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<span class="kw">let </span><font color="Maroon" title="c1">AFP</font> be   <a href="diraf.html#NM2" title="DIRAF:NM.2">AffinPlane</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">x</font>, <font color="Olive" title="b4">p</font>, <font color="Olive" title="b5">q</font>, <font color="Olive" title="b6">p9</font>, <font color="Olive" title="b7">q9</font>, <font color="Olive" title="b8">y</font>, <font color="Olive" title="b9">y9</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font>  st <font color="Maroon" title="c1">AFP</font> is  <a href="aff_2.html#V11" title="AFF_2:attr.11">translational</a>  &amp; <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">b</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">x</font> &amp; <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b4">p</font>,<font color="Olive" title="b5">q</font> &amp; <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b6">p9</font>,<font color="Olive" title="b7">q9</font> &amp; <font color="Olive" title="b1">a</font>,<font color="Olive" title="b4">p</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b2">b</font>,<font color="Olive" title="b5">q</font> &amp; <font color="Olive" title="b1">a</font>,<font color="Olive" title="b6">p9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b2">b</font>,<font color="Olive" title="b7">q9</font> &amp; <font color="Olive" title="b4">p</font>,<font color="Olive" title="b5">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b3">x</font>,<font color="Olive" title="b8">y</font> &amp; <font color="Olive" title="b6">p9</font>,<font color="Olive" title="b7">q9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b3">x</font>,<font color="Olive" title="b9">y9</font> &amp; <font color="Olive" title="b4">p</font>,<font color="Olive" title="b3">x</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b5">q</font>,<font color="Olive" title="b8">y</font> &amp; <font color="Olive" title="b6">p9</font>,<font color="Olive" title="b3">x</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b7">q9</font>,<font color="Olive" title="b9">y9</font> &amp; not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b4">p</font> &amp; not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b6">p9</font> holds <br/><font color="Olive" title="b8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b9">y9</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font>, <font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c5">p</font>, <font color="Maroon" title="c6">q</font>, <font color="Maroon" title="c7">p9</font>, <font color="Maroon" title="c8">q9</font>, <font color="Maroon" title="c9">y</font>, <font color="Maroon" title="c10">y9</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</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="c1">AFP</font> is  <a href="aff_2.html#V11" title="AFF_2:attr.11">translational</a>  &amp; <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> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">x</font> &amp; <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</font> &amp; <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">p9</font>,<font color="Maroon" title="c8">q9</font> &amp; <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">p</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c6">q</font> &amp; <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c7">p9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c8">q9</font> &amp; <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c9">y</font> &amp; <font color="Maroon" title="c7">p9</font>,<font color="Maroon" title="c8">q9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c10">y9</font> &amp; <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c4">x</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c9">y</font> &amp; <font color="Maroon" title="c7">p9</font>,<font color="Maroon" title="c4">x</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">q9</font>,<font color="Maroon" title="c10">y9</font> &amp; not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">p</font> &amp; not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c7">p9</font> implies <font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">y9</font> )</span><br/>



















<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:18"/><span class="lab"><font color="Green" title="E17">A1</font></span>: 
<font color="Maroon" title="c1">AFP</font> is  <a href="aff_2.html#V11" title="AFF_2:attr.11">translational</a> 
 <span class="kw">and </span><br/><a NAME="E2:18"/><span class="lab"><font color="Green" title="E18">A2</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E3:18"/><span class="lab"><font color="Green" title="E19">A3</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">x</font>
 <span class="kw">and </span><br/><a NAME="E4:18"/><span class="lab"><font color="Green" title="E20">A4</font></span>: 
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</font>
 <span class="kw">and </span><br/><a NAME="E5:18"/><span class="lab"><font color="Green" title="E21">A5</font></span>: 
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">p9</font>,<font color="Maroon" title="c8">q9</font>
 <span class="kw">and </span><br/><a NAME="E6:18"/><span class="lab"><font color="Green" title="E22">A6</font></span>: 
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">p</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c6">q</font>
 <span class="kw">and </span><br/><a NAME="E7:18"/><span class="lab"><font color="Green" title="E23">A7</font></span>: 
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c7">p9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c8">q9</font>
 <span class="kw">and </span><br/><a NAME="E8:18"/><span class="lab"><font color="Green" title="E24">A8</font></span>: 
<font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c9">y</font>
 <span class="kw">and </span><br/><a NAME="E9:18"/><span class="lab"><font color="Green" title="E25">A9</font></span>: 
<font color="Maroon" title="c7">p9</font>,<font color="Maroon" title="c8">q9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c10">y9</font>
 <span class="kw">and </span><br/><a NAME="E10:18"/><span class="lab"><font color="Green" title="E26">A10</font></span>: 
<font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c4">x</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c9">y</font>
 <span class="kw">and </span><br/><a NAME="E11:18"/><span class="lab"><font color="Green" title="E27">A11</font></span>: 
<font color="Maroon" title="c7">p9</font>,<font color="Maroon" title="c4">x</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">q9</font>,<font color="Maroon" title="c10">y9</font>
 <span class="kw">and </span><br/><a NAME="E12:18"/><span class="lab"><font color="Green" title="E28">A12</font></span>: 
not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">p</font>
 <span class="kw">and </span><br/><a NAME="E13:18"/><span class="lab"><font color="Green" title="E29">A13</font></span>: 
not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c7">p9</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="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">y9</font></span><br/>

<a NAME="E14:18"/><span class="lab"><font color="Green" title="E30">A14</font></span>: 
<font color="Maroon" title="c5">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">q</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:18_1"/>
<font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">q</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="E2:18_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c2">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c3">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:18"><span class="lab"><font color="Green" title="E22">A6</font></span></a>, <a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></span>;<br/>
<a NAME="E3:18_1"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:18_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:18"><span class="lab"><font color="Green" title="E28">A12</font></span></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</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="E15:18"/><span class="kw">then </span>
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c9">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:18"><span class="lab"><font color="Green" title="E20">A4</font></span></a>, <a class="txt" href="#E8:18"><span class="lab"><font color="Green" title="E24">A8</font></span></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></span>;<br/>
<a NAME="E16:18"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A15</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c9">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:18"><span class="lab"><font color="Green" title="E18">A2</font></span></a>, <a class="txt" href="#E3:18"><span class="lab"><font color="Green" title="E19">A3</font></span></a>, <a class="ref" href="aff_1.html#T18" title="AFF_1:th.18">AFF_1:18</a></span>;<br/>
<a NAME="E17:18"/><span class="lab"><font color="Green" title="E32">A16</font></span>: 
not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c4">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:18_2"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c4">x</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="E2:18_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c4">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></span>;<br/>
<a NAME="E3:18_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c4">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:18"><span class="lab"><font color="Green" title="E20">A4</font></span></a>, <a class="txt" href="#E14:18"><span class="lab"><font color="Green" title="E30">A14</font></span></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></span>;<br/>
<a NAME="E4:18_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:18_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:18"><span class="lab"><font color="Green" title="E18">A2</font></span></a>, <a class="txt" href="#E3:18"><span class="lab"><font color="Green" title="E19">A3</font></span></a>, <a class="txt" href="#E12:18"><span class="lab"><font color="Green" title="E28">A12</font></span></a>, <a class="ref" href="aff_1.html#T18" title="AFF_1:th.18">AFF_1:18</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="E33">A17</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="18_3"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:18_3"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y</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="E2:18_3"/><span class="kw">then </span>
<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">p</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c6">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:18"><span class="lab"><font color="Green" title="E26">A10</font></span></a>, <a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></span>;<br/><a NAME="E3:18_3"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E4:18_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:18"><span class="lab"><font color="Green" title="E32">A16</font></span></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</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="E19:18"/><span class="lab"><font color="Green" title="E34">A18</font></span>: 
<font color="Maroon" title="c7">p9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">q9</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:18_4"/>
<font color="Maroon" title="c7">p9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">q9</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="E2:18_4"/><span class="kw">then </span>
<font color="Maroon" title="c7">p9</font>,<font color="Maroon" title="c2">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">p9</font>,<font color="Maroon" title="c3">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:18"><span class="lab"><font color="Green" title="E23">A7</font></span></a>, <a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></span>;<br/>
<a NAME="E3:18_4"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c7">p9</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:18_4"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:18"><span class="lab"><font color="Green" title="E29">A13</font></span></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</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="E20:18"/><span class="lab"><font color="Green" title="E35">A19</font></span>: 
not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c3">b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_5"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:18_5"/><span class="lab"><font color="Green" title="E36">A20</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c5">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T16" title="AFF_1:th.16">AFF_1:16</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:18_5"/><span class="lab"><font color="Green" title="E37">A21</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c3">b</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="E3:18_5"/>
<font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c5">p</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:18"><span class="lab"><font color="Green" title="E20">A4</font></span></a>, <a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></span>;<br/>
<a NAME="E4:18_5"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c2">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:18"><span class="lab"><font color="Green" title="E30">A14</font></span></a>, <a class="txt" href="#E2:18_5"><span class="lab"><font color="Green" title="E37">A21</font></span></a>, <a class="ref" href="aff_1.html#T18" title="AFF_1:th.18">AFF_1:18</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:18_5"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:18"><span class="lab"><font color="Green" title="E28">A12</font></span></a>, <a class="txt" href="#E14:18"><span class="lab"><font color="Green" title="E30">A14</font></span></a>, <a class="txt" href="#E2:18_5"><span class="lab"><font color="Green" title="E37">A21</font></span></a>, <a class="txt" href="#E1:18_5"><span class="lab"><font color="Green" title="E36">A20</font></span></a>, <a class="ref" href="aff_1.html#T17" title="AFF_1:th.17">AFF_1:17</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><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_6"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:18_6"/><span class="lab"><font color="Green" title="E36">A22</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c7">p9</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="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">y9</font></span><br/><a NAME="E2:18_6"/>
<font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">p9</font>,<font color="Maroon" title="c8">q9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:18"><span class="lab"><font color="Green" title="E18">A2</font></span></a>, <a class="txt" href="#E4:18"><span class="lab"><font color="Green" title="E20">A4</font></span></a>, <a class="txt" href="#E5:18"><span class="lab"><font color="Green" title="E21">A5</font></span></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></span>;<br/><a NAME="E3:18_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A23</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c8">q9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:18"><span class="lab"><font color="Green" title="E30">A14</font></span></a>, <a class="txt" href="#E1:18_6"><span class="lab"><font color="Green" title="E36">A22</font></span></a>, <a class="ref" href="aff_1.html#T18" title="AFF_1:th.18">AFF_1:18</a></span>;<br/><div><span class="lab"><font color="Green" title="E38">A24</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="18_6_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:18_6_1"/><span class="lab"><font color="Green" title="E39">A25</font></span>: 
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">AFP</font>  holds <br/>(  not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">p</font>,<font color="Olive" title="b1">x</font> or <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> or <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">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="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">y9</font></span><br/><a NAME="E2:18_6_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A26</font></span>: 
for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font>, <font color="Olive" title="b3">r</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font>  st <font color="Olive" title="b1">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">q</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font>,<font color="Olive" title="b3">r</font> &amp; not <font color="Olive" title="b3">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> holds <br/><font color="Olive" title="b3">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">q</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="translac.html#T2" target="_self" title="TRANSLAC:th.2">Th2</a></span>;<br/><div><span class="lab"><font color="Green" title="E41">A27</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="18_6_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:18_6_1_1"/><span class="lab"><font color="Green" title="E42">A28</font></span>: 
<font color="Maroon" title="c6">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p9</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="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">y9</font></span><br/><div><span class="lab"><font color="Green" title="E43">A29</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="18_6_1_1_1"><span class="kw">now </span></a><div class="add"><a NAME="E1:18_6_1_1_1"/>
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c6">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:18"><span class="lab"><font color="Green" title="E20">A4</font></span></a>, <a class="txt" href="#E6:18"><span class="lab"><font color="Green" title="E22">A6</font></span></a>, <a class="txt" href="#E12:18"><span class="lab"><font color="Green" title="E28">A12</font></span></a>, <a class="txt" href="#E2:18_6_1"><span class="lab"><font color="Green" title="E40">A26</font></span></a>, <a class="ref" href="translac.html#T8" target="_self" title="TRANSLAC:th.8">Th8</a></span>;<br/><a NAME="E2:18_6_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A30</font></span>: 
<font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c2">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c3">b</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></span>;<br/><a NAME="E3:18_6_1_1_1"/><span class="lab"><font color="Green" title="E45">A31</font></span>: 
<font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c5">p</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:18"><span class="lab"><font color="Green" title="E20">A4</font></span></a>, <a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></span>;<br/><span class="kw">assume </span><a NAME="E4:18_6_1_1_1"/><span class="lab"><font color="Green" title="E46">A32</font></span>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">y9</font></span><br/><a NAME="E5:18_6_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A33</font></span>: 
( <font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c2">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c10">y9</font> &amp; not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c2">a</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:18"><span class="lab"><font color="Green" title="E27">A11</font></span></a>, <a class="txt" href="#E14:18"><span class="lab"><font color="Green" title="E30">A14</font></span></a>, <a class="txt" href="#E19:18"><span class="lab"><font color="Green" title="E34">A18</font></span></a>, <a class="txt" href="#E17:18"><span class="lab"><font color="Green" title="E32">A16</font></span></a>, <a class="txt" href="#E3:18_6"><span class="lab"><font color="Green" title="E37">A23</font></span></a>, <a class="txt" href="#E1:18_6_1"><span class="lab"><font color="Green" title="E39">A25</font></span></a>, <a class="txt" href="#E1:18_6_1_1"><span class="lab"><font color="Green" title="E42">A28</font></span></a>, <a class="ref" href="translac.html#T2" target="_self" title="TRANSLAC:th.2">Th2</a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></span>;<br/><a NAME="E6:18_6_1_1_1"/>
( <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y</font> &amp; <font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c5">p</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c10">y9</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:18"><span class="lab"><font color="Green" title="E18">A2</font></span></a>, <a class="txt" href="#E9:18"><span class="lab"><font color="Green" title="E25">A9</font></span></a>, <a class="txt" href="#E14:18"><span class="lab"><font color="Green" title="E30">A14</font></span></a>, <a class="txt" href="#E19:18"><span class="lab"><font color="Green" title="E34">A18</font></span></a>, <a class="txt" href="#E18:18"><span class="lab"><font color="Green" title="E33">A17</font></span></a>, <a class="txt" href="#E16:18"><span class="lab"><font color="Green" title="E31">A15</font></span></a>, <a class="txt" href="#E3:18_6"><span class="lab"><font color="Green" title="E37">A23</font></span></a>, <a class="txt" href="#E1:18_6_1"><span class="lab"><font color="Green" title="E39">A25</font></span></a>, <a class="txt" href="#E1:18_6_1_1"><span class="lab"><font color="Green" title="E42">A28</font></span></a>, <a class="txt" href="#E4:18_6_1_1_1"><span class="lab"><font color="Green" title="E46">A32</font></span></a>, <a class="ref" href="translac.html#T2" target="_self" title="TRANSLAC:th.2">Th2</a></span>;<br/><span class="kw">hence </span><a NAME="E7:18_6_1_1_1"/>
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">y9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:18_6_1_1_1"><span class="lab"><font color="Green" title="E45">A31</font></span></a>, <a class="txt" href="#E2:18_6_1_1_1"><span class="lab"><font color="Green" title="E44">A30</font></span></a>, <a class="txt" href="#E5:18_6_1_1_1"><span class="lab"><font color="Green" title="E47">A33</font></span></a>, <a class="ref" href="transgeo.html#T97" title="TRANSGEO:th.97">TRANSGEO:97</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><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_6_1_1_2"><span class="kw">now </span></a><div class="add"><a NAME="E1:18_6_1_1_2"/><span class="lab"><font color="Green" title="E44">A34</font></span>: 
( <font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c5">p</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">a</font> &amp; <font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c2">a</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:18"><span class="lab"><font color="Green" title="E20">A4</font></span></a>, <a class="txt" href="#E6:18"><span class="lab"><font color="Green" title="E22">A6</font></span></a>, <a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></span>;<br/><span class="kw">assume </span><a NAME="E2:18_6_1_1_2"/><span class="lab"><font color="Green" title="E45">A35</font></span>: 
<font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">y9</font></span><br/><a NAME="E3:18_6_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A36</font></span>: 
<font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c10">y9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:18"><span class="lab"><font color="Green" title="E27">A11</font></span></a>, <a class="txt" href="#E14:18"><span class="lab"><font color="Green" title="E30">A14</font></span></a>, <a class="txt" href="#E19:18"><span class="lab"><font color="Green" title="E34">A18</font></span></a>, <a class="txt" href="#E3:18_6"><span class="lab"><font color="Green" title="E37">A23</font></span></a>, <a class="txt" href="#E1:18_6_1"><span class="lab"><font color="Green" title="E39">A25</font></span></a>, <a class="txt" href="#E1:18_6_1_1"><span class="lab"><font color="Green" title="E42">A28</font></span></a>, <a class="ref" href="translac.html#T2" target="_self" title="TRANSLAC:th.2">Th2</a></span>;<br/><a NAME="E4:18_6_1_1_2"/>
( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y</font> &amp; <font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c5">p</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c10">y9</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:18"><span class="lab"><font color="Green" title="E18">A2</font></span></a>, <a class="txt" href="#E9:18"><span class="lab"><font color="Green" title="E25">A9</font></span></a>, <a class="txt" href="#E14:18"><span class="lab"><font color="Green" title="E30">A14</font></span></a>, <a class="txt" href="#E19:18"><span class="lab"><font color="Green" title="E34">A18</font></span></a>, <a class="txt" href="#E18:18"><span class="lab"><font color="Green" title="E33">A17</font></span></a>, <a class="txt" href="#E16:18"><span class="lab"><font color="Green" title="E31">A15</font></span></a>, <a class="txt" href="#E3:18_6"><span class="lab"><font color="Green" title="E37">A23</font></span></a>, <a class="txt" href="#E1:18_6_1"><span class="lab"><font color="Green" title="E39">A25</font></span></a>, <a class="txt" href="#E1:18_6_1_1"><span class="lab"><font color="Green" title="E42">A28</font></span></a>, <a class="txt" href="#E2:18_6_1_1_2"><span class="lab"><font color="Green" title="E45">A35</font></span></a>, <a class="ref" href="translac.html#T2" target="_self" title="TRANSLAC:th.2">Th2</a></span>;<br/><span class="kw">hence </span><a NAME="E5:18_6_1_1_2"/>
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">y9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:18"><span class="lab"><font color="Green" title="E35">A19</font></span></a>, <a class="txt" href="#E1:18_6_1_1_2"><span class="lab"><font color="Green" title="E44">A34</font></span></a>, <a class="txt" href="#E3:18_6_1_1_2"><span class="lab"><font color="Green" title="E46">A36</font></span></a>, <a class="ref" href="transgeo.html#T97" title="TRANSGEO:th.97">TRANSGEO:97</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><span class="kw">hence </span><a NAME="E4:18_6_1_1"/>
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">y9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:18"><span class="lab"><font color="Green" title="E18">A2</font></span></a>, <a class="txt" href="#E3:18"><span class="lab"><font color="Green" title="E19">A3</font></span></a>, <a class="txt" href="#E1:18_6_1"><span class="lab"><font color="Green" title="E39">A25</font></span></a>, <a class="txt" href="#E2:18_6_1_1"><span class="lab"><font color="Green" title="E43">A29</font></span></a>, <a class="ref" href="translac.html#T2" target="_self" title="TRANSLAC:th.2">Th2</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><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_6_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:18_6_1_2"/><span class="lab"><font color="Green" title="E42">A37</font></span>: 
<font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p9</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="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">y9</font></span><br/><a NAME="E2:18_6_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">q9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:18"><span class="lab"><font color="Green" title="E30">A14</font></span></a>, <a class="txt" href="#E19:18"><span class="lab"><font color="Green" title="E34">A18</font></span></a>, <a class="txt" href="#E3:18_6"><span class="lab"><font color="Green" title="E37">A23</font></span></a>, <a class="txt" href="#E1:18_6_1"><span class="lab"><font color="Green" title="E39">A25</font></span></a>, <a class="ref" href="translac.html#T2" target="_self" title="TRANSLAC:th.2">Th2</a></span>;<br/><span class="kw">hence </span><a NAME="E3:18_6_1_2"/>
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">y9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:18"><span class="lab"><font color="Green" title="E24">A8</font></span></a>, <a class="txt" href="#E9:18"><span class="lab"><font color="Green" title="E25">A9</font></span></a>, <a class="txt" href="#E10:18"><span class="lab"><font color="Green" title="E26">A10</font></span></a>, <a class="txt" href="#E11:18"><span class="lab"><font color="Green" title="E27">A11</font></span></a>, <a class="txt" href="#E17:18"><span class="lab"><font color="Green" title="E32">A16</font></span></a>, <a class="txt" href="#E1:18_6_1_2"><span class="lab"><font color="Green" title="E42">A37</font></span></a>, <a class="ref" href="transgeo.html#T97" title="TRANSGEO:th.97">TRANSGEO:97</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><span class="kw">hence </span><a NAME="E5:18_6_1"/>
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">y9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:18"><span class="lab"><font color="Green" title="E30">A14</font></span></a>, <a class="txt" href="#E1:18_6"><span class="lab"><font color="Green" title="E36">A22</font></span></a>, <a class="txt" href="#E1:18_6_1"><span class="lab"><font color="Green" title="E39">A25</font></span></a>, <a class="txt" href="#E3:18_6_1"><span class="lab"><font color="Green" title="E41">A27</font></span></a>, <a class="ref" href="translac.html#T2" target="_self" title="TRANSLAC:th.2">Th2</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><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_6_2"><span class="kw">now </span></a><div class="add"><span class="kw">given </span><font color="Maroon" title="c11">p99</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font><span class="kw"> such that </span><a NAME="E2:18_6_2"/><span class="lab"><font color="Green" title="E39">A38</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c11">p99</font>
 <span class="kw">and </span><br/><a NAME="E3:18_6_2"/><span class="lab"><font color="Green" title="E40">A39</font></span>: 
<font color="Maroon" title="c11">p99</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">a</font>
 <span class="kw">and </span><br/><a NAME="E4:18_6_2"/><span class="lab"><font color="Green" title="E41">A40</font></span>: 
<font color="Maroon" title="c11">p99</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">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="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">y9</font></span><br/><a NAME="E5:18_6_2"/><span class="lab"><font color="Green" title="E42">A41</font></span>: 
not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c11">p99</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_6_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:18_6_2_1"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c11">p99</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="E2:18_6_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A42</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c11">p99</font>,<font color="Maroon" title="c6">q</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></span>;<br/>
<a NAME="E3:18_6_2_1"/>
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c11">p99</font>,<font color="Maroon" title="c5">p</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c11">p99</font>,<font color="Maroon" title="c2">a</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:18_6_2"><span class="lab"><font color="Green" title="E39">A38</font></span></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a>, <a class="ref" href="aff_1.html#T16" title="AFF_1:th.16">AFF_1:16</a></span>;<br/>
<a NAME="E4:18_6_2_1"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c2">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:18_6_2"><span class="lab"><font color="Green" title="E41">A40</font></span></a>, <a class="txt" href="#E2:18_6_2_1"><span class="lab"><font color="Green" title="E43">A42</font></span></a>, <a class="ref" href="aff_1.html#T17" title="AFF_1:th.17">AFF_1:17</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:18_6_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:18"><span class="lab"><font color="Green" title="E20">A4</font></span></a>, <a class="txt" href="#E6:18"><span class="lab"><font color="Green" title="E22">A6</font></span></a>, <a class="txt" href="#E12:18"><span class="lab"><font color="Green" title="E28">A12</font></span></a>, <a class="txt" href="translac.html#E10"><span class="lab"><font color="Green" title="E8">Lm5</font></span></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="E6:18_6_2"/><span class="lab"><font color="Green" title="E43">A43</font></span>: 
not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c7">p9</font>,<font color="Maroon" title="c8">q9</font>,<font color="Maroon" title="c11">p99</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_6_2_2"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:18_6_2_2"/>
<font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">p9</font>,<font color="Maroon" title="c8">q9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:18"><span class="lab"><font color="Green" title="E18">A2</font></span></a>, <a class="txt" href="#E4:18"><span class="lab"><font color="Green" title="E20">A4</font></span></a>, <a class="txt" href="#E5:18"><span class="lab"><font color="Green" title="E21">A5</font></span></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></span>;<br/>
<a NAME="E2:18_6_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A44</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c8">q9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:18"><span class="lab"><font color="Green" title="E30">A14</font></span></a>, <a class="txt" href="#E1:18_6"><span class="lab"><font color="Green" title="E36">A22</font></span></a>, <a class="ref" href="aff_1.html#T18" title="AFF_1:th.18">AFF_1:18</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:18_6_2_2"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c7">p9</font>,<font color="Maroon" title="c8">q9</font>,<font color="Maroon" title="c11">p99</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/>

<span class="kw">hence </span><a NAME="E4:18_6_2_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:18"><span class="lab"><font color="Green" title="E34">A18</font></span></a>, <a class="txt" href="#E1:18_6"><span class="lab"><font color="Green" title="E36">A22</font></span></a>, <a class="txt" href="#E5:18_6_2"><span class="lab"><font color="Green" title="E42">A41</font></span></a>, <a class="txt" href="#E2:18_6_2_2"><span class="lab"><font color="Green" title="E44">A44</font></span></a>, <a class="ref" href="aff_1.html#T20" title="AFF_1:th.20">AFF_1:20</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><span class="kw">consider </span><font color="Maroon" title="c12">q99</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font><span class="kw"> such that </span><br/><a NAME="E8:18_6_2"/><span class="lab"><font color="Green" title="E44">A45</font></span>: 
( <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">p99</font>,<font color="Maroon" title="c12">q99</font> &amp; <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c11">p99</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c12">q99</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="diraf.html#T47" title="DIRAF:th.47">DIRAF:47</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c13">y99</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font><span class="kw"> such that </span><br/><a NAME="E10:18_6_2"/><span class="lab"><font color="Green" title="E45">A46</font></span>: 
( <font color="Maroon" title="c11">p99</font>,<font color="Maroon" title="c12">q99</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c13">y99</font> &amp; <font color="Maroon" title="c11">p99</font>,<font color="Maroon" title="c4">x</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c12">q99</font>,<font color="Maroon" title="c13">y99</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="diraf.html#T47" title="DIRAF:th.47">DIRAF:47</a></span>;<br/><a NAME="E11:18_6_2"/><span class="lab"><font color="Green" title="E46">A47</font></span>: 
not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c11">p99</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_6_2_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:18_6_2_3"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c11">p99</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="E2:18_6_2_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A48</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c11">p99</font>,<font color="Maroon" title="c3">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></span>;<br/>
<a NAME="E3:18_6_2_3"/>
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c11">p99</font>,<font color="Maroon" title="c2">a</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c11">p99</font>,<font color="Maroon" title="c5">p</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:18_6_2"><span class="lab"><font color="Green" title="E39">A38</font></span></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a>, <a class="ref" href="aff_1.html#T16" title="AFF_1:th.16">AFF_1:16</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:18_6_2_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:18"><span class="lab"><font color="Green" title="E28">A12</font></span></a>, <a class="txt" href="#E3:18_6_2"><span class="lab"><font color="Green" title="E40">A39</font></span></a>, <a class="txt" href="#E2:18_6_2_3"><span class="lab"><font color="Green" title="E47">A48</font></span></a>, <a class="ref" href="aff_1.html#T17" title="AFF_1:th.17">AFF_1:17</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="E12:18_6_2"/><span class="kw">then </span>
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">y99</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18"><span class="lab"><font color="Green" title="E17">A1</font></span></a>, <a class="txt" href="#E2:18"><span class="lab"><font color="Green" title="E18">A2</font></span></a>, <a class="txt" href="#E3:18"><span class="lab"><font color="Green" title="E19">A3</font></span></a>, <a class="txt" href="#E4:18"><span class="lab"><font color="Green" title="E20">A4</font></span></a>, <a class="txt" href="#E6:18"><span class="lab"><font color="Green" title="E22">A6</font></span></a>, <a class="txt" href="#E8:18"><span class="lab"><font color="Green" title="E24">A8</font></span></a>, <a class="txt" href="#E10:18"><span class="lab"><font color="Green" title="E26">A10</font></span></a>, <a class="txt" href="#E12:18"><span class="lab"><font color="Green" title="E28">A12</font></span></a>, <a class="txt" href="#E8:18_6_2"><span class="lab"><font color="Green" title="E44">A45</font></span></a>, <a class="txt" href="#E10:18_6_2"><span class="lab"><font color="Green" title="E45">A46</font></span></a>, <a class="txt" href="#E5:18_6_2"><span class="lab"><font color="Green" title="E42">A41</font></span></a>, <a class="txt" href="translac.html#E16"><span class="lab"><font color="Green" title="E14">Lm8</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E13:18_6_2"/>
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">y9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18"><span class="lab"><font color="Green" title="E17">A1</font></span></a>, <a class="txt" href="#E2:18"><span class="lab"><font color="Green" title="E18">A2</font></span></a>, <a class="txt" href="#E3:18"><span class="lab"><font color="Green" title="E19">A3</font></span></a>, <a class="txt" href="#E5:18"><span class="lab"><font color="Green" title="E21">A5</font></span></a>, <a class="txt" href="#E7:18"><span class="lab"><font color="Green" title="E23">A7</font></span></a>, <a class="txt" href="#E9:18"><span class="lab"><font color="Green" title="E25">A9</font></span></a>, <a class="txt" href="#E11:18"><span class="lab"><font color="Green" title="E27">A11</font></span></a>, <a class="txt" href="#E13:18"><span class="lab"><font color="Green" title="E29">A13</font></span></a>, <a class="txt" href="#E8:18_6_2"><span class="lab"><font color="Green" title="E44">A45</font></span></a>, <a class="txt" href="#E10:18_6_2"><span class="lab"><font color="Green" title="E45">A46</font></span></a>, <a class="txt" href="#E6:18_6_2"><span class="lab"><font color="Green" title="E43">A43</font></span></a>, <a class="txt" href="#E11:18_6_2"><span class="lab"><font color="Green" title="E46">A47</font></span></a>, <a class="txt" href="translac.html#E16"><span class="lab"><font color="Green" title="E14">Lm8</font></span></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><span class="kw">hence </span><a NAME="E6:18_6"/>
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">y9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:18_6"><span class="lab"><font color="Green" title="E38">A24</font></span></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>
<span class="kw">hence </span><a NAME="E22:18"/>
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">y9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18"><span class="lab"><font color="Green" title="E17">A1</font></span></a>, <a class="txt" href="#E2:18"><span class="lab"><font color="Green" title="E18">A2</font></span></a>, <a class="txt" href="#E3:18"><span class="lab"><font color="Green" title="E19">A3</font></span></a>, <a class="txt" href="#E4:18"><span class="lab"><font color="Green" title="E20">A4</font></span></a>, <a class="txt" href="#E5:18"><span class="lab"><font color="Green" title="E21">A5</font></span></a>, <a class="txt" href="#E6:18"><span class="lab"><font color="Green" title="E22">A6</font></span></a>, <a class="txt" href="#E7:18"><span class="lab"><font color="Green" title="E23">A7</font></span></a>, <a class="txt" href="#E8:18"><span class="lab"><font color="Green" title="E24">A8</font></span></a>, <a class="txt" href="#E9:18"><span class="lab"><font color="Green" title="E25">A9</font></span></a>, <a class="txt" href="#E10:18"><span class="lab"><font color="Green" title="E26">A10</font></span></a>, <a class="txt" href="#E11:18"><span class="lab"><font color="Green" title="E27">A11</font></span></a>, <a class="txt" href="#E12:18"><span class="lab"><font color="Green" title="E28">A12</font></span></a>, <a class="txt" href="#E13:18"><span class="lab"><font color="Green" title="E29">A13</font></span></a>, <a class="txt" href="translac.html#E16"><span class="lab"><font color="Green" title="E14">Lm8</font></span></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>
