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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> 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>  holds <br/>(  not <font color="Olive" title="b1">f</font> is <a href="transgeo.html#V4" title="TRANSGEO:attr.4">dilatation</a> or <font color="Olive" title="b1">f</font> is <a href="transgeo.html#V2" title="TRANSGEO:attr.2">positive_dilatation</a> or <font color="Olive" title="b1">f</font> is <a href="transgeo.html#V3" title="TRANSGEO:attr.3">negative_dilatation</a> )</span><br/><b>let </b><font color="Maroon" title="c2">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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c2">f</font> is <a href="transgeo.html#V4" title="TRANSGEO:attr.4">dilatation</a> or <font color="Maroon" title="c2">f</font> is <a href="transgeo.html#V2" title="TRANSGEO:attr.2">positive_dilatation</a> or <font color="Maroon" title="c2">f</font> is <a href="transgeo.html#V3" title="TRANSGEO:attr.3">negative_dilatation</a> )</span><br/>



<b>assume </b><a NAME="E1:81"/><i><font color="Green" title="E54">A1</font></i>: 
<font color="Maroon" title="c2">f</font> is <a href="transgeo.html#V4" title="TRANSGEO:attr.4">dilatation</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> is <a href="transgeo.html#V2" title="TRANSGEO:attr.2">positive_dilatation</a> or <font color="Maroon" title="c2">f</font> is <a href="transgeo.html#V3" title="TRANSGEO:attr.3">negative_dilatation</a> )</span><br/>

<div><i><font color="Green" title="E55">A2</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:81_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 <font color="Maroon" title="c2">f</font> <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>
 ; <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> is <a href="transgeo.html#V2" title="TRANSGEO:attr.2">positive_dilatation</a></span><br/><a NAME="E2:81_1"/><b>then </b>
<font color="Maroon" title="c2">f</font> is <a href="transgeo.html#V5" title="TRANSGEO:attr.5">translation</a>
 <b>by </b><i><a class="txt" href="#E1:81"><i><font color="Green" title="E54">A1</font></i></a>, <a class="ref" href="transgeo.html#D9" target="_self" title="TRANSGEO:def.9">Def9</a></i>;<br/><b>hence </b><a NAME="E3:81_1"/>
<font color="Maroon" title="c2">f</font> is <a href="transgeo.html#V2" title="TRANSGEO:attr.2">positive_dilatation</a>
 <b>by </b><i><a class="ref" href="transgeo.html#T73" target="_self" title="TRANSGEO:th.73">Th73</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>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_2"><b>now </b></a><div class="add"><b>given </b><font color="Maroon" title="c3">p</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="E2:81_2"/><i><font color="Green" title="E56">A3</font></i>: 
<font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p</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> is <a href="transgeo.html#V2" title="TRANSGEO:attr.2">positive_dilatation</a> or <font color="Maroon" title="c2">f</font> is <a href="transgeo.html#V3" title="TRANSGEO:attr.3">negative_dilatation</a> )</span><br/><div><i><font color="Green" title="E57">A4</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="81_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:81_2_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  <font color="Maroon" title="c3">p</font>,<font color="Olive" title="b1">x</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</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> is <a href="transgeo.html#V2" title="TRANSGEO:attr.2">positive_dilatation</a></span><br/><a NAME="E2:81_2_1"/><b>then </b>
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="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font>,<font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">y</font>
 <b>by </b><i><a class="txt" href="#E1:81"><i><font color="Green" title="E54">A1</font></i></a>, <a class="txt" href="#E2:81_2"><i><font color="Green" title="E56">A3</font></i></a>, <a class="ref" href="transgeo.html#T79" target="_self" title="TRANSGEO:th.79">Th79</a></i>;<br/><b>hence </b><a NAME="E3:81_2_1"/>
<font color="Maroon" title="c2">f</font> is <a href="transgeo.html#V2" title="TRANSGEO:attr.2">positive_dilatation</a>
 <b>by </b><i><a class="ref" href="transgeo.html#T42" target="_self" title="TRANSGEO:th.42">Th42</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><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_2_2"><b>now </b></a><div class="add"><b>given </b><font color="Maroon" title="c4">q</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="E2:81_2_2"/><i><font color="Green" title="E58">A5</font></i>: 
not <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">q</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> is <a href="transgeo.html#V3" title="TRANSGEO:attr.3">negative_dilatation</a></span><br/><a NAME="E3:81_2_2"/>
<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">q</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">q</font>
 <b>by </b><i><a class="txt" href="#E1:81"><i><font color="Green" title="E54">A1</font></i></a>, <a class="txt" href="#E2:81_2"><i><font color="Green" title="E56">A3</font></i></a>, <a class="ref" href="transgeo.html#T51" target="_self" title="TRANSGEO:th.51">Th51</a></i>;<br/><a NAME="E4:81_2_2"/><b>then </b>
<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c3">p</font>
 <b>by </b><i><a class="txt" href="#E2:81_2_2"><i><font color="Green" title="E58">A5</font></i></a>, <a class="ref" href="diraf.html#D4" title="DIRAF:def.4">DIRAF:def 4</a></i>;<br/><a NAME="E5:81_2_2"/><b>then </b>
<font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c3">p</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">q</font>
 <b>by </b><i><a class="ref" href="diraf.html#T5" title="DIRAF:th.5">DIRAF:5</a></i>;<br/><a NAME="E6:81_2_2"/><b>then </b>
(  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">q</font> &amp; <font color="Maroon" title="c3">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">q</font> )
 <b>by </b><i><a class="txt" href="#E2:81_2_2"><i><font color="Green" title="E58">A5</font></i></a>, <a class="ref" href="diraf.html#D3" title="DIRAF:def.3">DIRAF:def 3</a></i>;<br/><b>hence </b><a NAME="E7:81_2_2"/>
<font color="Maroon" title="c2">f</font> is <a href="transgeo.html#V3" title="TRANSGEO:attr.3">negative_dilatation</a>
 <b>by </b><i><a class="txt" href="#E1:81"><i><font color="Green" title="E54">A1</font></i></a>, <a class="txt" href="#E2:81_2"><i><font color="Green" title="E56">A3</font></i></a>, <a class="ref" href="transgeo.html#T78" target="_self" title="TRANSGEO:th.78">Th78</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="E5:81_2"/>
( <font color="Maroon" title="c2">f</font> is <a href="transgeo.html#V2" title="TRANSGEO:attr.2">positive_dilatation</a> or <font color="Maroon" title="c2">f</font> is <a href="transgeo.html#V3" title="TRANSGEO:attr.3">negative_dilatation</a> )
 <b>by </b><i><a class="txt" href="#E3:81_2"><i><font color="Green" title="E57">A4</font></i></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="E4:81"/>
( <font color="Maroon" title="c2">f</font> is <a href="transgeo.html#V2" title="TRANSGEO:attr.2">positive_dilatation</a> or <font color="Maroon" title="c2">f</font> is <a href="transgeo.html#V3" title="TRANSGEO:attr.3">negative_dilatation</a> )
 <b>by </b><i><a class="txt" href="#E2:81"><i><font color="Green" title="E55">A2</font></i></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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