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<b>let </b><font color="Maroon" title="c1">AS</font> be   <a href="diraf.html#NM1" title="DIRAF:NM.1">AffinSpace</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">K</font>, <font color="Olive" title="b2">P</font>, <font color="Olive" title="b3">Q</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AS</font>  st <font color="Olive" title="b1">K</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp; <font color="Olive" title="b2">P</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp; <font color="Olive" title="b3">Q</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp; not <font color="Olive" title="b1">K</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Olive" title="b2">P</font> &amp; not <font color="Olive" title="b1">K</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Olive" title="b3">Q</font> &amp; <font color="Olive" title="b3">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="aff_4.html#K1" title="AFF_4:func.1">Plane</a> <font color="Olive" title="b1">K</font>,<font color="Olive" title="b2">P</font> holds <br/> <a href="aff_4.html#K1" title="AFF_4:func.1">Plane</a> <font color="Olive" title="b1">K</font>,<font color="Olive" title="b3">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="aff_4.html#K1" title="AFF_4:func.1">Plane</a> <font color="Olive" title="b1">K</font>,<font color="Olive" title="b2">P</font></span><br/><b>let </b><font color="Maroon" title="c2">K</font>, <font color="Maroon" title="c3">P</font>, <font color="Maroon" title="c4">Q</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AS</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">K</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp; <font color="Maroon" title="c3">P</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp; <font color="Maroon" title="c4">Q</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp; not <font color="Maroon" title="c2">K</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c3">P</font> &amp; not <font color="Maroon" title="c2">K</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c4">Q</font> &amp; <font color="Maroon" title="c4">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="aff_4.html#K1" title="AFF_4:func.1">Plane</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c3">P</font> implies  <a href="aff_4.html#K1" title="AFF_4:func.1">Plane</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c4">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="aff_4.html#K1" title="AFF_4:func.1">Plane</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c3">P</font> )</span><br/>







<b>assume </b><b>that </b><br/><a NAME="E1:28"/><i><font color="Green" title="E23">A1</font></i>: 
( <font color="Maroon" title="c2">K</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp; <font color="Maroon" title="c3">P</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp; <font color="Maroon" title="c4">Q</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  )
 <b>and </b><br/><a NAME="E2:28"/><i><font color="Green" title="E24">A2</font></i>: 
( not <font color="Maroon" title="c2">K</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c3">P</font> &amp; not <font color="Maroon" title="c2">K</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c4">Q</font> )
 <b>and </b><br/><a NAME="E3:28"/><i><font color="Green" title="E25">A3</font></i>: 
<font color="Maroon" title="c4">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="aff_4.html#K1" title="AFF_4:func.1">Plane</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c3">P</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="aff_4.html#K1" title="AFF_4:func.1">Plane</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c4">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="aff_4.html#K1" title="AFF_4:func.1">Plane</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c3">P</font></span><br/>

<a NAME="E4:28"/><i><font color="Green" title="E26">A4</font></i>: 
 <a href="aff_4.html#K1" title="AFF_4:func.1">Plane</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c4">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="aff_4.html#K1" title="AFF_4:func.1">Plane</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c3">P</font>
 
<b>by </b><i><a class="txt" href="aff_4.html#E1:28"><i><font color="Green" title="E23">A1</font></i></a>, <a class="txt" href="aff_4.html#E3:28"><i><font color="Green" title="E25">A3</font></i></a>, <a class="txt" href="aff_4.html#E27"><i><font color="Green" title="E21">Lm6</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c6">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font><b> such that </b><br/><a NAME="E6:28"/><i><font color="Green" title="E27">A5</font></i>: 
( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Q</font> &amp; <font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Q</font> )
 <b>and </b><br/><a NAME="E7:28"/><i><font color="Green" title="E28">A6</font></i>: 
<font color="Maroon" title="c5">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b</font>
 <b>by </b><i><a class="txt" href="aff_4.html#E1:28"><i><font color="Green" title="E23">A1</font></i></a>, <a class="ref" href="aff_1.html#T31" title="AFF_1:th.31">AFF_1:31</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">a'</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font><b> such that </b><br/><a NAME="E9:28"/><i><font color="Green" title="E29">A7</font></i>: 
<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">a'</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c2">K</font>
 <b>and </b><br/><a NAME="E10:28"/><i><font color="Green" title="E30">A8</font></i>: 
<font color="Maroon" title="c7">a'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font>
 <b>by </b><i><a class="txt" href="aff_4.html#E3:28"><i><font color="Green" title="E25">A3</font></i></a>, <a class="txt" href="aff_4.html#E6:28"><i><font color="Green" title="E27">A5</font></i></a>, <a class="txt" href="aff_4.html#E17"><i><font color="Green" title="E14">Lm3</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">b'</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font><b> such that </b><br/><a NAME="E12:28"/><i><font color="Green" title="E31">A9</font></i>: 
<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c8">b'</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c2">K</font>
 <b>and </b><br/><a NAME="E13:28"/><i><font color="Green" title="E32">A10</font></i>: 
<font color="Maroon" title="c8">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font>
 <b>by </b><i><a class="txt" href="aff_4.html#E3:28"><i><font color="Green" title="E25">A3</font></i></a>, <a class="txt" href="aff_4.html#E6:28"><i><font color="Green" title="E27">A5</font></i></a>, <a class="txt" href="aff_4.html#E17"><i><font color="Green" title="E14">Lm3</font></i></a></i>;<br/>
<a NAME="E14:28"/><i><font color="Green" title="E33">A11</font></i>: 
<font color="Maroon" title="c7">a'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">b'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:28_1"/><i><font color="Green" title="E34">A12</font></i>: 
<font color="Maroon" title="c7">a'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<b>consider </b><font color="Maroon" title="c9">A</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AS</font><b> such that </b><br/><a NAME="E3:28_1"/><i><font color="Green" title="E35">A13</font></i>: 
<font color="Maroon" title="c7">a'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">A</font>
 <b>and </b><br/><a NAME="E4:28_1"/><i><font color="Green" title="E36">A14</font></i>: 
<font color="Maroon" title="c2">K</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c9">A</font>
 <b>by </b><i><a class="txt" href="aff_4.html#E1:28"><i><font color="Green" title="E23">A1</font></i></a>, <a class="ref" href="aff_1.html#T63" title="AFF_1:th.63">AFF_1:63</a></i>;<br/>
<a NAME="E5:28_1"/><i><font color="Green" title="E37">A15</font></i>: 
<font color="Maroon" title="c9">A</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a> 
 
<b>by </b><i><a class="txt" href="aff_4.html#E4:28_1"><i><font color="Green" title="E36">A14</font></i></a>, <a class="ref" href="aff_1.html#T50" title="AFF_1:th.50">AFF_1:50</a></i>;<br/>
<a NAME="E6:28_1"/>
( <font color="Maroon" title="c7">a'</font>,<font color="Maroon" title="c6">b</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c9">A</font> &amp; <font color="Maroon" title="c7">a'</font>,<font color="Maroon" title="c5">a</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c9">A</font> )
 
<b>by </b><i><a class="txt" href="aff_4.html#E9:28"><i><font color="Green" title="E29">A7</font></i></a>, <a class="txt" href="aff_4.html#E12:28"><i><font color="Green" title="E31">A9</font></i></a>, <a class="txt" href="aff_4.html#E1:28_1"><i><font color="Green" title="E34">A12</font></i></a>, <a class="txt" href="aff_4.html#E4:28_1"><i><font color="Green" title="E36">A14</font></i></a>, <a class="ref" href="aff_4.html#T3" target="_self" title="AFF_4:th.3">Th3</a></i>;<br/>
<a NAME="E7:28_1"/><b>then </b>
( <font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">A</font> &amp; <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">A</font> )
 
<b>by </b><i><a class="txt" href="aff_4.html#E3:28_1"><i><font color="Green" title="E35">A13</font></i></a>, <a class="ref" href="aff_4.html#T2" target="_self" title="AFF_4:th.2">Th2</a></i>;<br/>
<b>hence </b><a NAME="E8:28_1"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_4.html#E1:28"><i><font color="Green" title="E23">A1</font></i></a>, <a class="txt" href="aff_4.html#E2:28"><i><font color="Green" title="E24">A2</font></i></a>, <a class="txt" href="aff_4.html#E6:28"><i><font color="Green" title="E27">A5</font></i></a>, <a class="txt" href="aff_4.html#E7:28"><i><font color="Green" title="E28">A6</font></i></a>, <a class="txt" href="aff_4.html#E4:28_1"><i><font color="Green" title="E36">A14</font></i></a>, <a class="txt" href="aff_4.html#E5:28_1"><i><font color="Green" title="E37">A15</font></i></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E15:28"/>
( <font color="Maroon" title="c7">a'</font>,<font color="Maroon" title="c5">a</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c2">K</font> &amp; <font color="Maroon" title="c8">b'</font>,<font color="Maroon" title="c6">b</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c2">K</font> )
 
<b>by </b><i><a class="txt" href="aff_4.html#E9:28"><i><font color="Green" title="E29">A7</font></i></a>, <a class="txt" href="aff_4.html#E12:28"><i><font color="Green" title="E31">A9</font></i></a>, <a class="ref" href="aff_1.html#T48" title="AFF_1:th.48">AFF_1:48</a></i>;<br/>
<a NAME="E16:28"/><b>then </b>
( <font color="Maroon" title="c7">a'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_4.html#K1" title="AFF_4:func.1">Plane</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c4">Q</font> &amp; <font color="Maroon" title="c8">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_4.html#K1" title="AFF_4:func.1">Plane</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c4">Q</font> )
 
<b>by </b><i><a class="txt" href="aff_4.html#E6:28"><i><font color="Green" title="E27">A5</font></i></a></i>;<br/>
<a NAME="E17:28"/><b>then </b>
 <a href="aff_4.html#K1" title="AFF_4:func.1">Plane</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c3">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="aff_4.html#K1" title="AFF_4:func.1">Plane</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c4">Q</font>
 
<b>by </b><i><a class="txt" href="aff_4.html#E1:28"><i><font color="Green" title="E23">A1</font></i></a>, <a class="txt" href="aff_4.html#E10:28"><i><font color="Green" title="E30">A8</font></i></a>, <a class="txt" href="aff_4.html#E13:28"><i><font color="Green" title="E32">A10</font></i></a>, <a class="txt" href="aff_4.html#E14:28"><i><font color="Green" title="E33">A11</font></i></a>, <a class="txt" href="aff_4.html#E25"><i><font color="Green" title="E19">Lm5</font></i></a>, <a class="txt" href="aff_4.html#E27"><i><font color="Green" title="E21">Lm6</font></i></a></i>;<br/>
<b>hence </b><a NAME="E18:28"/>
 <a href="aff_4.html#K1" title="AFF_4:func.1">Plane</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c4">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="aff_4.html#K1" title="AFF_4:func.1">Plane</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c3">P</font>
 <b>by </b><i><a class="txt" href="aff_4.html#E4:28"><i><font color="Green" title="E26">A4</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
