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<span class="kw">set </span><font color="Maroon" title="c3">S</font> =  <a href="vectmetr.html#K6" title="VECTMETR:func.6">SubIsomGroupRel</a> <font color="Maroon" title="c2">G</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c4">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/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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> implies <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> )</span><br/><span class="kw">assume </span><a NAME="E1:34_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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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></span><br/><span class="kw">then </span><span class="kw">reconsider </span><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="E3:34_1_1_1"/>
 <a href="group_1.html#K1" title="GROUP_1:func.1">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vectmetr.html#T10" target="_self" title="VECTMETR:th.10">Th10</a></span>;<br/><a NAME="E4:34_1_1_1"/><span class="kw">then </span>
 <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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_2.html#T46" title="GROUP_2:th.46">GROUP_2:46</a></span>;<br/><a NAME="E5:34_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E12">A1</font></span>: 
 <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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></span>;<br/><a NAME="E6:34_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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T18" title="FUNCT_1:th.18">FUNCT_1:18</a></span>;<br/><span class="kw">hence </span><a NAME="E7:34_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:34_1_1_1"><span class="lab"><font color="Green" title="E12">A1</font></span></a>, <a class="ref" href="vectmetr.html#D10" target="_self" title="VECTMETR:def.10">Def10</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E2:34_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E12">A2</font></span>: 
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_2.html#D1" title="RELAT_2:def.1">RELAT_2:def 1</a></span>;<br/>
<a NAME="E3:34_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E13">A3</font></span>: 
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="orders_1.html#T13" title="ORDERS_1:th.13">ORDERS_1:13</a></span>;<br/>
<a NAME="E4:34_1_1"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:34_1_1"><span class="lab"><font color="Green" title="E12">A2</font></span></a>, <a class="ref" href="orders_1.html#T13" title="ORDERS_1:th.13">ORDERS_1:13</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:34_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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D2" title="PARTFUN1:def.2">PARTFUN1:def 2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <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>  &amp;  <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>  )</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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> &amp; <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> &amp; <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> implies <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> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:34_1_1_2"/><span class="lab"><font color="Green" title="E14">A4</font></span>: 
( <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> &amp; <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> )
 <span class="kw">and </span><br/><a NAME="E2:34_1_1_2"/><span class="lab"><font color="Green" title="E15">A5</font></span>: 
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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></span><br/><span class="kw">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34_1_1_2"><span class="lab"><font color="Green" title="E14">A4</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c9">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E5:34_1_1_2"/><span class="lab"><font color="Green" title="E16">A6</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E6:34_1_1_2"/><span class="lab"><font color="Green" title="E17">A7</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:34_1_1_2"><span class="lab"><font color="Green" title="E15">A5</font></span></a>, <a class="ref" href="vectmetr.html#D10" target="_self" title="VECTMETR:def.10">Def10</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c10">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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:34_1_1_2"><span class="lab"><font color="Green" title="E16">A6</font></span></a></span>;<br/><a NAME="E8:34_1_1_2"/><span class="lab"><font color="Green" title="E18">A8</font></span>: 
 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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_2.html#D5" title="GROUP_2:def.5">GROUP_2:def 5</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">f3</font> = <font color="Maroon" title="c10">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> <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/><a NAME="E10:34_1_1_2"/>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:34_1_1_2"><span class="lab"><font color="Green" title="E16">A6</font></span></a>, <a class="txt" href="#E8:34_1_1_2"><span class="lab"><font color="Green" title="E18">A8</font></span></a></span>;<br/><a NAME="E11:34_1_1_2"/><span class="kw">then </span>
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vectmetr.html#D9" target="_self" title="VECTMETR:def.9">Def9</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">f2</font> = <font color="Maroon" title="c9">f</font> as   <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>   <a href="vectmetr.html#V3" title="VECTMETR:attr.3">isometric</a>  <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> <span class="kw">by</span> <span class="lab"><a class="ref" href="vectmetr.html#D4" target="_self" title="VECTMETR:def.4">Def4</a></span>;<br/><a NAME="E13:34_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="E14:34_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c12">f2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/><a NAME="E15:34_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E19">A9</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:34_1_1_2"><span class="lab"><font color="Green" title="E17">A7</font></span></a>, <a class="ref" href="funct_1.html#T34" title="FUNCT_1:th.34">FUNCT_1:34</a></span>;<br/><a NAME="E16:34_1_1_2"/><span class="lab"><font color="Green" title="E20">A10</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></span>;<br/><font color="Maroon" title="c10">f1</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a>  = 
<font color="Maroon" title="c11">f3</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_2.html#T48" title="GROUP_2:th.48">GROUP_2:48</a></span>
<br/>.= 
<font color="Maroon" title="c12">f2</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectmetr.html#T11" target="_self" title="VECTMETR:th.11">Th11</a></span>
<br/>.= 
<font color="Maroon" title="c9">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:34_1_1_2"><span class="lab"><font color="Green" title="E20">A10</font></span></a>, <a class="ref" href="tops_2.html#D4" title="TOPS_2:def.4">TOPS_2:def 4</a></span>
;<br/><span class="kw">hence </span><a NAME="E18:34_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:34_1_1_2"><span class="lab"><font color="Green" title="E19">A9</font></span></a>, <a class="ref" href="vectmetr.html#D10" target="_self" title="VECTMETR:def.10">Def10</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E7:34_1_1"/><span class="kw">then </span>
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_2.html#D3" title="RELAT_2:def.3">RELAT_2:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:34_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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:34_1_1"><span class="lab"><font color="Green" title="E13">A3</font></span></a>, <a class="ref" href="relat_2.html#D11" title="RELAT_2:def.11">RELAT_2:def 11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <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> </span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_1_3"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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> &amp; <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> &amp; <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> &amp; <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> &amp; <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> implies <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> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:34_1_1_3"/><span class="lab"><font color="Green" title="E14">A11</font></span>: 
( <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> &amp; <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> &amp; <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> )
 <span class="kw">and </span><br/><a NAME="E2:34_1_1_3"/><span class="lab"><font color="Green" title="E15">A12</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E3:34_1_1_3"/><span class="lab"><font color="Green" title="E16">A13</font></span>: 
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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></span><br/><span class="kw">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34_1_1_3"><span class="lab"><font color="Green" title="E14">A11</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c11">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E6:34_1_1_3"/><span class="lab"><font color="Green" title="E17">A14</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E7:34_1_1_3"/><span class="lab"><font color="Green" title="E18">A15</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:34_1_1_3"><span class="lab"><font color="Green" title="E15">A12</font></span></a>, <a class="ref" href="vectmetr.html#D10" target="_self" title="VECTMETR:def.10">Def10</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c12">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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:34_1_1_3"><span class="lab"><font color="Green" title="E17">A14</font></span></a></span>;<br/><a NAME="E9:34_1_1_3"/><span class="lab"><font color="Green" title="E19">A16</font></span>: 
 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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_2.html#D5" title="GROUP_2:def.5">GROUP_2:def 5</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c13">f3</font> = <font color="Maroon" title="c12">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> <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/><a NAME="E11:34_1_1_3"/>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:34_1_1_3"><span class="lab"><font color="Green" title="E17">A14</font></span></a>, <a class="txt" href="#E9:34_1_1_3"><span class="lab"><font color="Green" title="E19">A16</font></span></a></span>;<br/><a NAME="E12:34_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A17</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vectmetr.html#D9" target="_self" title="VECTMETR:def.9">Def9</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c14">g</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E14:34_1_1_3"/><span class="lab"><font color="Green" title="E21">A18</font></span>: 
<font color="Maroon" title="c14">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>
 <span class="kw">and </span><br/><a NAME="E15:34_1_1_3"/><span class="lab"><font color="Green" title="E22">A19</font></span>: 
<font color="Maroon" title="c14">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:34_1_1_3"><span class="lab"><font color="Green" title="E16">A13</font></span></a>, <a class="ref" href="vectmetr.html#D10" target="_self" title="VECTMETR:def.10">Def10</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c15">g2</font> = <font color="Maroon" title="c14">g</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:34_1_1_3"><span class="lab"><font color="Green" title="E21">A18</font></span></a></span>;<br/><a NAME="E17:34_1_1_3"/><span class="lab"><font color="Green" title="E23">A20</font></span>: 
<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/><span class="kw">reconsider </span><font color="Maroon" title="c16">g3</font> = <font color="Maroon" title="c15">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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:34_1_1_3"><span class="lab"><font color="Green" title="E19">A16</font></span></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/><a NAME="E19:34_1_1_3"/>
( <font color="Maroon" title="c12">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="c15">g2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">G</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></span>;<br/><a NAME="E20:34_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A21</font></span>: 
<font color="Maroon" title="c16">g3</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c13">f3</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">G</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_2.html#T50" title="GROUP_2:th.50">GROUP_2:50</a></span>;<br/><a NAME="E21:34_1_1_3"/>
<font color="Maroon" title="c14">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:34_1_1_3"><span class="lab"><font color="Green" title="E21">A18</font></span></a>, <a class="txt" href="#E9:34_1_1_3"><span class="lab"><font color="Green" title="E19">A16</font></span></a></span>;<br/><a NAME="E22:34_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c14">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vectmetr.html#D9" target="_self" title="VECTMETR:def.9">Def9</a></span>;<br/><a NAME="E23:34_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c16">g3</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c13">f3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">g</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c11">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:34_1_1_3"><span class="lab"><font color="Green" title="E20">A17</font></span></a>, <a class="ref" href="vectmetr.html#D9" target="_self" title="VECTMETR:def.9">Def9</a></span>;<br/><a NAME="E24:34_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A22</font></span>: 
<font color="Maroon" title="c14">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:34_1_1_3"><span class="lab"><font color="Green" title="E24">A21</font></span></a>, <a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></span>;<br/><a NAME="E25:34_1_1_3"/>
<font color="Maroon" title="c11">f</font> is   <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>   <a href="vectmetr.html#V3" title="VECTMETR:attr.3">isometric</a>  <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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:34_1_1_3"><span class="lab"><font color="Green" title="E20">A17</font></span></a>, <a class="ref" href="vectmetr.html#D4" target="_self" title="VECTMETR:def.4">Def4</a></span>;<br/><a NAME="E26:34_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c8">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c11">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:34_1_1_3"><span class="lab"><font color="Green" title="E23">A20</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/><a NAME="E27:34_1_1_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c14">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:34_1_1_3"><span class="lab"><font color="Green" title="E18">A15</font></span></a>, <a class="txt" href="#E15:34_1_1_3"><span class="lab"><font color="Green" title="E22">A19</font></span></a>, <a class="ref" href="funct_1.html#T13" title="FUNCT_1:th.13">FUNCT_1:13</a></span>;<br/><span class="kw">hence </span><a NAME="E28:34_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:34_1_1_3"><span class="lab"><font color="Green" title="E25">A22</font></span></a>, <a class="ref" href="vectmetr.html#D10" target="_self" title="VECTMETR:def.10">Def10</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E10:34_1_1"/><span class="kw">then </span>
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_2.html#D8" title="RELAT_2:def.8">RELAT_2:def 8</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:34_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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:34_1_1"><span class="lab"><font color="Green" title="E13">A3</font></span></a>, <a class="ref" href="relat_2.html#D16" title="RELAT_2:def.16">RELAT_2:def 16</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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