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<div class="add">

<b>let </b><font color="Maroon" title="c1">AFV</font> be  <a href="analoaf.html#V1" title="ANALOAF:attr.1">strict</a> <a href="tdgroup.html#NM3" title="TDGROUP:NM.3">AffVect</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">o</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFV</font> holds  <font color="Maroon" title="c1">AFV</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tdgroup.html#K2" title="TDGROUP:func.2">AV</a> <span class="p1">(<span class="default"><a href="afvect0.html#K5" title="AFVECT0:func.5">GroupVect</a> <font color="Maroon" title="c1">AFV</font>,<font color="Olive" title="b1">o</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c2">o</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFV</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">AFV</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tdgroup.html#K2" title="TDGROUP:func.2">AV</a> <span class="p1">(<span class="default"><a href="afvect0.html#K5" title="AFVECT0:func.5">GroupVect</a> <font color="Maroon" title="c1">AFV</font>,<font color="Maroon" title="c2">o</font></span>)</span></span><br/>



<b>set </b><font color="Maroon" title="c3">X</font> =  <a href="afvect0.html#K5" title="AFVECT0:func.5">GroupVect</a> <font color="Maroon" title="c1">AFV</font>,<font color="Maroon" title="c2">o</font>;<br/>
<a NAME="E1:71"/><i><font color="Green" title="E44">A1</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="tdgroup.html#K2" title="TDGROUP:func.2">AV</a> <span class="p2">(<span class="default"><a href="afvect0.html#K5" title="AFVECT0:func.5">GroupVect</a> <font color="Maroon" title="c1">AFV</font>,<font color="Maroon" title="c2">o</font></span>)</span></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 <font color="Maroon" title="c1">AFV</font>
 
<b>by </b><i><a class="ref" href="tdgroup.html#T9" title="TDGROUP:th.9">TDGROUP:9</a></i>;<br/>
<a NAME="E2:71"/>
 <a href="tdgroup.html#K1" title="TDGROUP:func.1">CONGRD</a> <span class="p1">(<span class="default"><a href="afvect0.html#K5" title="AFVECT0:func.5">GroupVect</a> <font color="Maroon" title="c1">AFV</font>,<font color="Maroon" title="c2">o</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="analoaf.html#U1" title="ANALOAF:sel.1">CONGR</a> of <font color="Maroon" title="c1">AFV</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c5">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">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="tdgroup.html#K1" title="TDGROUP:func.1">CONGRD</a> <span class="p1">(<span class="default"><a href="afvect0.html#K5" title="AFVECT0:func.5">GroupVect</a> <font color="Maroon" title="c1">AFV</font>,<font color="Maroon" title="c2">o</font></span>)</span> iff <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">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> the <a href="analoaf.html#U1" title="ANALOAF:sel.1">CONGR</a> of <font color="Maroon" title="c1">AFV</font> )</span><br/><b>set </b><font color="Maroon" title="c6">xy</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>;<br/><div><i><font color="Green" title="E45">A2</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:71_1_1_1"/><i><font color="Green" title="E46">A3</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">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="tdgroup.html#K1" title="TDGROUP:func.1">CONGRD</a> <span class="p1">(<span class="default"><a href="afvect0.html#K5" title="AFVECT0:func.5">GroupVect</a> <font color="Maroon" title="c1">AFV</font>,<font color="Maroon" title="c2">o</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">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> the <a href="analoaf.html#U1" title="ANALOAF:sel.1">CONGR</a> of <font color="Maroon" title="c1">AFV</font></span><br/><b>set </b><font color="Maroon" title="c7">V</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="afvect0.html#K5" title="AFVECT0:func.5">GroupVect</a> <font color="Maroon" title="c1">AFV</font>,<font color="Maroon" title="c2">o</font></span>)</span>;<br/><b>set </b><font color="Maroon" title="c8">VV</font> = <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="afvect0.html#K5" title="AFVECT0:func.5">GroupVect</a> <font color="Maroon" title="c1">AFV</font>,<font color="Maroon" title="c2">o</font></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="afvect0.html#K5" title="AFVECT0:func.5">GroupVect</a> <font color="Maroon" title="c1">AFV</font>,<font color="Maroon" title="c2">o</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>;<br/><a NAME="E2:71_1_1_1"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">y</font> )
 <b>by </b><i><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>;<br/><a NAME="E3:71_1_1_1"/><b>then </b><i><font color="Green" title="E47">A4</font></i>: 
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="afvect0.html#K5" title="AFVECT0:func.5">GroupVect</a> <font color="Maroon" title="c1">AFV</font>,<font color="Maroon" title="c2">o</font></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="afvect0.html#K5" title="AFVECT0:func.5">GroupVect</a> <font color="Maroon" title="c1">AFV</font>,<font color="Maroon" title="c2">o</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp; <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="afvect0.html#K5" title="AFVECT0:func.5">GroupVect</a> <font color="Maroon" title="c1">AFV</font>,<font color="Maroon" title="c2">o</font></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="afvect0.html#K5" title="AFVECT0:func.5">GroupVect</a> <font color="Maroon" title="c1">AFV</font>,<font color="Maroon" title="c2">o</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )
 <b>by </b><i><a class="txt" href="#E1:71_1_1_1"><i><font color="Green" title="E46">A3</font></i></a>, <a class="ref" href="mcart_1.html#T10" title="MCART_1:th.10">MCART_1:10</a></i>;<br/><a NAME="E4:71_1_1_1"/><b>then </b><i><font color="Green" title="E48">A5</font></i>: 
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c4">x</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <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="afvect0.html#K5" title="AFVECT0:func.5">GroupVect</a> <font color="Maroon" title="c1">AFV</font>,<font color="Maroon" title="c2">o</font></span>)</span> &amp; <font color="Maroon" title="c4">x</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <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="afvect0.html#K5" title="AFVECT0:func.5">GroupVect</a> <font color="Maroon" title="c1">AFV</font>,<font color="Maroon" title="c2">o</font></span>)</span> &amp; <font color="Maroon" title="c5">y</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <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="afvect0.html#K5" title="AFVECT0:func.5">GroupVect</a> <font color="Maroon" title="c1">AFV</font>,<font color="Maroon" title="c2">o</font></span>)</span> &amp; <font color="Maroon" title="c5">y</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <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="afvect0.html#K5" title="AFVECT0:func.5">GroupVect</a> <font color="Maroon" title="c1">AFV</font>,<font color="Maroon" title="c2">o</font></span>)</span> )
 <b>by </b><i><a class="ref" href="mcart_1.html#T10" title="MCART_1:th.10">MCART_1:10</a>, <a class="ref" href="mcart_1.html#T23" title="MCART_1:th.23">MCART_1:23</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c9">x1'</font> = <font color="Maroon" title="c4">x</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> , <font color="Maroon" title="c10">x2'</font> = <font color="Maroon" title="c4">x</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> , <font color="Maroon" title="c11">y1'</font> = <font color="Maroon" title="c5">y</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> , <font color="Maroon" title="c12">y2'</font> = <font color="Maroon" title="c5">y</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="afvect0.html#K5" title="AFVECT0:func.5">GroupVect</a> <font color="Maroon" title="c1">AFV</font>,<font color="Maroon" title="c2">o</font></span>)</span> <b>by </b><i><a class="txt" href="#E3:71_1_1_1"><i><font color="Green" title="E47">A4</font></i></a>, <a class="ref" href="mcart_1.html#T10" title="MCART_1:th.10">MCART_1:10</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c13">x1</font> = <font color="Maroon" title="c9">x1'</font>, <font color="Maroon" title="c14">x2</font> = <font color="Maroon" title="c10">x2'</font>, <font color="Maroon" title="c15">y1</font> = <font color="Maroon" title="c11">y1'</font>, <font color="Maroon" title="c16">y2</font> = <font color="Maroon" title="c12">y2'</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFV</font> ;<br/><b>set </b><font color="Maroon" title="c17">z1'</font> = <font color="Maroon" title="c9">x1'</font> <a href="tdgroup.html#NK1" title="TDGROUP:NK.1">#</a> <font color="Maroon" title="c12">y2'</font>;<br/><b>set </b><font color="Maroon" title="c18">z2'</font> = <font color="Maroon" title="c10">x2'</font> <a href="tdgroup.html#NK1" title="TDGROUP:NK.1">#</a> <font color="Maroon" title="c11">y1'</font>;<br/><a NAME="E7:71_1_1_1"/><i><font color="Green" title="E49">A6</font></i>: 
<font color="Maroon" title="c9">x1'</font> <a href="tdgroup.html#NK1" title="TDGROUP:NK.1">#</a> <font color="Maroon" title="c12">y2'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">x2'</font> <a href="tdgroup.html#NK1" title="TDGROUP:NK.1">#</a> <font color="Maroon" title="c11">y1'</font>
 <b>by </b><i><a class="txt" href="#E1:71_1_1_1"><i><font color="Green" title="E46">A3</font></i></a>, <a class="txt" href="#E4:71_1_1_1"><i><font color="Green" title="E48">A5</font></i></a>, <a class="ref" href="tdgroup.html#D4" title="TDGROUP:def.4">TDGROUP:def 4</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c19">z1</font> = <font color="Maroon" title="c9">x1'</font> <a href="tdgroup.html#NK1" title="TDGROUP:NK.1">#</a> <font color="Maroon" title="c12">y2'</font>, <font color="Maroon" title="c20">z2</font> = <font color="Maroon" title="c10">x2'</font> <a href="tdgroup.html#NK1" title="TDGROUP:NK.1">#</a> <font color="Maroon" title="c11">y1'</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFV</font> ;<br/><a NAME="E9:71_1_1_1"/>
( <font color="Maroon" title="c19">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afvect0.html#K2" title="AFVECT0:func.2">Padd</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c13">x1</font>,<font color="Maroon" title="c16">y2</font> &amp; <font color="Maroon" title="c20">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afvect0.html#K2" title="AFVECT0:func.2">Padd</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c14">x2</font>,<font color="Maroon" title="c15">y1</font> )
 <b>by </b><i><a class="ref" href="afvect0.html#D7" target="_self" title="AFVECT0:def.7">Def7</a></i>;<br/><a NAME="E10:71_1_1_1"/><b>then </b>
( <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c13">x1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c16">y2</font>,<font color="Maroon" title="c19">z1</font> &amp; <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c14">x2</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c15">y1</font>,<font color="Maroon" title="c19">z1</font> )
 <b>by </b><i><a class="txt" href="#E7:71_1_1_1"><i><font color="Green" title="E49">A6</font></i></a>, <a class="ref" href="afvect0.html#D5" target="_self" title="AFVECT0:def.5">Def5</a></i>;<br/><a NAME="E11:71_1_1_1"/><b>then </b>
<font color="Maroon" title="c13">x1</font>,<font color="Maroon" title="c14">x2</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c15">y1</font>,<font color="Maroon" title="c16">y2</font>
 <b>by </b><i><a class="ref" href="afvect0.html#T13" target="_self" title="AFVECT0:th.13">Th13</a></i>;<br/><b>hence </b><a NAME="E12:71_1_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">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> the <a href="analoaf.html#U1" title="ANALOAF:sel.1">CONGR</a> of <font color="Maroon" title="c1">AFV</font>
 <b>by </b><i><a class="txt" href="#E4:71_1_1_1"><i><font color="Green" title="E48">A5</font></i></a>, <a class="ref" href="analoaf.html#D2" title="ANALOAF:def.2">ANALOAF:def 2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:71_1_1_2"/><i><font color="Green" title="E46">A7</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">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> the <a href="analoaf.html#U1" title="ANALOAF:sel.1">CONGR</a> of <font color="Maroon" title="c1">AFV</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">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="tdgroup.html#K1" title="TDGROUP:func.1">CONGRD</a> <span class="p1">(<span class="default"><a href="afvect0.html#K5" title="AFVECT0:func.5">GroupVect</a> <font color="Maroon" title="c1">AFV</font>,<font color="Maroon" title="c2">o</font></span>)</span></span><br/><b>set </b><font color="Maroon" title="c7">V</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">AFV</font>;<br/><b>set </b><font color="Maroon" title="c8">VV</font> = <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">AFV</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">AFV</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>;<br/><a NAME="E2:71_1_1_2"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">y</font> )
 <b>by </b><i><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>;<br/><a NAME="E3:71_1_1_2"/><b>then </b><i><font color="Green" title="E47">A8</font></i>: 
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">AFV</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">AFV</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp; <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">AFV</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">AFV</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )
 <b>by </b><i><a class="txt" href="#E1:71_1_1_2"><i><font color="Green" title="E46">A7</font></i></a>, <a class="ref" href="mcart_1.html#T10" title="MCART_1:th.10">MCART_1:10</a></i>;<br/><a NAME="E4:71_1_1_2"/><b>then </b><i><font color="Green" title="E48">A9</font></i>: 
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c4">x</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <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="c1">AFV</font> &amp; <font color="Maroon" title="c4">x</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <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="c1">AFV</font> &amp; <font color="Maroon" title="c5">y</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <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="c1">AFV</font> &amp; <font color="Maroon" title="c5">y</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <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="c1">AFV</font> )
 <b>by </b><i><a class="ref" href="mcart_1.html#T10" title="MCART_1:th.10">MCART_1:10</a>, <a class="ref" href="mcart_1.html#T23" title="MCART_1:th.23">MCART_1:23</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c9">x1</font> = <font color="Maroon" title="c4">x</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> , <font color="Maroon" title="c10">x2</font> = <font color="Maroon" title="c4">x</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> , <font color="Maroon" title="c11">y1</font> = <font color="Maroon" title="c5">y</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> , <font color="Maroon" title="c12">y2</font> = <font color="Maroon" title="c5">y</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFV</font> <b>by </b><i><a class="txt" href="#E3:71_1_1_2"><i><font color="Green" title="E47">A8</font></i></a>, <a class="ref" href="mcart_1.html#T10" title="MCART_1:th.10">MCART_1:10</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c13">x1'</font> = <font color="Maroon" title="c9">x1</font>, <font color="Maroon" title="c14">x2'</font> = <font color="Maroon" title="c10">x2</font>, <font color="Maroon" title="c15">y1'</font> = <font color="Maroon" title="c11">y1</font>, <font color="Maroon" title="c16">y2'</font> = <font color="Maroon" title="c12">y2</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="afvect0.html#K5" title="AFVECT0:func.5">GroupVect</a> <font color="Maroon" title="c1">AFV</font>,<font color="Maroon" title="c2">o</font></span>)</span> ;<br/><a NAME="E7:71_1_1_2"/><i><font color="Green" title="E49">A10</font></i>: 
<font color="Maroon" title="c9">x1</font>,<font color="Maroon" title="c10">x2</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">y1</font>,<font color="Maroon" title="c12">y2</font>
 <b>by </b><i><a class="txt" href="#E1:71_1_1_2"><i><font color="Green" title="E46">A7</font></i></a>, <a class="txt" href="#E4:71_1_1_2"><i><font color="Green" title="E48">A9</font></i></a>, <a class="ref" href="analoaf.html#D2" title="ANALOAF:def.2">ANALOAF:def 2</a></i>;<br/><a NAME="E8:71_1_1_2"/>
<font color="Maroon" title="c13">x1'</font> <a href="tdgroup.html#NK1" title="TDGROUP:NK.1">#</a> <font color="Maroon" title="c16">y2'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">x2'</font> <a href="tdgroup.html#NK1" title="TDGROUP:NK.1">#</a> <font color="Maroon" title="c15">y1'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1_1_2_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon" title="c17">z1</font> = <font color="Maroon" title="c13">x1'</font> <a href="tdgroup.html#NK1" title="TDGROUP:NK.1">#</a> <font color="Maroon" title="c16">y2'</font>, <font color="Maroon" title="c18">z2</font> = <font color="Maroon" title="c14">x2'</font> <a href="tdgroup.html#NK1" title="TDGROUP:NK.1">#</a> <font color="Maroon" title="c15">y1'</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFV</font> ;<br/>
<a NAME="E2:71_1_1_2_1"/><i><font color="Green" title="E50">A11</font></i>: 
( <font color="Maroon" title="c17">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afvect0.html#K2" title="AFVECT0:func.2">Padd</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c9">x1</font>,<font color="Maroon" title="c12">y2</font> &amp; <font color="Maroon" title="c18">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afvect0.html#K2" title="AFVECT0:func.2">Padd</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c10">x2</font>,<font color="Maroon" title="c11">y1</font> )
 
<b>by </b><i><a class="ref" href="afvect0.html#D7" target="_self" title="AFVECT0:def.7">Def7</a></i>;<br/>
<a NAME="E3:71_1_1_2_1"/><b>then </b>
( <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c9">x1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c12">y2</font>,<font color="Maroon" title="c17">z1</font> &amp; <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c10">x2</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">y1</font>,<font color="Maroon" title="c18">z2</font> )
 
<b>by </b><i><a class="ref" href="afvect0.html#D5" target="_self" title="AFVECT0:def.5">Def5</a></i>;<br/>
<a NAME="E4:71_1_1_2_1"/><b>then </b>
( <font color="Maroon" title="c9">x1</font>,<font color="Maroon" title="c2">o</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c17">z1</font>,<font color="Maroon" title="c12">y2</font> &amp; <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c10">x2</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">y1</font>,<font color="Maroon" title="c18">z2</font> )
 
<b>by </b><i><a class="ref" href="afvect0.html#T8" target="_self" title="AFVECT0:th.8">Th8</a></i>;<br/>
<a NAME="E5:71_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c10">x2</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">y1</font>,<font color="Maroon" title="c17">z1</font>
 
<b>by </b><i><a class="txt" href="#E7:71_1_1_2"><i><font color="Green" title="E49">A10</font></i></a>, <a class="ref" href="afvect0.html#T13" target="_self" title="AFVECT0:th.13">Th13</a></i>;<br/>
<b>hence </b><a NAME="E6:71_1_1_2_1"/>
<font color="Maroon" title="c13">x1'</font> <a href="tdgroup.html#NK1" title="TDGROUP:NK.1">#</a> <font color="Maroon" title="c16">y2'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">x2'</font> <a href="tdgroup.html#NK1" title="TDGROUP:NK.1">#</a> <font color="Maroon" title="c15">y1'</font>
 <b>by </b><i><a class="txt" href="#E2:71_1_1_2_1"><i><font color="Green" title="E50">A11</font></i></a>, <a class="ref" href="afvect0.html#D5" target="_self" title="AFVECT0:def.5">Def5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><b>hence </b><a NAME="E9:71_1_1_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">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="tdgroup.html#K1" title="TDGROUP:func.1">CONGRD</a> <span class="p1">(<span class="default"><a href="afvect0.html#K5" title="AFVECT0:func.5">GroupVect</a> <font color="Maroon" title="c1">AFV</font>,<font color="Maroon" title="c2">o</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E4:71_1_1_2"><i><font color="Green" title="E48">A9</font></i></a>, <a class="ref" href="tdgroup.html#D4" title="TDGROUP:def.4">TDGROUP:def 4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>hence </b><a NAME="E3:71_1_1"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">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="tdgroup.html#K1" title="TDGROUP:func.1">CONGRD</a> <span class="p1">(<span class="default"><a href="afvect0.html#K5" title="AFVECT0:func.5">GroupVect</a> <font color="Maroon" title="c1">AFV</font>,<font color="Maroon" title="c2">o</font></span>)</span> iff <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">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> the <a href="analoaf.html#U1" title="ANALOAF:sel.1">CONGR</a> of <font color="Maroon" title="c1">AFV</font> )
 <b>by </b><i><a class="txt" href="#E1:71_1_1"><i><font color="Green" title="E45">A2</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:71_1"/>
 <a href="tdgroup.html#K1" title="TDGROUP:func.1">CONGRD</a> <span class="p1">(<span class="default"><a href="afvect0.html#K5" title="AFVECT0:func.5">GroupVect</a> <font color="Maroon" title="c1">AFV</font>,<font color="Maroon" title="c2">o</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="analoaf.html#U1" title="ANALOAF:sel.1">CONGR</a> of <font color="Maroon" title="c1">AFV</font>
 <b>by </b><i><a class="ref" href="relat_1.html#D2" title="RELAT_1:def.2">RELAT_1:def 2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:71"/>
<font color="Maroon" title="c1">AFV</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tdgroup.html#K2" title="TDGROUP:func.2">AV</a> <span class="p1">(<span class="default"><a href="afvect0.html#K5" title="AFVECT0:func.5">GroupVect</a> <font color="Maroon" title="c1">AFV</font>,<font color="Maroon" title="c2">o</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:71"><i><font color="Green" title="E44">A1</font></i></a>, <a class="ref" href="tdgroup.html#T9" title="TDGROUP:th.9">TDGROUP:9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
