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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">p</font>, <font color="Olive" title="b4">q</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font>  st ( for <font color="Olive" title="b5">p</font>, <font color="Olive" title="b6">q</font>, <font color="Olive" title="b7">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="b5">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b6">q</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Olive" title="b5">p</font>,<font color="Olive" title="b6">q</font>,<font color="Olive" title="b7">r</font> &amp; not <font color="Olive" title="b7">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">p</font> holds <br/><font color="Olive" title="b7">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">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="b3">p</font>,<font color="Olive" title="b4">q</font> &amp; <font color="Olive" title="b1">a</font>,<font color="Olive" title="b3">p</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b2">b</font>,<font color="Olive" title="b4">q</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="b3">p</font> holds <br/><font color="Olive" title="b1">a</font>,<font color="Olive" title="b4">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">p</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">p</font>, <font color="Maroon" title="c5">q</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"> ( ( 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> ) &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="c4">p</font>,<font color="Maroon" title="c5">q</font> &amp; <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">p</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">q</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="c4">p</font> implies <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">p</font> )</span><br/>









<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:17"/><span class="lab"><font color="Green" title="E16">A1</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">and </span><br/><a NAME="E2:17"/><span class="lab"><font color="Green" title="E17">A2</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="c4">p</font>,<font color="Maroon" title="c5">q</font>
 <span class="kw">and </span><br/><a NAME="E3:17"/><span class="lab"><font color="Green" title="E18">A3</font></span>: 
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">p</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">q</font>
 <span class="kw">and </span><br/><a NAME="E4:17"/><span class="lab"><font color="Green" title="E19">A4</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="c4">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="c2">a</font>,<font color="Maroon" title="c5">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">p</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c6">z</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="E6:17"/><span class="lab"><font color="Green" title="E20">A5</font></span>: 
<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c4">p</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c6">z</font>
 <span class="kw">and </span><br/><a NAME="E7:17"/><span class="lab"><font color="Green" title="E21">A6</font></span>: 
<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c2">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c6">z</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="E8:17"/><span class="lab"><font color="Green" title="E22">A7</font></span>: 
<font color="Maroon" title="c4">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">q</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:17_1"/>
<font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">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="E2:17_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c3">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:17"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></span>;<br/>
<a NAME="E3:17_1"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c4">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="E4:17_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:17"><span class="lab"><font color="Green" title="E19">A4</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="E9:17"/><span class="lab"><font color="Green" title="E23">A8</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="c4">p</font>,<font color="Maroon" title="c5">q</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:17_2"/><span class="lab"><font color="Green" title="E24">A9</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c4">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="E2:17_2"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">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="E3:17_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A10</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c2">a</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="E4:17_2"/>
<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</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="#E2:17"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></span>;<br/>
<a NAME="E5:17_2"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c3">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:17"><span class="lab"><font color="Green" title="E22">A7</font></span></a>, <a class="txt" href="#E3:17_2"><span class="lab"><font color="Green" title="E25">A10</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="E6:17_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:17"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="txt" href="#E8:17"><span class="lab"><font color="Green" title="E22">A7</font></span></a>, <a class="txt" href="#E3:17_2"><span class="lab"><font color="Green" title="E25">A10</font></span></a>, <a class="txt" href="#E1:17_2"><span class="lab"><font color="Green" title="E24">A9</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><span class="lab"><font color="Green" title="E24">A11</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="17_3"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:17_3"/>
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">z</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="E2:17_3"/><span class="kw">then </span>
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">p</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:17"><span class="lab"><font color="Green" title="E21">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">hence </span><a NAME="E3:17_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:17"><span class="lab"><font color="Green" title="E23">A8</font></span></a>, <a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</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="E11:17"/>
<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c6">z</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:17"><span class="lab"><font color="Green" title="E20">A5</font></span></a>, <a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></span>;<br/>
<a NAME="E12:17"/><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="c2">a</font>,<font color="Maroon" title="c6">z</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:17"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="txt" href="#E8:17"><span class="lab"><font color="Green" title="E22">A7</font></span></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></span>;<br/>
<a NAME="E13:17"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A12</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="c6">z</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="E14:17"/>
<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">by</span> <span class="lab"><a class="txt" href="#E4:17"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="ref" href="aff_1.html#T16" title="AFF_1:th.16">AFF_1:16</a></span>;<br/>
<a NAME="E15:17"/><span class="kw">then </span>
( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">z</font> or <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">z</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:17"><span class="lab"><font color="Green" title="E16">A1</font></span></a>, <a class="txt" href="#E13:17"><span class="lab"><font color="Green" title="E25">A12</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E16:17"/>
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:17"><span class="lab"><font color="Green" title="E21">A6</font></span></a>, <a class="txt" href="#E10:17"><span class="lab"><font color="Green" title="E24">A11</font></span></a>, <a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</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>
