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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">a</font>, <font color="Olive" title="b2">b</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#V4" title="TRANSGEO:attr.4">dilatation</a> &amp; <font color="Olive" title="b4">g</font> is <a href="transgeo.html#V4" title="TRANSGEO:attr.4">dilatation</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; <font color="Olive" title="b3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">b</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">b</font> &amp; not <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> holds <br/><font color="Olive" title="b3">f</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b4">g</font></span><br/><b>let </b><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</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:()"><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> is <a href="transgeo.html#V4" title="TRANSGEO:attr.4">dilatation</a> &amp; <font color="Olive" title="b2">g</font> is <a href="transgeo.html#V4" title="TRANSGEO:attr.4">dilatation</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; <font color="Olive" title="b1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">b</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">b</font> &amp; not <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> holds <br/><font color="Olive" title="b1">f</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b2">g</font></span><br/><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">f</font> is <a href="transgeo.html#V4" title="TRANSGEO:attr.4">dilatation</a> &amp; <font color="Maroon" title="c5">g</font> is <a href="transgeo.html#V4" title="TRANSGEO:attr.4">dilatation</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; <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">b</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">b</font> &amp; not <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> implies <font color="Maroon" title="c4">f</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c5">g</font> )</span><br/>









<b>assume </b><b>that </b><br/><a NAME="E1:62"/><i><font color="Green" title="E37">A1</font></i>: 
( <font color="Maroon" title="c4">f</font> is <a href="transgeo.html#V4" title="TRANSGEO:attr.4">dilatation</a> &amp; <font color="Maroon" title="c5">g</font> is <a href="transgeo.html#V4" title="TRANSGEO:attr.4">dilatation</a> )
 <b>and </b><br/><a NAME="E2:62"/><i><font color="Green" title="E38">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">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; <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">b</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">b</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">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> or <font color="Maroon" title="c4">f</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c5">g</font> )</span><br/>

<b>assume </b><a NAME="E3:62"/><i><font color="Green" title="E39">A3</font></i>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">f</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c5">g</font></span><br/>

<i><font color="Green" title="E40">A4</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a> </span>)</span> <a href="transgeo.html#K1" title="TRANSGEO:func.1">*</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">a</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">g</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> <span class="p1">(<span class="default"><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>)</span>
<b>by </b><i><a class="txt" href="#E2:62"><i><font color="Green" title="E38">A2</font></i></a>, <a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>
<br/>.= 
<font color="Maroon" title="c2">a</font>
<b>by </b><i><a class="ref" href="transgeo.html#T4" target="_self" title="TRANSGEO:th.4">Th4</a></i>
;<br/>
<i><font color="Green" title="E41">A5</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a> </span>)</span> <a href="transgeo.html#K1" title="TRANSGEO:func.1">*</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">b</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">g</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> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">b</font></span>)</span>
<b>by </b><i><a class="txt" href="#E2:62"><i><font color="Green" title="E38">A2</font></i></a>, <a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>
<br/>.= 
<font color="Maroon" title="c3">b</font>
<b>by </b><i><a class="ref" href="transgeo.html#T4" target="_self" title="TRANSGEO:th.4">Th4</a></i>
;<br/>
<a NAME="E6:62"/>
<font color="Maroon" title="c5">g</font> <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a>  is <a href="transgeo.html#V4" title="TRANSGEO:attr.4">dilatation</a>
 
<b>by </b><i><a class="txt" href="#E1:62"><i><font color="Green" title="E37">A1</font></i></a>, <a class="ref" href="transgeo.html#T56" target="_self" title="TRANSGEO:th.56">Th56</a></i>;<br/>
<a NAME="E7:62"/><b>then </b><i><font color="Green" title="E42">A6</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a> </span>)</span> <a href="transgeo.html#K1" title="TRANSGEO:func.1">*</a> <font color="Maroon" title="c4">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">OAS</font>
 
<b>by </b><i><a class="txt" href="#E1:62"><i><font color="Green" title="E37">A1</font></i></a>, <a class="txt" href="#E3:62"><i><font color="Green" title="E39">A3</font></i></a>, <a class="txt" href="#E4:62"><i><font color="Green" title="E40">A4</font></i></a>, <a class="txt" href="#E5:62"><i><font color="Green" title="E41">A5</font></i></a>, <a class="ref" href="transgeo.html#T57" target="_self" title="TRANSGEO:th.57">Th57</a>, <a class="ref" href="transgeo.html#T64" target="_self" title="TRANSGEO:th.64">Th64</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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">x</font></span><br/><a NAME="E1:62_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">g</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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a> </span>)</span> <a href="transgeo.html#K1" title="TRANSGEO:func.1">*</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>;<br/><a NAME="E2:62_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">g</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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font>
 <b>by </b><i><a class="txt" href="#E7:62"><i><font color="Green" title="E42">A6</font></i></a>, <a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</a></i>;<br/><b>hence </b><a NAME="E3:62_3"/>
<font color="Maroon" title="c5">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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">x</font>
 <b>by </b><i><a class="ref" href="transgeo.html#T4" target="_self" title="TRANSGEO:th.4">Th4</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="E9:62"/>
<font color="Maroon" title="c4">f</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c5">g</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</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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