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

<b>set </b><font color="Maroon" title="c3">X</font> = 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#K4" title="VECTMETR:func.4">RLMSpace</a> <font color="Maroon" title="c1">n</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c4">S</font> =  <a href="vectmetr.html#K6" title="VECTMETR:func.6">SubIsomGroupRel</a> <font color="Maroon" title="c2">G</font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:36_1_1_1"/>
<font color="Maroon" title="c5">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#K4" title="VECTMETR:func.4">RLMSpace</a> <font color="Maroon" title="c1">n</font></span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c6">x1</font> = <font color="Maroon" title="c5">x</font> 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#K4" title="VECTMETR:func.4">RLMSpace</a> <font color="Maroon" title="c1">n</font></span>)</span> ;<br/><a NAME="E3:36_1_1_1"/>
 <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="struct_0.html#K3" title="STRUCT_0:func.3">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="vectmetr.html#T10" target="_self" title="VECTMETR:th.10">Th10</a></i>;<br/><a NAME="E4:36_1_1_1"/><b>then </b>
 <a href="struct_0.html#K3" title="STRUCT_0:func.3">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="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">G</font>
 <b>by </b><i><a class="ref" href="group_2.html#T55" title="GROUP_2:th.55">GROUP_2:55</a></i>;<br/><a NAME="E5:36_1_1_1"/><b>then </b><i><font color="Green" title="E16">A1</font></i>: 
 <a href="struct_0.html#K3" title="STRUCT_0:func.3">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> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font>
 <b>by </b><i><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/><a NAME="E6:36_1_1_1"/>
<span class="p1">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <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="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x1</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</a></i>;<br/><b>hence </b><a NAME="E7:36_1_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c5">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="vectmetr.html#K6" title="VECTMETR:func.6">SubIsomGroupRel</a> <font color="Maroon" title="c2">G</font>
 <b>by </b><i><a class="txt" href="vectmetr.html#E5:36_1_1_1"><i><font color="Green" title="E16">A1</font></i></a>, <a class="ref" href="vectmetr.html#D10" target="_self" title="VECTMETR:def.10">Def10</a></i>;<br/></div><b>end;</b></div>
<a NAME="E2:36_1_1"/><b>then </b>
 <a href="vectmetr.html#K6" title="VECTMETR:func.6">SubIsomGroupRel</a> <font color="Maroon" title="c2">G</font> <a href="relat_2.html#R1" title="RELAT_2:pred.1">is_reflexive_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#K4" title="VECTMETR:func.4">RLMSpace</a> <font color="Maroon" title="c1">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_2.html#D1" title="RELAT_2:def.1">RELAT_2:def 1</a></i>;<br/>
<a NAME="E3:36_1_1"/><b>then </b><i><font color="Green" title="E16">A2</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="vectmetr.html#K6" title="VECTMETR:func.6">SubIsomGroupRel</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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#K4" title="VECTMETR:func.4">RLMSpace</a> <font color="Maroon" title="c1">n</font></span>)</span> &amp;  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><a href="vectmetr.html#K6" title="VECTMETR:func.6">SubIsomGroupRel</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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#K4" title="VECTMETR:func.4">RLMSpace</a> <font color="Maroon" title="c1">n</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="orders_1.html#T98" title="ORDERS_1:th.98">ORDERS_1:98</a></i>;<br/>
<b>hence </b><a NAME="E4:36_1_1"/>
 <a href="vectmetr.html#K6" title="VECTMETR:func.6">SubIsomGroupRel</a> <font color="Maroon" title="c2">G</font> is <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>
 <b>by </b><i><a class="ref" href="partfun1.html#D4" title="PARTFUN1:def.4">PARTFUN1:def 4</a></i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c6">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E1:36_1_1_2"/><i><font color="Green" title="E17">A3</font></i>: 
<font color="Maroon" title="c5">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#K4" title="VECTMETR:func.4">RLMSpace</a> <font color="Maroon" title="c1">n</font></span>)</span>
 <b>and </b><br/><a NAME="E2:36_1_1_2"/><i><font color="Green" title="E18">A4</font></i>: 
<font color="Maroon" title="c6">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#K4" title="VECTMETR:func.4">RLMSpace</a> <font color="Maroon" title="c1">n</font></span>)</span>
 <b>and </b><br/><a NAME="E3:36_1_1_2"/><i><font color="Green" title="E19">A5</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="vectmetr.html#K6" title="VECTMETR:func.6">SubIsomGroupRel</a> <font color="Maroon" title="c2">G</font>
 ;<br/><b>reconsider </b><font color="Maroon" title="c7">x1</font> = <font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c8">y1</font> = <font color="Maroon" title="c6">y</font> 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#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#E1:36_1_1_2"><i><font color="Green" title="E17">A3</font></i></a>, <a class="txt" href="vectmetr.html#E2:36_1_1_2"><i><font color="Green" title="E18">A4</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c9">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E6:36_1_1_2"/><i><font color="Green" title="E20">A6</font></i>: 
<font color="Maroon" title="c9">f</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 <font color="Maroon" title="c2">G</font>
 <b>and </b><br/><a NAME="E7:36_1_1_2"/><i><font color="Green" title="E21">A7</font></i>: 
<font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">y1</font>
 <b>by </b><i><a class="txt" href="vectmetr.html#E3:36_1_1_2"><i><font color="Green" title="E19">A5</font></i></a>, <a class="ref" href="vectmetr.html#D10" target="_self" title="VECTMETR:def.10">Def10</a></i>;<br/><a NAME="E8:36_1_1_2"/><i><font color="Green" title="E22">A8</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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>
 <b>by </b><i><a class="ref" href="group_2.html#D5" title="GROUP_2:def.5">GROUP_2:def 5</a></i>;<br/><a NAME="E9:36_1_1_2"/><b>then </b>
<font color="Maroon" title="c9">f</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>
 <b>by </b><i><a class="txt" href="vectmetr.html#E6:36_1_1_2"><i><font color="Green" title="E20">A6</font></i></a></i>;<br/><a NAME="E10:36_1_1_2"/><b>then </b>
<font color="Maroon" title="c9">f</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="c10">f2</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="E12:36_1_1_2"/><i><font color="Green" title="E23">A9</font></i>: 
<font color="Maroon" title="c10">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">f</font>
 <b>and </b><br/><a NAME="E13:36_1_1_2"/><i><font color="Green" title="E24">A10</font></i>: 
<font color="Maroon" title="c10">f2</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>reconsider </b><font color="Maroon" title="c11">f1</font> = <font color="Maroon" title="c9">f</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font> <b>by </b><i><a class="txt" href="vectmetr.html#E6:36_1_1_2"><i><font color="Green" title="E20">A6</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c12">f3</font> = <font color="Maroon" title="c11">f1</font> 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="txt" href="vectmetr.html#E8:36_1_1_2"><i><font color="Green" title="E22">A8</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><a NAME="E16:36_1_1_2"/><i><font color="Green" title="E25">A11</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c10">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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#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#E13:36_1_1_2"><i><font color="Green" title="E24">A10</font></i></a>, <a class="ref" href="vectmetr.html#D3" target="_self" title="VECTMETR:def.3">Def3</a></i>;<br/><a NAME="E17:36_1_1_2"/><i><font color="Green" title="E26">A12</font></i>: 
<font color="Maroon" title="c10">f2</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#E13:36_1_1_2"><i><font color="Green" title="E24">A10</font></i></a>, <a class="ref" href="vectmetr.html#T2" target="_self" title="VECTMETR:th.2">Th2</a></i>;<br/><i><font color="Green" title="E27">A13</font></i>: <font color="Maroon" title="c11">f1</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a>  = 
<font color="Maroon" title="c12">f3</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> 
<b>by </b><i><a class="ref" href="group_2.html#T57" title="GROUP_2:th.57">GROUP_2:57</a></i>
<br/>.= 
<font color="Maroon" title="c10">f2</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> 
<b>by </b><i><a class="txt" href="vectmetr.html#E12:36_1_1_2"><i><font color="Green" title="E23">A9</font></i></a>, <a class="ref" href="vectmetr.html#T11" target="_self" title="VECTMETR:th.11">Th11</a></i>
<br/>.= 
<font color="Maroon" title="c9">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> 
<b>by </b><i><a class="txt" href="vectmetr.html#E12:36_1_1_2"><i><font color="Green" title="E23">A9</font></i></a>, <a class="txt" href="vectmetr.html#E16:36_1_1_2"><i><font color="Green" title="E25">A11</font></i></a>, <a class="txt" href="vectmetr.html#E17:36_1_1_2"><i><font color="Green" title="E26">A12</font></i></a>, <a class="ref" href="tops_2.html#D4" title="TOPS_2:def.4">TOPS_2:def 4</a></i>
;<br/><a NAME="E19:36_1_1_2"/>
<font color="Maroon" title="c7">x1</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#K4" title="VECTMETR:func.4">RLMSpace</a> <font color="Maroon" title="c1">n</font></span>)</span>
 ;<br/><a NAME="E20:36_1_1_2"/><b>then </b>
<font color="Maroon" title="c7">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c9">f</font>
 <b>by </b><i><a class="txt" href="vectmetr.html#E12:36_1_1_2"><i><font color="Green" title="E23">A9</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E21:36_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x1</font>
 <b>by </b><i><a class="txt" href="vectmetr.html#E7:36_1_1_2"><i><font color="Green" title="E21">A7</font></i></a>, <a class="txt" href="vectmetr.html#E12:36_1_1_2"><i><font color="Green" title="E23">A9</font></i></a>, <a class="txt" href="vectmetr.html#E17:36_1_1_2"><i><font color="Green" title="E26">A12</font></i></a>, <a class="ref" href="funct_1.html#T56" title="FUNCT_1:th.56">FUNCT_1:56</a></i>;<br/><b>hence </b><a NAME="E22:36_1_1_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c5">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="vectmetr.html#K6" title="VECTMETR:func.6">SubIsomGroupRel</a> <font color="Maroon" title="c2">G</font>
 <b>by </b><i><a class="txt" href="vectmetr.html#E18:36_1_1_2"><i><font color="Green" title="E27">A13</font></i></a>, <a class="ref" href="vectmetr.html#D10" target="_self" title="VECTMETR:def.10">Def10</a></i>;<br/></div><b>end;</b></div>
<a NAME="E6:36_1_1"/><b>then </b>
 <a href="vectmetr.html#K6" title="VECTMETR:func.6">SubIsomGroupRel</a> <font color="Maroon" title="c2">G</font> <a href="relat_2.html#R3" title="RELAT_2:pred.3">is_symmetric_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#K4" title="VECTMETR:func.4">RLMSpace</a> <font color="Maroon" title="c1">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_2.html#D3" title="RELAT_2:def.3">RELAT_2:def 3</a></i>;<br/>
<b>hence </b><a NAME="E7:36_1_1"/>
 <a href="vectmetr.html#K6" title="VECTMETR:func.6">SubIsomGroupRel</a> <font color="Maroon" title="c2">G</font> is <a href="relat_2.html#V3" title="RELAT_2:attr.3">symmetric</a>
 <b>by </b><i><a class="txt" href="vectmetr.html#E3:36_1_1"><i><font color="Green" title="E16">A2</font></i></a>, <a class="ref" href="relat_2.html#D11" title="RELAT_2:def.11">RELAT_2:def 11</a></i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c6">y</font>, <font color="Maroon" title="c7">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E1:36_1_1_3"/><i><font color="Green" title="E17">A14</font></i>: 
<font color="Maroon" title="c5">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#K4" title="VECTMETR:func.4">RLMSpace</a> <font color="Maroon" title="c1">n</font></span>)</span>
 <b>and </b><br/><a NAME="E2:36_1_1_3"/><i><font color="Green" title="E18">A15</font></i>: 
<font color="Maroon" title="c6">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#K4" title="VECTMETR:func.4">RLMSpace</a> <font color="Maroon" title="c1">n</font></span>)</span>
 <b>and </b><br/><a NAME="E3:36_1_1_3"/><i><font color="Green" title="E19">A16</font></i>: 
<font color="Maroon" title="c7">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#K4" title="VECTMETR:func.4">RLMSpace</a> <font color="Maroon" title="c1">n</font></span>)</span>
 <b>and </b><br/><a NAME="E4:36_1_1_3"/><i><font color="Green" title="E20">A17</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="vectmetr.html#K6" title="VECTMETR:func.6">SubIsomGroupRel</a> <font color="Maroon" title="c2">G</font>
 <b>and </b><br/><a NAME="E5:36_1_1_3"/><i><font color="Green" title="E21">A18</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c7">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="vectmetr.html#K6" title="VECTMETR:func.6">SubIsomGroupRel</a> <font color="Maroon" title="c2">G</font>
 ;<br/><b>reconsider </b><font color="Maroon" title="c8">x1</font> = <font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c9">y1</font> = <font color="Maroon" title="c6">y</font>, <font color="Maroon" title="c10">z1</font> = <font color="Maroon" title="c7">z</font> 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#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#E1:36_1_1_3"><i><font color="Green" title="E17">A14</font></i></a>, <a class="txt" href="vectmetr.html#E2:36_1_1_3"><i><font color="Green" title="E18">A15</font></i></a>, <a class="txt" href="vectmetr.html#E3:36_1_1_3"><i><font color="Green" title="E19">A16</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c11">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E8:36_1_1_3"/><i><font color="Green" title="E22">A19</font></i>: 
<font color="Maroon" title="c11">f</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 <font color="Maroon" title="c2">G</font>
 <b>and </b><br/><a NAME="E9:36_1_1_3"/><i><font color="Green" title="E23">A20</font></i>: 
<font color="Maroon" title="c11">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y1</font>
 <b>by </b><i><a class="txt" href="vectmetr.html#E4:36_1_1_3"><i><font color="Green" title="E20">A17</font></i></a>, <a class="ref" href="vectmetr.html#D10" target="_self" title="VECTMETR:def.10">Def10</a></i>;<br/><b>consider </b><font color="Maroon" title="c12">g</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E11:36_1_1_3"/><i><font color="Green" title="E24">A21</font></i>: 
<font color="Maroon" title="c12">g</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 <font color="Maroon" title="c2">G</font>
 <b>and </b><br/><a NAME="E12:36_1_1_3"/><i><font color="Green" title="E25">A22</font></i>: 
<font color="Maroon" title="c12">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">z1</font>
 <b>by </b><i><a class="txt" href="vectmetr.html#E5:36_1_1_3"><i><font color="Green" title="E21">A18</font></i></a>, <a class="ref" href="vectmetr.html#D10" target="_self" title="VECTMETR:def.10">Def10</a></i>;<br/><a NAME="E13:36_1_1_3"/><i><font color="Green" title="E26">A23</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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>
 <b>by </b><i><a class="ref" href="group_2.html#D5" title="GROUP_2:def.5">GROUP_2:def 5</a></i>;<br/><a NAME="E14:36_1_1_3"/><b>then </b>
<font color="Maroon" title="c11">f</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>
 <b>by </b><i><a class="txt" href="vectmetr.html#E8:36_1_1_3"><i><font color="Green" title="E22">A19</font></i></a></i>;<br/><a NAME="E15:36_1_1_3"/><b>then </b><i><font color="Green" title="E27">A24</font></i>: 
<font color="Maroon" title="c11">f</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="c13">f1</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="E17:36_1_1_3"/><i><font color="Green" title="E28">A25</font></i>: 
<font color="Maroon" title="c13">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">f</font>
 <b>and </b><br/><a NAME="E18:36_1_1_3"/>
<font color="Maroon" title="c13">f1</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>reconsider </b><font color="Maroon" title="c14">f2</font> = <font color="Maroon" title="c11">f</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font> <b>by </b><i><a class="txt" href="vectmetr.html#E8:36_1_1_3"><i><font color="Green" title="E22">A19</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c15">f3</font> = <font color="Maroon" title="c14">f2</font> 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="txt" href="vectmetr.html#E13:36_1_1_3"><i><font color="Green" title="E26">A23</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><a NAME="E21:36_1_1_3"/>
<font color="Maroon" title="c12">g</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>
 <b>by </b><i><a class="txt" href="vectmetr.html#E11:36_1_1_3"><i><font color="Green" title="E24">A21</font></i></a>, <a class="txt" href="vectmetr.html#E13:36_1_1_3"><i><font color="Green" title="E26">A23</font></i></a></i>;<br/><a NAME="E22:36_1_1_3"/><b>then </b><i><font color="Green" title="E29">A26</font></i>: 
<font color="Maroon" title="c12">g</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>reconsider </b><font color="Maroon" title="c16">g2</font> = <font color="Maroon" title="c12">g</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font> <b>by </b><i><a class="txt" href="vectmetr.html#E11:36_1_1_3"><i><font color="Green" title="E24">A21</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c17">g3</font> = <font color="Maroon" title="c16">g2</font> 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="txt" href="vectmetr.html#E13:36_1_1_3"><i><font color="Green" title="E26">A23</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><i><font color="Green" title="E30">A27</font></i>: <font color="Maroon" title="c17">g3</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c15">f3</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#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c12">g</font>,<font color="Maroon" title="c11">f</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="c12">g</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c11">f</font>
<b>by </b><i><a class="txt" href="vectmetr.html#E15:36_1_1_3"><i><font color="Green" title="E27">A24</font></i></a>, <a class="txt" href="vectmetr.html#E22:36_1_1_3"><i><font color="Green" title="E29">A26</font></i></a>, <a class="ref" href="vectmetr.html#D9" target="_self" title="VECTMETR:def.9">Def9</a></i>
;<br/><a NAME="E26:36_1_1_3"/>
( <font color="Maroon" title="c14">f2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c16">g2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">G</font> )
 <b>by </b><i><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/><a NAME="E27:36_1_1_3"/><b>then </b>
<font color="Maroon" title="c17">g3</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c15">f3</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">G</font>
 <b>by </b><i><a class="ref" href="group_2.html#T59" title="GROUP_2:th.59">GROUP_2:59</a></i>;<br/><a NAME="E28:36_1_1_3"/><b>then </b><i><font color="Green" title="E31">A28</font></i>: 
<font color="Maroon" title="c12">g</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c11">f</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 <font color="Maroon" title="c2">G</font>
 <b>by </b><i><a class="txt" href="vectmetr.html#E25:36_1_1_3"><i><font color="Green" title="E30">A27</font></i></a>, <a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/><a NAME="E29:36_1_1_3"/>
<font color="Maroon" title="c8">x1</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#K4" title="VECTMETR:func.4">RLMSpace</a> <font color="Maroon" title="c1">n</font></span>)</span>
 ;<br/><a NAME="E30:36_1_1_3"/><b>then </b>
<font color="Maroon" title="c8">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c11">f</font>
 <b>by </b><i><a class="txt" href="vectmetr.html#E17:36_1_1_3"><i><font color="Green" title="E28">A25</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E31:36_1_1_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c12">g</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">z1</font>
 <b>by </b><i><a class="txt" href="vectmetr.html#E9:36_1_1_3"><i><font color="Green" title="E23">A20</font></i></a>, <a class="txt" href="vectmetr.html#E12:36_1_1_3"><i><font color="Green" title="E25">A22</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>;<br/><b>hence </b><a NAME="E32:36_1_1_3"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c7">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="vectmetr.html#K6" title="VECTMETR:func.6">SubIsomGroupRel</a> <font color="Maroon" title="c2">G</font>
 <b>by </b><i><a class="txt" href="vectmetr.html#E28:36_1_1_3"><i><font color="Green" title="E31">A28</font></i></a>, <a class="ref" href="vectmetr.html#D10" target="_self" title="VECTMETR:def.10">Def10</a></i>;<br/></div><b>end;</b></div>
<a NAME="E9:36_1_1"/><b>then </b>
 <a href="vectmetr.html#K6" title="VECTMETR:func.6">SubIsomGroupRel</a> <font color="Maroon" title="c2">G</font> <a href="relat_2.html#R8" title="RELAT_2:pred.8">is_transitive_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#K4" title="VECTMETR:func.4">RLMSpace</a> <font color="Maroon" title="c1">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_2.html#D8" title="RELAT_2:def.8">RELAT_2:def 8</a></i>;<br/>
<b>hence </b><a NAME="E10:36_1_1"/>
 <a href="vectmetr.html#K6" title="VECTMETR:func.6">SubIsomGroupRel</a> <font color="Maroon" title="c2">G</font> is <a href="relat_2.html#V8" title="RELAT_2:attr.8">transitive</a>
 <b>by </b><i><a class="txt" href="vectmetr.html#E3:36_1_1"><i><font color="Green" title="E16">A2</font></i></a>, <a class="ref" href="relat_2.html#D16" title="RELAT_2:def.16">RELAT_2:def 16</a></i>;<br/>


</div>
