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<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</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>  &amp; <font color="Olive" title="b2">g</font> is  <a href="transgeo.html#V7" title="TRANSGEO:attr.7">translation</a>  holds <br/><font color="Olive" title="b1">f</font> <a href="transgeo.html#K1" title="TRANSGEO:func.1">*</a> <font color="Olive" title="b2">g</font> is  <a href="transgeo.html#V7" title="TRANSGEO:attr.7">translation</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">f</font>, <font color="Maroon" title="c3">g</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">f</font> is  <a href="transgeo.html#V7" title="TRANSGEO:attr.7">translation</a>  &amp; <font color="Maroon" title="c3">g</font> is  <a href="transgeo.html#V7" title="TRANSGEO:attr.7">translation</a>  implies <font color="Maroon" title="c2">f</font> <a href="transgeo.html#K1" title="TRANSGEO:func.1">*</a> <font color="Maroon" title="c3">g</font> is  <a href="transgeo.html#V7" title="TRANSGEO:attr.7">translation</a>  )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:127"/><span class="lab"><font color="Green" title="E72">A1</font></span>: 
<font color="Maroon" title="c2">f</font> is  <a href="transgeo.html#V7" title="TRANSGEO:attr.7">translation</a> 
 <span class="kw">and </span><br/><a NAME="E2:127"/><span class="lab"><font color="Green" title="E73">A2</font></span>: 
<font color="Maroon" title="c3">g</font> is  <a href="transgeo.html#V7" title="TRANSGEO:attr.7">translation</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">f</font> <a href="transgeo.html#K1" title="TRANSGEO:func.1">*</a> <font color="Maroon" title="c3">g</font> is  <a href="transgeo.html#V7" title="TRANSGEO:attr.7">translation</a> </span><br/>

<a NAME="E3:127"/>
( <font color="Maroon" title="c2">f</font> is  <a href="transgeo.html#V6" title="TRANSGEO:attr.6">dilatation</a>  &amp; <font color="Maroon" title="c3">g</font> is  <a href="transgeo.html#V6" title="TRANSGEO:attr.6">dilatation</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:127"><span class="lab"><font color="Green" title="E72">A1</font></span></a>, <a class="txt" href="#E2:127"><span class="lab"><font color="Green" title="E73">A2</font></span></a>, <a class="ref" href="transgeo.html#D11" target="_self" title="TRANSGEO:def.11">Def11</a></span>;<br/>
<a NAME="E4:127"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A3</font></span>: 
<font color="Maroon" title="c2">f</font> <a href="transgeo.html#K1" title="TRANSGEO:func.1">*</a> <font color="Maroon" title="c3">g</font> is  <a href="transgeo.html#V6" title="TRANSGEO:attr.6">dilatation</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="transgeo.html#T88" target="_self" title="TRANSGEO:th.88">Th88</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="127_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:127_1"/><span class="lab"><font color="Green" title="E75">A4</font></span>: 
<font color="Maroon" title="c2">f</font> <a href="transgeo.html#K1" title="TRANSGEO:func.1">*</a> <font color="Maroon" title="c3">g</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">f</font> <a href="transgeo.html#K1" title="TRANSGEO:func.1">*</a> <font color="Maroon" title="c3">g</font> is  <a href="transgeo.html#V7" title="TRANSGEO:attr.7">translation</a> </span><br/><a NAME="E2:127_1"/>
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 <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="transgeo.html#K1" title="TRANSGEO:func.1">*</a> <font color="Maroon" title="c3">g</font></span>)</span> <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>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="127_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">given </span><font color="Maroon" title="c4">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFS</font><span class="kw"> such that </span><a NAME="E2:127_1_1"/><span class="lab"><font color="Green" title="E76">A5</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="transgeo.html#K1" title="TRANSGEO:func.1">*</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E3:127_1_1"/>
<font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:127_1_1"><span class="lab"><font color="Green" title="E76">A5</font></span></a>, <a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></span>;<br/>
<a NAME="E4:127_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A6</font></span>: 
<font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="c4">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="transgeo.html#T4" target="_self" title="TRANSGEO:th.4">Th4</a></span>;<br/>
<a NAME="E5:127_1_1"/>
<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 class="kw">by</span> <span class="lab"><a class="txt" href="#E1:127"><span class="lab"><font color="Green" title="E72">A1</font></span></a>, <a class="ref" href="transgeo.html#T104" target="_self" title="TRANSGEO:th.104">Th104</a></span>;<br/>
<a NAME="E6:127_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">f</font> <a href="transgeo.html#K1" title="TRANSGEO:func.1">*</a> <font color="Maroon" title="c3">g</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c2">f</font> <a href="transgeo.html#K1" title="TRANSGEO:func.1">*</a> <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:127"><span class="lab"><font color="Green" title="E73">A2</font></span></a>, <a class="txt" href="#E4:127_1_1"><span class="lab"><font color="Green" title="E77">A6</font></span></a>, <a class="ref" href="transgeo.html#T103" target="_self" title="TRANSGEO:th.103">Th103</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:127_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:127_1"><span class="lab"><font color="Green" title="E75">A4</font></span></a>, <a class="ref" href="funct_2.html#T88" title="FUNCT_2:th.88">FUNCT_2:88</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:127_1"/>
<font color="Maroon" title="c2">f</font> <a href="transgeo.html#K1" title="TRANSGEO:func.1">*</a> <font color="Maroon" title="c3">g</font> is  <a href="transgeo.html#V7" title="TRANSGEO:attr.7">translation</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:127"><span class="lab"><font color="Green" title="E74">A3</font></span></a>, <a class="ref" href="transgeo.html#D11" target="_self" title="TRANSGEO:def.11">Def11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E6:127"/>
<font color="Maroon" title="c2">f</font> <a href="transgeo.html#K1" title="TRANSGEO:func.1">*</a> <font color="Maroon" title="c3">g</font> is  <a href="transgeo.html#V7" title="TRANSGEO:attr.7">translation</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:127"><span class="lab"><font color="Green" title="E74">A3</font></span></a>, <a class="ref" href="transgeo.html#D11" target="_self" title="TRANSGEO:def.11">Def11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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