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<span class="kw">let </span><font color="Maroon" title="c1">AP</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="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">C</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AP</font>  st <font color="Olive" title="b1">A</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Olive" title="b2">C</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp;  not <font color="Olive" title="b1">A</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Olive" title="b2">C</font> holds <br/> ex <font color="Olive" title="b3">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AP</font> st <br/>( <font color="Olive" title="b3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">A</font> &amp; <font color="Olive" title="b3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">C</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">A</font>, <font color="Maroon" title="c3">C</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AP</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">A</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Maroon" title="c3">C</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp;  not <font color="Maroon" title="c2">A</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c3">C</font> implies  ex <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">AP</font> st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font> ) )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:68"/><span class="lab"><font color="Green" title="E54">A1</font></span>: 
<font color="Maroon" title="c2">A</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 <span class="kw">and </span><br/><a NAME="E2:68"/><span class="lab"><font color="Green" title="E55">A2</font></span>: 
<font color="Maroon" title="c3">C</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 <span class="kw">and </span><br/><a NAME="E3:68"/><span class="lab"><font color="Green" title="E56">A3</font></span>: 
 not <font color="Maroon" title="c2">A</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c3">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <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">AP</font> st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font> )</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c5">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AP</font><span class="kw"> such that </span><br/><a NAME="E5:68"/><span class="lab"><font color="Green" title="E57">A4</font></span>: 
<font color="Maroon" title="c4">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">b</font>
 <span class="kw">and </span><br/><a NAME="E6:68"/><span class="lab"><font color="Green" title="E58">A5</font></span>: 
<font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:68"><span class="lab"><font color="Green" title="E54">A1</font></span></a>, <a class="ref" href="aff_1.html#D3" target="_self" title="AFF_1:def.3">Def3</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">c</font>, <font color="Maroon" title="c7">d</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AP</font><span class="kw"> such that </span><br/><a NAME="E8:68"/><span class="lab"><font color="Green" title="E59">A6</font></span>: 
<font color="Maroon" title="c6">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">d</font>
 <span class="kw">and </span><br/><a NAME="E9:68"/><span class="lab"><font color="Green" title="E60">A7</font></span>: 
<font color="Maroon" title="c3">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">d</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:68"><span class="lab"><font color="Green" title="E55">A2</font></span></a>, <a class="ref" href="aff_1.html#D3" target="_self" title="AFF_1:def.3">Def3</a></span>;<br/>
<a NAME="E10:68"/>
 not <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">d</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:68"><span class="lab"><font color="Green" title="E56">A3</font></span></a>, <a class="txt" href="#E5:68"><span class="lab"><font color="Green" title="E57">A4</font></span></a>, <a class="txt" href="#E6:68"><span class="lab"><font color="Green" title="E58">A5</font></span></a>, <a class="txt" href="#E8:68"><span class="lab"><font color="Green" title="E59">A6</font></span></a>, <a class="txt" href="#E9:68"><span class="lab"><font color="Green" title="E60">A7</font></span></a>, <a class="ref" href="aff_1.html#T37" target="_self" title="AFF_1:th.37">Th51</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AP</font><span class="kw"> such that </span><br/><a NAME="E12:68"/><span class="lab"><font color="Green" title="E61">A8</font></span>: 
<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c8">x</font>
 <span class="kw">and </span><br/><a NAME="E13:68"/><span class="lab"><font color="Green" title="E62">A9</font></span>: 
<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">d</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c8">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="diraf.html#T46" title="DIRAF:th.46">DIRAF:46</a></span>;<br/>
<a NAME="E14:68"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">d</font>,<font color="Maroon" title="c8">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:68"><span class="lab"><font color="Green" title="E62">A9</font></span></a>, <a class="ref" href="aff_1.html#D1" target="_self" title="AFF_1:def.1">Def1</a></span>;<br/>
<a NAME="E15:68"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A10</font></span>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:68"><span class="lab"><font color="Green" title="E60">A7</font></span></a>, <a class="ref" href="aff_1.html#D2" target="_self" title="AFF_1:def.2">Def2</a></span>;<br/>
<a NAME="E16:68"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c8">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:68"><span class="lab"><font color="Green" title="E61">A8</font></span></a>, <a class="ref" href="aff_1.html#D1" target="_self" title="AFF_1:def.1">Def1</a></span>;<br/>
<a NAME="E17:68"/><span class="kw">then </span>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:68"><span class="lab"><font color="Green" title="E58">A5</font></span></a>, <a class="ref" href="aff_1.html#D2" target="_self" title="AFF_1:def.2">Def2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E18:68"/>
 ex <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">AP</font> st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:68"><span class="lab"><font color="Green" title="E63">A10</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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