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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">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/><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>











<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:97"/><span class="lab"><font color="Green" title="E52">A1</font></span>: 
<font color="Maroon" title="c6">f</font> is  <a href="transgeo.html#V4" title="TRANSGEO:attr.4">dilatation</a> 
 <span class="kw">and </span><br/><a NAME="E2:97"/><span class="lab"><font color="Green" title="E53">A2</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E3:97"/><span class="lab"><font color="Green" title="E54">A3</font></span>: 
( <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> )
 <span class="kw">and </span><br/><a NAME="E4:97"/><span class="lab"><font color="Green" title="E55">A4</font></span>: 
 <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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="E5:97"/><span class="lab"><font color="Green" title="E56">A5</font></span>: 
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:97"><span class="lab"><font color="Green" title="E52">A1</font></span></a>, <a class="txt" href="#E2:97"><span class="lab"><font color="Green" title="E53">A2</font></span></a>, <a class="txt" href="#E3:97"><span class="lab"><font color="Green" title="E54">A3</font></span></a>, <a class="ref" href="transgeo.html#T75" target="_self" title="TRANSGEO:th.75">Th75</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E57">A6</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="97_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:97_1"/><span class="lab"><font color="Green" title="E58">A7</font></span>: 
<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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="E2:97_1"/><span class="kw">then </span>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:97"><span class="lab"><font color="Green" title="E56">A5</font></span></a>, <a class="ref" href="diraf.html#D3" title="DIRAF:def.3">DIRAF:def 3</a></span>;<br/><span class="kw">hence </span><a NAME="E3:97_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:97"><span class="lab"><font color="Green" title="E53">A2</font></span></a>, <a class="txt" href="#E1:97_1"><span class="lab"><font color="Green" title="E58">A7</font></span></a>, <a class="ref" href="diraf.html#T5" title="DIRAF:th.5">DIRAF:5</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/></div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E58">A8</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="97_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:97_2"/><span class="lab"><font color="Green" title="E59">A9</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E2:97_2"/>
<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>
 <span class="kw">and </span><br/><a NAME="E3:97_2"/><span class="lab"><font color="Green" title="E60">A10</font></span>: 
<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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E5:97_2"/><span class="lab"><font color="Green" title="E61">A11</font></span>: 
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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:97_2"><span class="lab"><font color="Green" title="E59">A9</font></span></a>, <a class="ref" href="diraf.html#T43" title="DIRAF:th.43">DIRAF:43</a></span>;<br/><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E7:97_2"/><span class="lab"><font color="Green" title="E62">A12</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E8:97_2"/><span class="lab"><font color="Green" title="E63">A13</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="diraf.html#T31" title="DIRAF:th.31">DIRAF:31</a></span>;<br/><a NAME="E9:97_2"/><span class="lab"><font color="Green" title="E64">A14</font></span>: 
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="97_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:97_2_1"/><span class="lab"><font color="Green" title="E65">A15</font></span>: 
 <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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:97_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="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> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:97"><span class="lab"><font color="Green" title="E55">A4</font></span></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></span>;<br/>
<span class="kw">hence </span><a NAME="E3:97_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:97_2"><span class="lab"><font color="Green" title="E60">A10</font></span></a>, <a class="txt" href="#E5:97_2"><span class="lab"><font color="Green" title="E61">A11</font></span></a>, <a class="txt" href="#E1:97_2_1"><span class="lab"><font color="Green" title="E65">A15</font></span></a>, <a class="ref" href="diraf.html#T38" title="DIRAF:th.38">DIRAF:38</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/>


</div><span class="kw">end;</span></div><a NAME="E10:97_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A16</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:97_2"><span class="lab"><font color="Green" title="E62">A12</font></span></a>, <a class="txt" href="#E8:97_2"><span class="lab"><font color="Green" title="E63">A13</font></span></a>, <a class="ref" href="pasch.html#T21" title="PASCH:th.21">PASCH:21</a></span>;<br/><a NAME="E11:97_2"/><span class="lab"><font color="Green" title="E66">A17</font></span>: 
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="97_2_2"><span class="kw">proof </span></a><div class="add">

<div><span class="lab"><font color="Green" title="E67">A18</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="97_2_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:97_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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:97_2_2_1"/><span class="kw">then </span>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:97_2"><span class="lab"><font color="Green" title="E63">A13</font></span></a>, <a class="ref" href="diraf.html#T27" title="DIRAF:th.27">DIRAF:27</a></span>;<br/><a NAME="E3:97_2_2_1"/><span class="kw">then </span>
 <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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="diraf.html#D5" title="DIRAF:def.5">DIRAF:def 5</a></span>;<br/><span class="kw">hence </span><a NAME="E4:97_2_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:97_2"><span class="lab"><font color="Green" title="E64">A14</font></span></a>, <a class="ref" href="diraf.html#T36" title="DIRAF:th.36">DIRAF:36</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/></div><span class="kw">end;</span></div>
<a NAME="E2:97_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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:97"><span class="lab"><font color="Green" title="E55">A4</font></span></a>, <a class="ref" href="diraf.html#T36" title="DIRAF:th.36">DIRAF:36</a></span>;<br/>
<a NAME="E3:97_2_2"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="diraf.html#D5" title="DIRAF:def.5">DIRAF:def 5</a></span>;<br/>
<a NAME="E4:97_2_2"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="diraf.html#T27" title="DIRAF:th.27">DIRAF:27</a></span>;<br/>
<a NAME="E5:97_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A19</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:97_2"><span class="lab"><font color="Green" title="E60">A10</font></span></a>, <a class="txt" href="#E7:97_2"><span class="lab"><font color="Green" title="E62">A12</font></span></a>, <a class="ref" href="diraf.html#T28" title="DIRAF:th.28">DIRAF:28</a></span>;<br/>
<a NAME="E6:97_2_2"/><span class="lab"><font color="Green" title="E69">A20</font></span>: 
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="diraf.html#T37" title="DIRAF:th.37">DIRAF:37</a></span>;<br/>
<span class="kw">assume </span><a NAME="E7:97_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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E8:97_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A21</font></span>: 
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="diraf.html#T36" title="DIRAF:th.36">DIRAF:36</a></span>;<br/>
<a NAME="E9:97_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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:97"><span class="lab"><font color="Green" title="E55">A4</font></span></a>, <a class="ref" href="diraf.html#D5" title="DIRAF:def.5">DIRAF:def 5</a></span>;<br/>
<a NAME="E10:97_2_2"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:97_2"><span class="lab"><font color="Green" title="E59">A9</font></span></a>, <a class="txt" href="#E5:97_2_2"><span class="lab"><font color="Green" title="E68">A19</font></span></a>, <a class="ref" href="diraf.html#T28" title="DIRAF:th.28">DIRAF:28</a></span>;<br/>
<a NAME="E11:97_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A22</font></span>: 
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:97_2_2"><span class="lab"><font color="Green" title="E67">A18</font></span></a>, <a class="txt" href="#E8:97_2_2"><span class="lab"><font color="Green" title="E70">A21</font></span></a>, <a class="ref" href="diraf.html#T39" title="DIRAF:th.39">DIRAF:39</a></span>;<br/>
<a NAME="E12:97_2_2"/>
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:97_2_2"><span class="lab"><font color="Green" title="E67">A18</font></span></a>, <a class="txt" href="#E8:97_2_2"><span class="lab"><font color="Green" title="E70">A21</font></span></a>, <a class="txt" href="#E5:97_2_2"><span class="lab"><font color="Green" title="E68">A19</font></span></a>, <a class="ref" href="diraf.html#T39" title="DIRAF:th.39">DIRAF:39</a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:97_2_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:97_2"><span class="lab"><font color="Green" title="E64">A14</font></span></a>, <a class="txt" href="#E1:97_2_2"><span class="lab"><font color="Green" title="E67">A18</font></span></a>, <a class="txt" href="#E11:97_2_2"><span class="lab"><font color="Green" title="E71">A22</font></span></a>, <a class="txt" href="#E6:97_2_2"><span class="lab"><font color="Green" title="E69">A20</font></span></a>, <a class="ref" href="diraf.html#T38" title="DIRAF:th.38">DIRAF:38</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/>


</div><span class="kw">end;</span></div><a NAME="E12:97_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A23</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="diraf.html#T37" title="DIRAF:th.37">DIRAF:37</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c9">u9</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/><span class="kw">set </span><font color="Maroon" title="c10">r9</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/><span class="kw">set </span><font color="Maroon" title="c11">x9</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/><span class="kw">set </span><font color="Maroon" title="c12">y9</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="E13:97_2"/><span class="lab"><font color="Green" title="E68">A24</font></span>: 
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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:97"><span class="lab"><font color="Green" title="E52">A1</font></span></a>, <a class="txt" href="#E9:97_2"><span class="lab"><font color="Green" title="E64">A14</font></span></a>, <a class="ref" href="transgeo.html#T59" target="_self" title="TRANSGEO:th.59">Th59</a></span>;<br/><a NAME="E14:97_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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:97"><span class="lab"><font color="Green" title="E52">A1</font></span></a>, <a class="txt" href="#E7:97_2"><span class="lab"><font color="Green" title="E62">A12</font></span></a>, <a class="txt" href="#E8:97_2"><span class="lab"><font color="Green" title="E63">A13</font></span></a>, <a class="ref" href="transgeo.html#T58" target="_self" title="TRANSGEO:th.58">Th58</a></span>;<br/><a NAME="E15:97_2"/><span class="kw">then </span>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:97_2"><span class="lab"><font color="Green" title="E68">A24</font></span></a>, <a class="ref" href="pasch.html#T21" title="PASCH:th.21">PASCH:21</a></span>;<br/><a NAME="E16:97_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A25</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="diraf.html#T5" title="DIRAF:th.5">DIRAF:5</a></span>;<br/><a NAME="E17:97_2"/>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:97"><span class="lab"><font color="Green" title="E52">A1</font></span></a>, <a class="txt" href="#E2:97"><span class="lab"><font color="Green" title="E53">A2</font></span></a>, <a class="txt" href="#E3:97"><span class="lab"><font color="Green" title="E54">A3</font></span></a>, <a class="txt" href="#E11:97_2"><span class="lab"><font color="Green" title="E66">A17</font></span></a>, <a class="ref" href="transgeo.html#T76" target="_self" title="TRANSGEO:th.76">Th76</a></span>;<br/><a NAME="E18:97_2"/><span class="kw">then </span>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:97_2"><span class="lab"><font color="Green" title="E65">A16</font></span></a>, <a class="txt" href="#E12:97_2"><span class="lab"><font color="Green" title="E67">A23</font></span></a>, <a class="ref" href="diraf.html#T6" title="DIRAF:th.6">DIRAF:6</a></span>;<br/><span class="kw">hence </span><a NAME="E19:97_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:97_2"><span class="lab"><font color="Green" title="E69">A25</font></span></a>, <a class="txt" href="#E12:97_2"><span class="lab"><font color="Green" title="E67">A23</font></span></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></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/></div><span class="kw">end;</span></div>
<a NAME="E8:97"/>
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:97"><span class="lab"><font color="Green" title="E52">A1</font></span></a>, <a class="txt" href="#E2:97"><span class="lab"><font color="Green" title="E53">A2</font></span></a>, <a class="txt" href="#E3:97"><span class="lab"><font color="Green" title="E54">A3</font></span></a>, <a class="ref" href="transgeo.html#T75" target="_self" title="TRANSGEO:th.75">Th75</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:97"/>
<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 class="kw">by</span> <span class="lab"><a class="txt" href="#E2:97"><span class="lab"><font color="Green" title="E53">A2</font></span></a>, <a class="txt" href="#E6:97"><span class="lab"><font color="Green" title="E57">A6</font></span></a>, <a class="txt" href="#E7:97"><span class="lab"><font color="Green" title="E58">A8</font></span></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></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/>


</div>
