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<b>let </b><font color="Maroon" title="c1">OAS</font> be   <a href="analoaf.html#NM1" title="ANALOAF:NM.1">OAffinSpace</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>, <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">OAS</font>  st <font color="Olive" title="b1">f</font> <a href="transgeo.html#NR7" title="TRANSGEO:NR.7">is_Tr</a>  &amp; <font color="Olive" title="b2">g</font> <a href="transgeo.html#NR7" title="TRANSGEO:NR.7">is_Tr</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> <a href="transgeo.html#NR7" title="TRANSGEO:NR.7">is_Tr</a> </span><br/><b>let </b><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">OAS</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> <a href="transgeo.html#NR7" title="TRANSGEO:NR.7">is_Tr</a>  &amp; <font color="Maroon" title="c3">g</font> <a href="transgeo.html#NR7" title="TRANSGEO:NR.7">is_Tr</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> <a href="transgeo.html#NR7" title="TRANSGEO:NR.7">is_Tr</a>  )</span><br/>





<b>assume </b><b>that </b><br/><a NAME="E1:72"/><i><font color="Green" title="E42">A1</font></i>: 
<font color="Maroon" title="c2">f</font> <a href="transgeo.html#NR7" title="TRANSGEO:NR.7">is_Tr</a> 
 <b>and </b><br/><a NAME="E2:72"/><i><font color="Green" title="E43">A2</font></i>: 
<font color="Maroon" title="c3">g</font> <a href="transgeo.html#NR7" title="TRANSGEO:NR.7">is_Tr</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="transgeo.html#K1" title="TRANSGEO:func.1">*</a> <font color="Maroon" title="c3">g</font> <a href="transgeo.html#NR7" title="TRANSGEO:NR.7">is_Tr</a> </span><br/>

<a NAME="E3:72"/><i><font color="Green" title="E44">A3</font></i>: 
<font color="Maroon" title="c2">f</font> <a href="transgeo.html#NR6" title="TRANSGEO:NR.6">is_Dil</a> 
 
<b>by </b><i><a class="txt" href="transgeo.html#E1:72"><i><font color="Green" title="E42">A1</font></i></a>, <a class="ref" href="transgeo.html#D9" target="_self" title="TRANSGEO:def.9">Def9</a></i>;<br/>
<a NAME="E4:72"/>
<font color="Maroon" title="c3">g</font> <a href="transgeo.html#NR6" title="TRANSGEO:NR.6">is_Dil</a> 
 
<b>by </b><i><a class="txt" href="transgeo.html#E2:72"><i><font color="Green" title="E43">A2</font></i></a>, <a class="ref" href="transgeo.html#D9" target="_self" title="TRANSGEO:def.9">Def9</a></i>;<br/>
<a NAME="E5:72"/><b>then </b><i><font color="Green" title="E45">A4</font></i>: 
<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="transgeo.html#NR6" title="TRANSGEO:NR.6">is_Dil</a> 
 
<b>by </b><i><a class="txt" href="transgeo.html#E3:72"><i><font color="Green" title="E44">A3</font></i></a>, <a class="ref" href="transgeo.html#T57" target="_self" title="TRANSGEO:th.57">Th57</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:72_1"/><i><font color="Green" title="E46">A5</font></i>: 
<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">OAS</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> <a href="transgeo.html#K1" title="TRANSGEO:func.1">*</a> <font color="Maroon" title="c3">g</font> <a href="transgeo.html#NR7" title="TRANSGEO:NR.7">is_Tr</a> </span><br/><a NAME="E2:72_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">OAS</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="72_1_1"><b>proof </b></a><div class="add">

<b>given </b><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">OAS</font><b> such that </b><a NAME="E2:72_1_1"/><i><font color="Green" title="E47">A6</font></i>: 
<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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E3:72_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>
 
<b>by </b><i><a class="txt" href="transgeo.html#E2:72_1_1"><i><font color="Green" title="E47">A6</font></i></a>, <a class="ref" href="funct_2.html#T70" title="FUNCT_2:th.70">FUNCT_2:70</a></i>;<br/>
<a NAME="E4:72_1_1"/><b>then </b>
( <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> &amp; <font color="Maroon" title="c2">f</font> <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a>  <a href="transgeo.html#NR7" title="TRANSGEO:NR.7">is_Tr</a>  )
 
<b>by </b><i><a class="txt" href="transgeo.html#E1:72"><i><font color="Green" title="E42">A1</font></i></a>, <a class="ref" href="transgeo.html#T4" target="_self" title="TRANSGEO:th.4">Th4</a>, <a class="ref" href="transgeo.html#T71" target="_self" title="TRANSGEO:th.71">Th71</a></i>;<br/>
<a NAME="E5:72_1_1"/><b>then </b>
<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>
 
<b>by </b><i><a class="txt" href="transgeo.html#E2:72"><i><font color="Green" title="E43">A2</font></i></a>, <a class="ref" href="transgeo.html#T70" target="_self" title="TRANSGEO:th.70">Th70</a></i>;<br/>
<b>hence </b><a NAME="E6:72_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="transgeo.html#E1:72_1"><i><font color="Green" title="E46">A5</font></i></a>, <a class="ref" href="funct_2.html#T88" title="FUNCT_2:th.88">FUNCT_2:88</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="E3:72_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> <a href="transgeo.html#NR7" title="TRANSGEO:NR.7">is_Tr</a> 
 <b>by </b><i><a class="txt" href="transgeo.html#E5:72"><i><font color="Green" title="E45">A4</font></i></a>, <a class="ref" href="transgeo.html#D9" target="_self" title="TRANSGEO:def.9">Def9</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="E7:72"/>
<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="transgeo.html#NR7" title="TRANSGEO:NR.7">is_Tr</a> 
 <b>by </b><i><a class="txt" href="transgeo.html#E5:72"><i><font color="Green" title="E45">A4</font></i></a>, <a class="ref" href="transgeo.html#D9" target="_self" title="TRANSGEO:def.9">Def9</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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