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<b>let </b><font color="Maroon" title="c1">AFS</font> be   <a href="diraf.html#NM1" title="DIRAF:NM.1">AffinSpace</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">f</font> being  <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">AFS</font>  st <font color="Olive" title="b1">f</font> is <a href="transgeo.html#V7" title="TRANSGEO:attr.7">translation</a> holds <br/><font color="Olive" title="b1">f</font> <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a>  is <a href="transgeo.html#V7" title="TRANSGEO:attr.7">translation</a></span><br/><b>let </b><font color="Maroon" title="c2">f</font> be   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">AFS</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">f</font> is <a href="transgeo.html#V7" title="TRANSGEO:attr.7">translation</a> implies <font color="Maroon" title="c2">f</font> <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a>  is <a href="transgeo.html#V7" title="TRANSGEO:attr.7">translation</a> )</span><br/>



<b>assume </b><a NAME="E1:103"/><i><font color="Green" title="E72">A1</font></i>: 
<font color="Maroon" title="c2">f</font> is <a href="transgeo.html#V7" title="TRANSGEO:attr.7">translation</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">f</font> <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a>  is <a href="transgeo.html#V7" title="TRANSGEO:attr.7">translation</a></span><br/>

<a NAME="E2:103"/><b>then </b>
<font color="Maroon" title="c2">f</font> is <a href="transgeo.html#V6" title="TRANSGEO:attr.6">dilatation</a>
 
<b>by </b><i><a class="ref" href="transgeo.html#D11" target="_self" title="TRANSGEO:def.11">Def11</a></i>;<br/>
<a NAME="E3:103"/><b>then </b><i><font color="Green" title="E73">A2</font></i>: 
<font color="Maroon" title="c2">f</font> <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a>  is <a href="transgeo.html#V6" title="TRANSGEO:attr.6">dilatation</a>
 
<b>by </b><i><a class="ref" href="transgeo.html#T87" target="_self" title="TRANSGEO:th.87">Th87</a></i>;<br/>
<a NAME="E4:103"/><i><font color="Green" title="E74">A3</font></i>: 
( <font color="Maroon" title="c2">f</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">AFS</font> implies <font color="Maroon" title="c2">f</font> <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a>  <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">AFS</font> )
 
<b>by </b><i><a class="ref" href="funct_1.html#T67" title="FUNCT_1:th.67">FUNCT_1:67</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="103_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:103_1"/><i><font color="Green" title="E75">A4</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFS</font>  holds <font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">x</font>
 ; <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">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFS</font>  holds not <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">y</font></span><br/><b>given </b><font color="Maroon" title="c3">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFS</font><b> such that </b><a NAME="E3:103_1"/><i><font color="Green" title="E76">A5</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E4:103_1"/>
<font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">y</font>
 <b>by </b><i><a class="txt" href="#E3:103_1"><i><font color="Green" title="E76">A5</font></i></a>, <a class="ref" href="transgeo.html#T4" target="_self" title="TRANSGEO:th.4">Th4</a></i>;<br/><b>hence </b><a NAME="E5:103_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:103_1"><i><font color="Green" title="E75">A4</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="E6:103"/>
<font color="Maroon" title="c2">f</font> <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a>  is <a href="transgeo.html#V7" title="TRANSGEO:attr.7">translation</a>
 <b>by </b><i><a class="txt" href="#E1:103"><i><font color="Green" title="E72">A1</font></i></a>, <a class="txt" href="#E3:103"><i><font color="Green" title="E73">A2</font></i></a>, <a class="txt" href="#E4:103"><i><font color="Green" title="E74">A3</font></i></a>, <a class="ref" href="transgeo.html#D11" target="_self" title="TRANSGEO:def.11">Def11</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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