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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">p</font>, <font color="Olive" title="b2">q</font>, <font color="Olive" title="b3">x</font>, <font color="Olive" title="b4">y</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="b5">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>  st <font color="Olive" title="b5">f</font> is <a href="transgeo.html#V4" title="TRANSGEO:attr.4">dilatation</a> &amp; <font color="Olive" title="b5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> &amp; <font color="Olive" title="b2">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">p</font> &amp;  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Olive" title="b2">q</font>,<font color="Olive" title="b1">p</font>,<font color="Olive" title="b5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">q</font> &amp;  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Olive" title="b1">p</font>,<font color="Olive" title="b3">x</font>,<font color="Olive" title="b4">y</font> holds <br/><font color="Olive" title="b3">x</font>,<font color="Olive" title="b4">y</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">y</font>,<font color="Olive" title="b5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">x</font></span><br/><b>let </b><font color="Maroon" title="c2">p</font>, <font color="Maroon" title="c3">q</font>, <font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c5">y</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> 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="b1">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> &amp; <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> &amp;  <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="Olive" title="b1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">q</font> &amp;  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font> holds <br/><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">y</font>,<font color="Olive" title="b1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x</font></span><br/><b>let </b><font color="Maroon" title="c6">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"> ( <font color="Maroon" title="c6">f</font> is <a href="transgeo.html#V4" title="TRANSGEO:attr.4">dilatation</a> &amp; <font color="Maroon" title="c6">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> &amp; <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> &amp;  <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="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">q</font> &amp;  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font> implies <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x</font> )</span><br/>











<b>assume </b><b>that </b><br/><a NAME="E1:78"/><i><font color="Green" title="E52">A1</font></i>: 
<font color="Maroon" title="c6">f</font> is <a href="transgeo.html#V4" title="TRANSGEO:attr.4">dilatation</a>
 <b>and </b><br/><a NAME="E2:78"/><i><font color="Green" title="E53">A2</font></i>: 
<font color="Maroon" title="c6">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="E3:78"/><i><font color="Green" title="E54">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="E4:78"/><i><font color="Green" title="E55">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="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">q</font>
 <b>and </b><br/><a NAME="E5:78"/><i><font color="Green" title="E56">A5</font></i>: 
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</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">x</font>,<font color="Maroon" title="c5">y</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x</font></span><br/>

<a NAME="E6:78"/><i><font color="Green" title="E57">A6</font></i>: 
 <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x</font>
 
<b>by </b><i><a class="txt" href="#E1:78"><i><font color="Green" title="E52">A1</font></i></a>, <a class="txt" href="#E2:78"><i><font color="Green" title="E53">A2</font></i></a>, <a class="txt" href="#E3:78"><i><font color="Green" title="E54">A3</font></i></a>, <a class="txt" href="#E4:78"><i><font color="Green" title="E55">A4</font></i></a>, <a class="ref" href="transgeo.html#T75" target="_self" title="TRANSGEO:th.75">Th75</a></i>;<br/>
<a NAME="E7:78"/><i><font color="Green" title="E58">A7</font></i>: 
 <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">y</font>
 
<b>by </b><i><a class="txt" href="#E1:78"><i><font color="Green" title="E52">A1</font></i></a>, <a class="txt" href="#E2:78"><i><font color="Green" title="E53">A2</font></i></a>, <a class="txt" href="#E3:78"><i><font color="Green" title="E54">A3</font></i></a>, <a class="txt" href="#E4:78"><i><font color="Green" title="E55">A4</font></i></a>, <a class="ref" href="transgeo.html#T75" target="_self" title="TRANSGEO:th.75">Th75</a></i>;<br/>
<div><i><font color="Green" title="E59">A8</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="78_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:78_1"/><i><font color="Green" title="E60">A9</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">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="c4">x</font>,<font color="Maroon" title="c5">y</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x</font></span><br/><a NAME="E2:78_1"/><b>then </b>
<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c4">x</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">y</font>
 <b>by </b><i><a class="txt" href="#E7:78"><i><font color="Green" title="E58">A7</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="E3:78_1"/>
<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x</font>
 <b>by </b><i><a class="txt" href="#E2:78"><i><font color="Green" title="E53">A2</font></i></a>, <a class="txt" href="#E1:78_1"><i><font color="Green" title="E60">A9</font></i></a>, <a class="ref" href="diraf.html#T5" title="DIRAF:th.5">DIRAF:5</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="78_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:78_2"/><i><font color="Green" title="E60">A10</font></i>: 
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">p</font> &amp; <font color="Maroon" title="c5">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">p</font> &amp; <font color="Maroon" title="c4">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">y</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">x</font>,<font color="Maroon" title="c5">y</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x</font></span><br/><b>then consider </b><font color="Maroon" title="c7">u</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><br/><a NAME="E3:78_2"/><i><font color="Green" title="E61">A11</font></i>: 
not  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c7">u</font>
 <b>by </b><i><a class="ref" href="diraf.html#T43" title="DIRAF:th.43">DIRAF:43</a></i>;<br/><b>consider </b><font color="Maroon" title="c8">r</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><br/><a NAME="E5:78_2"/><i><font color="Green" title="E62">A12</font></i>: 
<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c7">u</font>,<font color="Maroon" title="c8">r</font>
 <b>and </b><br/><a NAME="E6:78_2"/><i><font color="Green" title="E63">A13</font></i>: 
<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c7">u</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c8">r</font>
 <b>by </b><i><a class="ref" href="diraf.html#T31" title="DIRAF:th.31">DIRAF:31</a></i>;<br/><a NAME="E7:78_2"/><i><font color="Green" title="E64">A14</font></i>: 
not  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c7">u</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="78_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:78_2_1"/>
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c7">u</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="E2:78_2_1"/><b>then </b>
(  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c7">u</font> &amp;  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c2">p</font> &amp;  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c4">x</font> )
 
<b>by </b><i><a class="txt" href="#E5:78"><i><font color="Green" title="E56">A5</font></i></a>, <a class="ref" href="diraf.html#T36" title="DIRAF:th.36">DIRAF:36</a>, <a class="ref" href="diraf.html#T37" title="DIRAF:th.37">DIRAF:37</a></i>;<br/>
<b>hence </b><a NAME="E3:78_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:78_2"><i><font color="Green" title="E60">A10</font></i></a>, <a class="txt" href="#E3:78_2"><i><font color="Green" title="E61">A11</font></i></a>, <a class="ref" href="diraf.html#T38" title="DIRAF:th.38">DIRAF:38</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><a NAME="E8:78_2"/><b>then </b><i><font color="Green" title="E65">A15</font></i>: 
<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">u</font>,<font color="Maroon" title="c8">r</font>
 <b>by </b><i><a class="txt" href="#E5:78_2"><i><font color="Green" title="E62">A12</font></i></a>, <a class="txt" href="#E6:78_2"><i><font color="Green" title="E63">A13</font></i></a>, <a class="ref" href="pasch.html#T21" title="PASCH:th.21">PASCH:21</a></i>;<br/><a NAME="E9:78_2"/><i><font color="Green" title="E66">A16</font></i>: 
not  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c7">u</font>,<font color="Maroon" title="c8">r</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="78_2_2"><b>proof </b></a><div class="add">

<div><i><font color="Green" title="E67">A17</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="78_2_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:78_2_2_1"/>
<font color="Maroon" title="c7">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">r</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="E2:78_2_2_1"/><b>then </b>
<font color="Maroon" title="c7">u</font>,<font color="Maroon" title="c4">x</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c7">u</font>,<font color="Maroon" title="c5">y</font>
 <b>by </b><i><a class="txt" href="#E6:78_2"><i><font color="Green" title="E63">A13</font></i></a>, <a class="ref" href="diraf.html#T27" title="DIRAF:th.27">DIRAF:27</a></i>;<br/><a NAME="E3:78_2_2_1"/><b>then </b>
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c7">u</font>,<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font>
 <b>by </b><i><a class="ref" href="diraf.html#D5" title="DIRAF:def.5">DIRAF:def 5</a></i>;<br/><b>hence </b><a NAME="E4:78_2_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E7:78_2"><i><font color="Green" title="E64">A14</font></i></a>, <a class="ref" href="diraf.html#T36" title="DIRAF:th.36">DIRAF:36</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>assume </b><a NAME="E2:78_2_2"/>
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c7">u</font>,<font color="Maroon" title="c8">r</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="E3:78_2_2"/><b>then </b><i><font color="Green" title="E68">A18</font></i>: 
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c7">u</font>,<font color="Maroon" title="c8">r</font>,<font color="Maroon" title="c2">p</font>
 
<b>by </b><i><a class="ref" href="diraf.html#T36" title="DIRAF:th.36">DIRAF:36</a></i>;<br/>
<a NAME="E4:78_2_2"/>
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c2">p</font>
 
<b>by </b><i><a class="txt" href="#E5:78"><i><font color="Green" title="E56">A5</font></i></a>, <a class="ref" href="diraf.html#T36" title="DIRAF:th.36">DIRAF:36</a></i>;<br/>
<a NAME="E5:78_2_2"/><b>then </b>
<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c2">p</font>
 
<b>by </b><i><a class="ref" href="diraf.html#D5" title="DIRAF:def.5">DIRAF:def 5</a></i>;<br/>
<a NAME="E6:78_2_2"/><b>then </b>
<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">x</font>
 
<b>by </b><i><a class="ref" href="diraf.html#T27" title="DIRAF:th.27">DIRAF:27</a></i>;<br/>
<a NAME="E7:78_2_2"/><b>then </b><i><font color="Green" title="E69">A19</font></i>: 
<font color="Maroon" title="c7">u</font>,<font color="Maroon" title="c8">r</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">x</font>
 
<b>by </b><i><a class="txt" href="#E1:78_2"><i><font color="Green" title="E60">A10</font></i></a>, <a class="txt" href="#E5:78_2"><i><font color="Green" title="E62">A12</font></i></a>, <a class="ref" href="diraf.html#T28" title="DIRAF:th.28">DIRAF:28</a></i>;<br/>
<a NAME="E8:78_2_2"/><b>then </b><i><font color="Green" title="E70">A20</font></i>: 
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c7">u</font>,<font color="Maroon" title="c8">r</font>,<font color="Maroon" title="c4">x</font>
 
<b>by </b><i><a class="txt" href="#E1:78_2_2"><i><font color="Green" title="E67">A17</font></i></a>, <a class="txt" href="#E3:78_2_2"><i><font color="Green" title="E68">A18</font></i></a>, <a class="ref" href="diraf.html#T39" title="DIRAF:th.39">DIRAF:39</a></i>;<br/>
<a NAME="E9:78_2_2"/>
<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">x</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c5">y</font>
 
<b>by </b><i><a class="txt" href="#E5:78"><i><font color="Green" title="E56">A5</font></i></a>, <a class="ref" href="diraf.html#D5" title="DIRAF:def.5">DIRAF:def 5</a></i>;<br/>
<a NAME="E10:78_2_2"/><b>then </b>
<font color="Maroon" title="c7">u</font>,<font color="Maroon" title="c8">r</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c5">y</font>
 
<b>by </b><i><a class="txt" href="#E1:78_2"><i><font color="Green" title="E60">A10</font></i></a>, <a class="txt" href="#E7:78_2_2"><i><font color="Green" title="E69">A19</font></i></a>, <a class="ref" href="diraf.html#T28" title="DIRAF:th.28">DIRAF:28</a></i>;<br/>
<a NAME="E11:78_2_2"/><b>then </b>
(  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c7">u</font>,<font color="Maroon" title="c8">r</font>,<font color="Maroon" title="c5">y</font> &amp;  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c7">u</font>,<font color="Maroon" title="c8">r</font>,<font color="Maroon" title="c7">u</font> )
 
<b>by </b><i><a class="txt" href="#E1:78_2_2"><i><font color="Green" title="E67">A17</font></i></a>, <a class="txt" href="#E3:78_2_2"><i><font color="Green" title="E68">A18</font></i></a>, <a class="ref" href="diraf.html#T37" title="DIRAF:th.37">DIRAF:37</a>, <a class="ref" href="diraf.html#T39" title="DIRAF:th.39">DIRAF:39</a></i>;<br/>
<b>hence </b><a NAME="E12:78_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="#E7:78_2"><i><font color="Green" title="E64">A14</font></i></a>, <a class="txt" href="#E1:78_2_2"><i><font color="Green" title="E67">A17</font></i></a>, <a class="txt" href="#E8:78_2_2"><i><font color="Green" title="E70">A20</font></i></a>, <a class="ref" href="diraf.html#T38" title="DIRAF:th.38">DIRAF:38</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><a NAME="E10:78_2"/><b>then </b><i><font color="Green" title="E67">A21</font></i>: 
<font color="Maroon" title="c7">u</font>,<font color="Maroon" title="c8">r</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">r</font>,<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">u</font>
 <b>by </b><i><a class="txt" href="#E1:78"><i><font color="Green" title="E52">A1</font></i></a>, <a class="txt" href="#E2:78"><i><font color="Green" title="E53">A2</font></i></a>, <a class="txt" href="#E3:78"><i><font color="Green" title="E54">A3</font></i></a>, <a class="txt" href="#E4:78"><i><font color="Green" title="E55">A4</font></i></a>, <a class="ref" href="transgeo.html#T76" target="_self" title="TRANSGEO:th.76">Th76</a></i>;<br/><b>set </b><font color="Maroon" title="c9">u'</font> = <font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">u</font>;<br/><b>set </b><font color="Maroon" title="c10">r'</font> = <font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">r</font>;<br/><b>set </b><font color="Maroon" title="c11">x'</font> = <font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x</font>;<br/><b>set </b><font color="Maroon" title="c12">y'</font> = <font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">y</font>;<br/><a NAME="E11:78_2"/>
( <font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">y</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">u</font>,<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">r</font> &amp; <font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">u</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">r</font> &amp; not  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">u</font> )
 <b>by </b><i><a class="txt" href="#E1:78"><i><font color="Green" title="E52">A1</font></i></a>, <a class="txt" href="#E5:78_2"><i><font color="Green" title="E62">A12</font></i></a>, <a class="txt" href="#E6:78_2"><i><font color="Green" title="E63">A13</font></i></a>, <a class="txt" href="#E7:78_2"><i><font color="Green" title="E64">A14</font></i></a>, <a class="ref" href="transgeo.html#T58" target="_self" title="TRANSGEO:th.58">Th58</a>, <a class="ref" href="transgeo.html#T59" target="_self" title="TRANSGEO:th.59">Th59</a></i>;<br/><a NAME="E12:78_2"/><b>then </b>
<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">y</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">u</font>,<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">r</font>
 <b>by </b><i><a class="ref" href="pasch.html#T21" title="PASCH:th.21">PASCH:21</a></i>;<br/><a NAME="E13:78_2"/><b>then </b><i><font color="Green" title="E68">A22</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">r</font>,<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">u</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x</font>
 <b>by </b><i><a class="ref" href="diraf.html#T5" title="DIRAF:th.5">DIRAF:5</a></i>;<br/><a NAME="E14:78_2"/><i><font color="Green" title="E69">A23</font></i>: 
<font color="Maroon" title="c7">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">r</font>
 <b>by </b><i><a class="txt" href="#E9:78_2"><i><font color="Green" title="E66">A16</font></i></a>, <a class="ref" href="diraf.html#T37" title="DIRAF:th.37">DIRAF:37</a></i>;<br/><a NAME="E15:78_2"/><b>then </b>
<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">r</font>,<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">u</font>
 <b>by </b><i><a class="txt" href="#E8:78_2"><i><font color="Green" title="E65">A15</font></i></a>, <a class="txt" href="#E10:78_2"><i><font color="Green" title="E67">A21</font></i></a>, <a class="ref" href="diraf.html#T6" title="DIRAF:th.6">DIRAF:6</a></i>;<br/><b>hence </b><a NAME="E16:78_2"/>
<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x</font>
 <b>by </b><i><a class="txt" href="#E13:78_2"><i><font color="Green" title="E68">A22</font></i></a>, <a class="txt" href="#E14:78_2"><i><font color="Green" title="E69">A23</font></i></a>, <a class="ref" href="diraf.html#T6" title="DIRAF:th.6">DIRAF:6</a>, <a class="ref" href="funct_2.html#T85" title="FUNCT_2:th.85">FUNCT_2:85</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="E10:78"/>
<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x</font>
 <b>by </b><i><a class="txt" href="#E2:78"><i><font color="Green" title="E53">A2</font></i></a>, <a class="txt" href="#E6:78"><i><font color="Green" title="E57">A6</font></i></a>, <a class="txt" href="#E8:78"><i><font color="Green" title="E59">A8</font></i></a>, <a class="ref" href="diraf.html#T7" title="DIRAF:th.7">DIRAF:7</a>, <a class="ref" href="diraf.html#D3" title="DIRAF:def.3">DIRAF:def 3</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>
