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<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp; <font color="Olive" title="b2">C</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp; not <font color="Olive" title="b1">A</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</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/><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">A</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp; <font color="Maroon" title="c3">C</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp; not <font color="Maroon" title="c2">A</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</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/>





<b>assume </b><a NAME="E1:67"/><i><font color="Green" title="E54">A1</font></i>: 
( <font color="Maroon" title="c2">A</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp; <font color="Maroon" title="c3">C</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp; not <font color="Maroon" title="c2">A</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c3">C</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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/>

<b>then consider </b><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><b> such that </b><br/><a NAME="E3:67"/><i><font color="Green" title="E55">A2</font></i>: 
( <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> &amp; <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> )
 <b>by </b><i><a class="ref" href="aff_1.html#D3" target="_self" title="AFF_1:def.3">Def3</a></i>;<br/>
<b>consider </b><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><b> such that </b><br/><a NAME="E5:67"/><i><font color="Green" title="E56">A3</font></i>: 
( <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> &amp; <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> )
 <b>by </b><i><a class="txt" href="aff_1.html#E1:67"><i><font color="Green" title="E54">A1</font></i></a>, <a class="ref" href="aff_1.html#D3" target="_self" title="AFF_1:def.3">Def3</a></i>;<br/>
<a NAME="E6:67"/>
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>
 
<b>by </b><i><a class="txt" href="aff_1.html#E1:67"><i><font color="Green" title="E54">A1</font></i></a>, <a class="txt" href="aff_1.html#E3:67"><i><font color="Green" title="E55">A2</font></i></a>, <a class="txt" href="aff_1.html#E5:67"><i><font color="Green" title="E56">A3</font></i></a>, <a class="ref" href="aff_1.html#T51" target="_self" title="AFF_1:th.51">Th51</a></i>;<br/>
<b>then consider </b><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><b> such that </b><br/><a NAME="E8:67"/><i><font color="Green" title="E57">A4</font></i>: 
( <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> &amp; <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> )
 <b>by </b><i><a class="ref" href="diraf.html#T54" title="DIRAF:th.54">DIRAF:54</a></i>;<br/>
<a NAME="E9:67"/>
(  <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> &amp;  <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> )
 
<b>by </b><i><a class="txt" href="aff_1.html#E8:67"><i><font color="Green" title="E57">A4</font></i></a>, <a class="ref" href="aff_1.html#D1" target="_self" title="AFF_1:def.1">Def1</a></i>;<br/>
<a NAME="E10:67"/><b>then </b>
( <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> &amp; <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> )
 
<b>by </b><i><a class="txt" href="aff_1.html#E3:67"><i><font color="Green" title="E55">A2</font></i></a>, <a class="txt" href="aff_1.html#E5:67"><i><font color="Green" title="E56">A3</font></i></a>, <a class="ref" href="aff_1.html#D2" target="_self" title="AFF_1:def.2">Def2</a></i>;<br/>
<b>hence </b><a NAME="E11:67"/>
 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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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