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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>;<br/><b>let </b><font color="Maroon" title="c2">p</font>, <font color="Maroon" title="c3">q</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">OAS</font>;<br/><b>let </b><font color="Maroon" title="c4">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">OAS</font>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:83"/><i><font color="Green" title="E53">A1</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="transgeo.html#NR6" title="TRANSGEO:NR.6">is_Dil</a> 
 <b>and </b><br/><a NAME="E2:83"/><i><font color="Green" title="E54">A2</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font>
 <b>and </b><br/><a NAME="E3:83"/><i><font color="Green" title="E55">A3</font></i>: 
<font color="Maroon" title="c3">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">p</font>
 <b>and </b><br/><a NAME="E4:83"/><i><font color="Green" title="E56">A4</font></i>: 
 <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">q</font>
 ;<br/>

<a NAME="E5:83"/>
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">OAS</font> holds  <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">y</font>,<font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c6">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">OAS</font>;<br/>



<a NAME="E1:83_1"/>
( not  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font> implies <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x</font> )
 
<b>by </b><i><a class="txt" href="transgeo.html#E1:83"><i><font color="Green" title="E53">A1</font></i></a>, <a class="txt" href="transgeo.html#E2:83"><i><font color="Green" title="E54">A2</font></i></a>, <a class="txt" href="transgeo.html#E3:83"><i><font color="Green" title="E55">A3</font></i></a>, <a class="txt" href="transgeo.html#E4:83"><i><font color="Green" title="E56">A4</font></i></a>, <a class="ref" href="transgeo.html#T76" target="_self" title="TRANSGEO:th.76">Th76</a></i>;<br/>
<b>hence </b><a NAME="E2:83_1"/>
<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x</font>
 <b>by </b><i><a class="txt" href="transgeo.html#E1:83"><i><font color="Green" title="E53">A1</font></i></a>, <a class="txt" href="transgeo.html#E2:83"><i><font color="Green" title="E54">A2</font></i></a>, <a class="txt" href="transgeo.html#E3:83"><i><font color="Green" title="E55">A3</font></i></a>, <a class="txt" href="transgeo.html#E4:83"><i><font color="Green" title="E56">A4</font></i></a>, <a class="ref" href="transgeo.html#T77" target="_self" title="TRANSGEO:th.77">Th77</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:83"/>
<font color="Maroon" title="c4">f</font> <a href="transgeo.html#NR5" title="TRANSGEO:NR.5">is_SDil</a> 
 <b>by </b><i><a class="ref" href="transgeo.html#D7" target="_self" title="TRANSGEO:def.7">Def7</a></i>;<br/>


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