color="Green" title="E18">A6</font></i>: 
<font color="Maroon" title="c6">y1</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/><a NAME="E11:32_1_1_1"/>
<font color="Maroon" title="c4">z</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="E12:32_1_1_1"/><b>then </b><i><font color="Green" title="E19">A7</font></i>: 
<font color="Maroon" title="c4">z</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="c7">z1</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="E14:32_1_1_1"/><i><font color="Green" title="E20">A8</font></i>: 
<font color="Maroon" title="c7">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">z</font>
 <b>and </b><br/><a NAME="E15:32_1_1_1"/><i><font color="Green" title="E21">A9</font></i>: 
<font color="Maroon" title="c7">z1</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/><a NAME="E16:32_1_1_1"/>
<font color="Maroon" title="c5">x1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c6">y1</font> is <a href="vectmetr.html#V3" title="VECTMETR:attr.3">isometric</a>
 <b>by </b><i><a class="txt" href="vectmetr.html#E5:32_1_1_1"><i><font color="Green" title="E15">A3</font></i></a>, <a class="txt" href="vectmetr.html#E10:32_1_1_1"><i><font color="Green" title="E18">A6</font></i></a>, <a class="ref" href="vectmetr.html#T4" target="_self" title="VECTMETR:th.4">Th4</a></i>;<br/><a NAME="E17:32_1_1_1"/><b>then </b><i><font color="Green" title="E22">A10</font></i>: 
<font color="Maroon" title="c5">x1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c6">y1</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#D4" target="_self" title="VECTMETR:def.4">Def4</a></i>;<br/><a NAME="E18:32_1_1_1"/>
<font color="Maroon" title="c6">y1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">z1</font> is <a href="vectmetr.html#V3" title="VECTMETR:attr.3">isometric</a>
 <b>by </b><i><a class="txt" href="vectmetr.html#E10:32_1_1_1"><i><font color="Green" title="E18">A6</font></i></a>, <a class="txt" href="vectmetr.html#E15:32_1_1_1"><i><font color="Green" title="E21">A9</font></i></a>, <a class="ref" href="vectmetr.html#T4" target="_self" title="VECTMETR:th.4">Th4</a></i>;<br/><a NAME="E19:32_1_1_1"/><b>then </b><i><font color="Green" title="E23">A11</font></i>: 
<font color="Maroon" title="c6">y1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">z1</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#D4" target="_self" title="VECTMETR:def.4">Def4</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c4">z</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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c3">y</font></span>)</span>,<font color="Maroon" title="c4">z</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/>.= 
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> <span class="p1">(<span class="default">the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <span class="p2">(<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">x</font>,<font color="Maroon" title="c3">y</font></span>)</span>,<font color="Maroon" title="c4">z</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/>.= 
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#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">x1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c6">y1</font></span>)</span>,<font color="Maroon" title="c4">z</font>
<b>by </b><i><a class="txt" href="vectmetr.html#E2:32_1_1_1"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="vectmetr.html#E4:32_1_1_1"><i><font color="Green" title="E14">A2</font></i></a>, <a class="txt" href="vectmetr.html#E7:32_1_1_1"><i><font color="Green" title="E16">A4</font></i></a>, <a class="txt" href="vectmetr.html#E9:32_1_1_1"><i><font color="Green" title="E17">A5</font></i></a>, <a class="ref" href="vectmetr.html#D9" target="_self" title="VECTMETR:def.9">Def9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">x1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c6">y1</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">z1</font>
<b>by </b><i><a class="txt" href="vectmetr.html#E12:32_1_1_1"><i><font color="Green" title="E19">A7</font></i></a>, <a class="txt" href="vectmetr.html#E14:32_1_1_1"><i><font color="Green" title="E20">A8</font></i></a>, <a class="txt" href="vectmetr.html#E17:32_1_1_1"><i><font color="Green" title="E22">A10</font></i></a>, <a class="ref" href="vectmetr.html#D9" target="_self" title="VECTMETR:def.9">Def9</a></i>
<br/>.= 
<font color="Maroon" title="c5">x1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">y1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">z1</font></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T55" title="RELAT_1:th.55">RELAT_1:55</a></i>
<br/>.= 
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#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c2">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">y1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">z1</font></span>)</span>
<b>by </b><i><a class="txt" href="vectmetr.html#E2:32_1_1_1"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="vectmetr.html#E4:32_1_1_1"><i><font color="Green" title="E14">A2</font></i></a>, <a class="txt" href="vectmetr.html#E19:32_1_1_1"><i><font color="Green" title="E23">A11</font></i></a>, <a class="ref" href="vectmetr.html#D9" target="_self" title="VECTMETR:def.9">Def9</a></i>
<br/>.= 
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">x</font>,<span class="p1">(<span class="default">the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <span class="p2">(<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">y</font>,<font color="Maroon" title="c4">z</font></span>)</span>
<b>by </b><i><a class="txt" href="vectmetr.html#E7:32_1_1_1"><i><font color="Green" title="E16">A4</font></i></a>, <a class="txt" href="vectmetr.html#E9:32_1_1_1"><i><font color="Green" title="E17">A5</font></i></a>, <a class="txt" href="vectmetr.html#E12:32_1_1_1"><i><font color="Green" title="E19">A7</font></i></a>, <a class="txt" href="vectmetr.html#E14:32_1_1_1"><i><font color="Green" title="E20">A8</font></i></a>, <a class="ref" href="vectmetr.html#D9" target="_self" title="VECTMETR:def.9">Def9</a></i>
<br/>.= 
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">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c4">z</font></span>)</span>
<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="c2">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c4">z</font></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D1" title="GROUP_1:def.1">GROUP_1:def 1</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:32_1_1"/>
 <a href="vectmetr.html#K5" title="VECTMETR:func.5">IsomGroup</a> <font color="Maroon" title="c1">n</font> is <a href="group_1.html#V4" title="GROUP_1:attr.4">associative</a>
 <b>by </b><i><a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a></i>;<br/>

<a NAME="E3:32_1_1"/>
 ex <font color="Olive" title="b1">e</font> being  <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> st <br/>for <font color="Olive" title="b2">h</font> being  <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> holds <br/> ( <font color="Olive" title="b2">h</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b1">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">h</font> &amp; <font color="Olive" title="b1">e</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b2">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">h</font> &amp;  ex <font color="Olive" title="b3">g</font> being  <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> st <br/>( <font color="Olive" title="b2">h</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">e</font> &amp; <font color="Olive" title="b3">g</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b2">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">e</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:32_1_1_2"/>
 <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="E2:32_1_1_2"/><b>then </b><i><font color="Green" title="E13">A12</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/>
<b>take </b>
<font color="Maroon" title="c2">e</font>
;<br/>

<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="E4:32_1_1_2"/>
<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="E5:32_1_1_2"/><b>then </b><i><font color="Green" title="E14">A13</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="E7:32_1_1_2"/><i><font color="Green" title="E15">A14</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="E8:32_1_1_2"/><i><font color="Green" title="E16">A15</font></i>: 
<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#E2:32_1_1_2"><i><font color="Green" title="E13">A12</font></i></a>, <a class="txt" href="vectmetr.html#E5:32_1_1_2"><i><font color="Green" title="E14">A13</font></i></a>, <a class="txt" href="vectmetr.html#E7:32_1_1_2"><i><font color="Green" title="E15">A14</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#E7:32_1_1_2"><i><font color="Green" title="E15">A14</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#E2:32_1_1_2"><i><font color="Green" title="E13">A12</font></i></a>, <a class="txt" href="vectmetr.html#E5:32_1_1_2"><i><font color="Green" title="E14">A13</font></i></a>, <a class="txt" href="vectmetr.html#E7:32_1_1_2"><i><font color="Green" title="E15">A14</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#E7:32_1_1_2"><i><font color="Green" title="E15">A14</font></i></a>, <a class="ref" href="funct_2.html#T23" title="FUNCT_2:th.23">FUNCT_2:23</a></i>
;<br/>

<a NAME="E11:32_1_1_2"/>
<font color="Maroon" title="c4">h1</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a>  is <a href="vectmetr.html#V3" title="VECTMETR:attr.3">isometric</a>
 
<b>by </b><i><a class="txt" href="vectmetr.html#E8:32_1_1_2"><i><font color="Green" title="E16">A15</font></i></a>, <a class="ref" href="vectmetr.html#T3" target="_self" title="VECTMETR:th.3">Th3</a></i>;<br/>
<a NAME="E12:32_1_1_2"/><b>then </b><i><font color="Green" title="E17">A16</font></i>: 
<font color="Maroon" title="c4">h1</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a>  <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="c5">g</font> = <font color="Maroon" title="c4">h1</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a>  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/>
<b>take </b>
<font color="Maroon" title="c5">g</font>
;<br/>

<a NAME="E14:32_1_1_2"/><i><font color="Green" title="E18">A17</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</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="txt" href="vectmetr.html#E8:32_1_1_2"><i><font color="Green" title="E16">A15</font></i></a>, <a class="ref" href="vectmetr.html#D3" target="_self" title="VECTMETR:def.3">Def3</a></i>;<br/>
<a NAME="E15:32_1_1_2"/><i><font color="Green" title="E19">A18</font></i>: 
<font color="Maroon" title="c4">h1</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="vectmetr.html#E8:32_1_1_2"><i><font color="Green" title="E16">A15</font></i></a>, <a class="ref" href="vectmetr.html#T2" target="_self" title="VECTMETR:th.2">Th2</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="c5">g</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="c5">g</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"><font color="Maroon" title="c4">h1</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span>
<b>by </b><i><a class="txt" href="vectmetr.html#E5:32_1_1_2"><i><font color="Green" title="E14">A13</font></i></a>, <a class="txt" href="vectmetr.html#E7:32_1_1_2"><i><font color="Green" title="E15">A14</font></i></a>, <a class="txt" href="vectmetr.html#E12:32_1_1_2"><i><font color="Green" title="E17">A16</font></i></a>, <a class="ref" href="vectmetr.html#D9" target="_self" title="VECTMETR:def.9">Def9</a></i>
<br/>.= 
<font color="Maroon" title="c2">e</font>
<b>by </b><i><a class="txt" href="vectmetr.html#E14:32_1_1_2"><i><font color="Green" title="E18">A17</font></i></a>, <a class="txt" href="vectmetr.html#E15:32_1_1_2"><i><font color="Green" title="E19">A18</font></i></a>, <a class="ref" href="tops_2.html#T65" title="TOPS_2:th.65">TOPS_2:65</a></i>
;<br/>

<b>thus </b><font color="Maroon" title="c5">g</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="c5">g</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"><font color="Maroon" title="c4">h1</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </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#E5:32_1_1_2"><i><font color="Green" title="E14">A13</font></i></a>, <a class="txt" href="vectmetr.html#E7:32_1_1_2"><i><font color="Green" title="E15">A14</font></i></a>, <a class="txt" href="vectmetr.html#E12:32_1_1_2"><i><font color="Green" title="E17">A16</font></i></a>, <a class="ref" href="vectmetr.html#D9" target="_self" title="VECTMETR:def.9">Def9</a></i>
<br/>.= 
 <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">h1</font></span>)</span>
<b>by </b><i><a class="txt" href="vectmetr.html#E14:32_1_1_2"><i><font color="Green" title="E18">A17</font></i></a>, <a class="txt" href="vectmetr.html#E15:32_1_1_2"><i><font color="Green" title="E19">A18</font></i></a>, <a class="ref" href="tops_2.html#T65" title="TOPS_2:th.65">TOPS_2:65</a></i>
<br/>.= 
<font color="Maroon" title="c2">e</font>
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>
;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:32_1_1"/>
 <a href="vectmetr.html#K5" title="VECTMETR:func.5">IsomGroup</a> <font color="Maroon" title="c1">n</font> is <a href="group_1.html#V3" title="GROUP_1:attr.3">Group-like</a>
 <b>by </b><i><a class="ref" href="group_1.html#D3" title="GROUP_1:def.3">GROUP_1:def 3</a></i>;<br/>


</div>
