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<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">OAS</font><br/> for <font color="Olive" title="b3">f</font>, <font color="Olive" title="b4">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="b3">f</font> is  <a href="transgeo.html#V5" title="TRANSGEO:attr.5">translation</a>  &amp; <font color="Olive" title="b4">g</font> is  <a href="transgeo.html#V5" title="TRANSGEO:attr.5">translation</a>  &amp; <font color="Olive" title="b3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font> &amp; not  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">x</font> holds <br/><font color="Olive" title="b3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">x</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">OAS</font>; <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">OAS</font>  st <font color="Olive" title="b1">f</font> is  <a href="transgeo.html#V5" title="TRANSGEO:attr.5">translation</a>  &amp; <font color="Olive" title="b2">g</font> is  <a href="transgeo.html#V5" title="TRANSGEO:attr.5">translation</a>  &amp; <font color="Olive" title="b1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">a</font> &amp; not  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c2">a</font>,<font color="Olive" title="b1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">x</font> holds <br/><font color="Olive" title="b1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">x</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">f</font>, <font color="Maroon" title="c5">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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">f</font> is  <a href="transgeo.html#V5" title="TRANSGEO:attr.5">translation</a>  &amp; <font color="Maroon" title="c5">g</font> is  <a href="transgeo.html#V5" title="TRANSGEO:attr.5">translation</a>  &amp; <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">a</font> &amp; not  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">x</font> implies <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">x</font> )</span><br/>









<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:84"/><span class="lab"><font color="Green" title="E40">A1</font></span>: 
<font color="Maroon" title="c4">f</font> is  <a href="transgeo.html#V5" title="TRANSGEO:attr.5">translation</a> 
 <span class="kw">and </span><br/><a NAME="E2:84"/><span class="lab"><font color="Green" title="E41">A2</font></span>: 
<font color="Maroon" title="c5">g</font> is  <a href="transgeo.html#V5" title="TRANSGEO:attr.5">translation</a> 
 <span class="kw">and </span><br/><a NAME="E3:84"/><span class="lab"><font color="Green" title="E42">A3</font></span>: 
<font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">a</font>
 <span class="kw">and </span><br/><a NAME="E4:84"/><span class="lab"><font color="Green" title="E43">A4</font></span>: 
not  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">x</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="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">x</font></span><br/>

<span class="kw">set </span><font color="Maroon" title="c6">b</font> = <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">a</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">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="c3">x</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">z</font> = <font color="Maroon" title="c5">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">x</font>;<br/>
<a NAME="E5:84"/>
<font color="Maroon" title="c5">g</font> is  <a href="transgeo.html#V4" title="TRANSGEO:attr.4">dilatation</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:84"><span class="lab"><font color="Green" title="E41">A2</font></span></a>, <a class="ref" href="transgeo.html#D9" target="_self" title="TRANSGEO:def.9">Def9</a></span>;<br/>
<a NAME="E6:84"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A5</font></span>: 
( <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">x</font> <a href="diraf.html#R2" title="DIRAF: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="c2">a</font>,<font color="Maroon" title="c5">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">a</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c5">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">x</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:84"><span class="lab"><font color="Green" title="E41">A2</font></span></a>, <a class="txt" href="#E3:84"><span class="lab"><font color="Green" title="E42">A3</font></span></a>, <a class="ref" href="transgeo.html#T51" target="_self" title="TRANSGEO:th.51">Th51</a>, <a class="ref" href="transgeo.html#T67" target="_self" title="TRANSGEO:th.67">Th67</a></span>;<br/>
<a NAME="E7:84"/><span class="lab"><font color="Green" title="E45">A6</font></span>: 
<font color="Maroon" title="c4">f</font> is  <a href="transgeo.html#V4" title="TRANSGEO:attr.4">dilatation</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:84"><span class="lab"><font color="Green" title="E40">A1</font></span></a>, <a class="ref" href="transgeo.html#D9" target="_self" title="TRANSGEO:def.9">Def9</a></span>;<br/>
<a NAME="E8:84"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A7</font></span>: 
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">x</font> <a href="diraf.html#R2" title="DIRAF: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="c2">a</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">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="transgeo.html#T51" target="_self" title="TRANSGEO:th.51">Th51</a></span>;<br/>
<a NAME="E9:84"/>
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">a</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c3">x</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">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:84"><span class="lab"><font color="Green" title="E40">A1</font></span></a>, <a class="txt" href="#E7:84"><span class="lab"><font color="Green" title="E45">A6</font></span></a>, <a class="ref" href="transgeo.html#T67" target="_self" title="TRANSGEO:th.67">Th67</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:84"/>
<font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:84"><span class="lab"><font color="Green" title="E43">A4</font></span></a>, <a class="txt" href="#E8:84"><span class="lab"><font color="Green" title="E46">A7</font></span></a>, <a class="txt" href="#E6:84"><span class="lab"><font color="Green" title="E44">A5</font></span></a>, <a class="ref" href="pasch.html#T12" title="PASCH:th.12">PASCH:12</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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