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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">a</font>, <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font><br/> for <font color="Olive" title="b3">K</font>, <font color="Olive" title="b4">P</font>, <font color="Olive" title="b5">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="b3">K</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp; <font color="Olive" title="b4">P</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp; <font color="Olive" title="b5">Q</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp; <font color="Olive" title="b1">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="Olive" title="b3">K</font>,<font color="Olive" title="b4">P</font> &amp; <font color="Olive" title="b2">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="Olive" title="b3">K</font>,<font color="Olive" title="b4">P</font> &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; <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b5">Q</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b5">Q</font> holds <br/><font color="Olive" title="b5">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="b3">K</font>,<font color="Olive" title="b4">P</font></span><br/><b>let </b><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</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"> 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> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp; <font color="Olive" title="b2">P</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp; <font color="Olive" title="b3">Q</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp; <font color="Maroon" title="c2">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="Olive" title="b1">K</font>,<font color="Olive" title="b2">P</font> &amp; <font color="Maroon" title="c3">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="Olive" title="b1">K</font>,<font color="Olive" title="b2">P</font> &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; <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">Q</font> &amp; <font color="Maroon" title="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">Q</font> holds <br/><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></span><br/><b>let </b><font color="Maroon" title="c4">K</font>, <font color="Maroon" title="c5">P</font>, <font color="Maroon" title="c6">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="c4">K</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp; <font color="Maroon" title="c5">P</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp; <font color="Maroon" title="c6">Q</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp; <font color="Maroon" title="c2">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="c4">K</font>,<font color="Maroon" title="c5">P</font> &amp; <font color="Maroon" title="c3">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="c4">K</font>,<font color="Maroon" title="c5">P</font> &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; <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Q</font> &amp; <font color="Maroon" title="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Q</font> implies <font color="Maroon" title="c6">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="c4">K</font>,<font color="Maroon" title="c5">P</font> )</span><br/>











<b>assume </b><a NAME="E1:24"/><i><font color="Green" title="E20">A1</font></i>: 
( <font color="Maroon" title="c4">K</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp; <font color="Maroon" title="c5">P</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp; <font color="Maroon" title="c6">Q</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp; <font color="Maroon" title="c2">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="c4">K</font>,<font color="Maroon" title="c5">P</font> &amp; <font color="Maroon" title="c3">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="c4">K</font>,<font color="Maroon" title="c5">P</font> &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; <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Q</font> &amp; <font color="Maroon" title="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Q</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">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="c4">K</font>,<font color="Maroon" title="c5">P</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Q</font> implies <font color="Maroon" title="c7">x</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="c4">K</font>,<font color="Maroon" title="c5">P</font> )</span><br/><b>assume </b><a NAME="E1:24_1"/><i><font color="Green" title="E21">A2</font></i>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">x</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="c4">K</font>,<font color="Maroon" title="c5">P</font></span><br/><b>then reconsider </b><font color="Maroon" title="c8">c</font> = <font color="Maroon" title="c7">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">AS</font> ;<br/><b>consider </b><font color="Maroon" title="c9">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="E4:24_1"/><i><font color="Green" title="E22">A3</font></i>: 
( <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c9">a'</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c4">K</font> &amp; <font color="Maroon" title="c9">a'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font> )
 <b>by </b><i><a class="txt" href="#E1:24"><i><font color="Green" title="E20">A1</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="c10">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="E6:24_1"/><i><font color="Green" title="E23">A4</font></i>: 
( <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c10">b'</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c4">K</font> &amp; <font color="Maroon" title="c10">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font> )
 <b>by </b><i><a class="txt" href="#E1:24"><i><font color="Green" title="E20">A1</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="c11">X</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="E8:24_1"/><i><font color="Green" title="E24">A5</font></i>: 
( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">X</font> &amp; <font color="Maroon" title="c4">K</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c11">X</font> )
 <b>by </b><i><a class="txt" href="#E1:24"><i><font color="Green" title="E20">A1</font></i></a>, <a class="ref" href="aff_1.html#T63" title="AFF_1:th.63">AFF_1:63</a></i>;<br/><b>consider </b><font color="Maroon" title="c12">Y</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="E10:24_1"/><i><font color="Green" title="E25">A6</font></i>: 
( <font color="Maroon" title="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">Y</font> &amp; <font color="Maroon" title="c4">K</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c12">Y</font> )
 <b>by </b><i><a class="txt" href="#E1:24"><i><font color="Green" title="E20">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="E11:24_1"/><i><font color="Green" title="E26">A7</font></i>: 
<font color="Maroon" title="c11">X</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c12">Y</font>
 <b>by </b><i><a class="txt" href="#E8:24_1"><i><font color="Green" title="E24">A5</font></i></a>, <a class="txt" href="#E10:24_1"><i><font color="Green" title="E25">A6</font></i></a>, <a class="ref" href="aff_1.html#T58" title="AFF_1:th.58">AFF_1:58</a></i>;<br/><a NAME="E12:24_1"/>
( <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c9">a'</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c11">X</font> &amp; <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c10">b'</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c12">Y</font> )
 <b>by </b><i><a class="txt" href="#E4:24_1"><i><font color="Green" title="E22">A3</font></i></a>, <a class="txt" href="#E6:24_1"><i><font color="Green" title="E23">A4</font></i></a>, <a class="txt" href="#E8:24_1"><i><font color="Green" title="E24">A5</font></i></a>, <a class="txt" href="#E10:24_1"><i><font color="Green" title="E25">A6</font></i></a>, <a class="ref" href="aff_1.html#T57" title="AFF_1:th.57">AFF_1:57</a></i>;<br/><a NAME="E13:24_1"/><b>then </b><i><font color="Green" title="E27">A8</font></i>: 
( <font color="Maroon" title="c9">a'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">X</font> &amp; <font color="Maroon" title="c10">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">Y</font> )
 <b>by </b><i><a class="txt" href="#E8:24_1"><i><font color="Green" title="E24">A5</font></i></a>, <a class="txt" href="#E10:24_1"><i><font color="Green" title="E25">A6</font></i></a>, <a class="ref" href="aff_4.html#T2" target="_self" title="AFF_4:th.2">Th2</a></i>;<br/><a NAME="E14:24_1"/><i><font color="Green" title="E28">A9</font></i>: 
( <font color="Maroon" title="c11">X</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp; <font color="Maroon" title="c12">Y</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> )
 <b>by </b><i><a class="txt" href="#E4:24_1"><i><font color="Green" title="E22">A3</font></i></a>, <a class="txt" href="#E8:24_1"><i><font color="Green" title="E24">A5</font></i></a>, <a class="txt" href="#E10:24_1"><i><font color="Green" title="E25">A6</font></i></a>, <a class="ref" href="aff_1.html#T40" title="AFF_1:th.40">AFF_1:40</a>, <a class="ref" href="aff_1.html#T57" title="AFF_1:th.57">AFF_1:57</a></i>;<br/><div><i><font color="Green" title="E29">A10</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:24_1_1"/>
<font color="Maroon" title="c11">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">Y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">x</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="c4">K</font>,<font color="Maroon" title="c5">P</font></span><br/><a NAME="E2:24_1_1"/><b>then </b>
<font color="Maroon" title="c6">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">X</font>
 <b>by </b><i><a class="txt" href="#E1:24"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="#E8:24_1"><i><font color="Green" title="E24">A5</font></i></a>, <a class="txt" href="#E10:24_1"><i><font color="Green" title="E25">A6</font></i></a>, <a class="txt" href="#E14:24_1"><i><font color="Green" title="E28">A9</font></i></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></i>;<br/><a NAME="E3:24_1_1"/><b>then </b>
<font color="Maroon" title="c6">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="c4">K</font>,<font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="#E1:24"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="#E8:24_1"><i><font color="Green" title="E24">A5</font></i></a>, <a class="txt" href="aff_4.html#E24"><i><font color="Green" title="E18">Lm4</font></i></a></i>;<br/><b>hence </b><a NAME="E4:24_1_1"/>
<font color="Maroon" title="c7">x</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="c4">K</font>,<font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="#E1:24_1"><i><font color="Green" title="E21">A2</font></i></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><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:24_1_2"/>
<font color="Maroon" title="c11">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">Y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">x</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="c4">K</font>,<font color="Maroon" title="c5">P</font></span><br/><a NAME="E2:24_1_2"/><b>then </b><i><font color="Green" title="E30">A11</font></i>: 
( <font color="Maroon" title="c5">P</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c6">Q</font> or  ex <font color="Olive" title="b1">q</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font> st <br/>( <font color="Olive" title="b1">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font> &amp; <font color="Olive" title="b1">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Q</font> ) )
 <b>by </b><i><a class="txt" href="#E1:24"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="#E4:24_1"><i><font color="Green" title="E22">A3</font></i></a>, <a class="txt" href="#E6:24_1"><i><font color="Green" title="E23">A4</font></i></a>, <a class="txt" href="#E8:24_1"><i><font color="Green" title="E24">A5</font></i></a>, <a class="txt" href="#E10:24_1"><i><font color="Green" title="E25">A6</font></i></a>, <a class="txt" href="#E11:24_1"><i><font color="Green" title="E26">A7</font></i></a>, <a class="txt" href="#E13:24_1"><i><font color="Green" title="E27">A8</font></i></a>, <a class="ref" href="aff_4.html#T18" target="_self" title="AFF_4:th.18">Th18</a></i>;<br/><div><i><font color="Green" title="E31">A12</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:24_1_2_1"/><i><font color="Green" title="E32">A13</font></i>: 
<font color="Maroon" title="c5">P</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c6">Q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">x</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="c4">K</font>,<font color="Maroon" title="c5">P</font></span><br/><div><i><font color="Green" title="E33">A14</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:24_1_2_1_1"/><i><font color="Green" title="E34">A15</font></i>: 
<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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">x</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="c4">K</font>,<font color="Maroon" title="c5">P</font></span><br/><a NAME="E2:24_1_2_1_1"/>
<font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c8">c</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c4">K</font>
 <b>by </b><i><a class="txt" href="#E1:24"><i><font color="Green" title="E20">A1</font></i></a>, <a class="ref" href="aff_1.html#T47" title="AFF_1:th.47">AFF_1:47</a></i>;<br/><b>hence </b><a NAME="E3:24_1_2_1_1"/>
<font color="Maroon" title="c7">x</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="c4">K</font>,<font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="#E1:24_1"><i><font color="Green" title="E21">A2</font></i></a>, <a class="txt" href="#E1:24_1_2_1_1"><i><font color="Green" title="E34">A15</font></i></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><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_2_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:24_1_2_1_2"/>
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">x</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="c4">K</font>,<font color="Maroon" title="c5">P</font></span><br/><a NAME="E2:24_1_2_1_2"/><b>then </b><i><font color="Green" title="E34">A16</font></i>: 
<font color="Maroon" title="c3">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">b'</font>
 <b>by </b><i><a class="txt" href="#E1:24"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="#E6:24_1"><i><font color="Green" title="E23">A4</font></i></a>, <a class="txt" href="#E1:24_1_2_1"><i><font color="Green" title="E32">A13</font></i></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_2_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:24_1_2_1_2_1"/><i><font color="Green" title="E35">A17</font></i>: 
<font color="Maroon" title="c8">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">x</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="c4">K</font>,<font color="Maroon" title="c5">P</font></span><br/><b>consider </b><font color="Maroon" title="c13">c'</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="E3:24_1_2_1_2_1"/><i><font color="Green" title="E36">A18</font></i>: 
( <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c8">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">b'</font>,<font color="Maroon" title="c13">c'</font> &amp; <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c10">b'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c13">c'</font> )
 <b>by </b><i><a class="ref" href="diraf.html#T47" title="DIRAF:th.47">DIRAF:47</a></i>;<br/><a NAME="E4:24_1_2_1_2_1"/>
<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c8">c</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c6">Q</font>
 <b>by </b><i><a class="txt" href="#E1:24"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="#E1:24_1"><i><font color="Green" title="E21">A2</font></i></a>, <a class="ref" href="aff_1.html#T37" title="AFF_1:th.37">AFF_1:37</a></i>;<br/><a NAME="E5:24_1_2_1_2_1"/><b>then </b>
<font color="Maroon" title="c10">b'</font>,<font color="Maroon" title="c13">c'</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c6">Q</font>
 <b>by </b><i><a class="txt" href="#E1:24_1_2_1_2_1"><i><font color="Green" title="E35">A17</font></i></a>, <a class="txt" href="#E3:24_1_2_1_2_1"><i><font color="Green" title="E36">A18</font></i></a>, <a class="ref" href="aff_1.html#T46" title="AFF_1:th.46">AFF_1:46</a></i>;<br/><a NAME="E6:24_1_2_1_2_1"/><b>then </b>
<font color="Maroon" title="c10">b'</font>,<font color="Maroon" title="c13">c'</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="#E1:24_1_2_1"><i><font color="Green" title="E32">A13</font></i></a>, <a class="ref" href="aff_1.html#T57" title="AFF_1:th.57">AFF_1:57</a></i>;<br/><a NAME="E7:24_1_2_1_2_1"/><b>then </b><i><font color="Green" title="E37">A19</font></i>: 
<font color="Maroon" title="c13">c'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="#E6:24_1"><i><font color="Green" title="E23">A4</font></i></a>, <a class="ref" href="aff_4.html#T2" target="_self" title="AFF_4:th.2">Th2</a></i>;<br/><a NAME="E8:24_1_2_1_2_1"/>
<font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c13">c'</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c4">K</font>
 <b>by </b><i><a class="txt" href="#E6:24_1"><i><font color="Green" title="E23">A4</font></i></a>, <a class="txt" href="#E2:24_1_2_1_2"><i><font color="Green" title="E34">A16</font></i></a>, <a class="txt" href="#E3:24_1_2_1_2_1"><i><font color="Green" title="E36">A18</font></i></a>, <a class="ref" href="aff_1.html#T46" title="AFF_1:th.46">AFF_1:46</a></i>;<br/><b>hence </b><a NAME="E9:24_1_2_1_2_1"/>
<font color="Maroon" title="c7">x</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="c4">K</font>,<font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="#E7:24_1_2_1_2_1"><i><font color="Green" title="E37">A19</font></i></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><b>hence </b><a NAME="E4:24_1_2_1_2"/>
<font color="Maroon" title="c7">x</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="c4">K</font>,<font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="#E1:24"><i><font color="Green" title="E20">A1</font></i></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><b>hence </b><a NAME="E4:24_1_2_1"/>
<font color="Maroon" title="c7">x</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="c4">K</font>,<font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="#E2:24_1_2_1"><i><font color="Green" title="E33">A14</font></i></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><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_2_2"><b>now </b></a><div class="add"><b>given </b><font color="Maroon" title="c13">q</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><a NAME="E2:24_1_2_2"/><i><font color="Green" title="E32">A20</font></i>: 
( <font color="Maroon" title="c13">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font> &amp; <font color="Maroon" title="c13">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Q</font> )
 <b>and </b><br/><a NAME="E3:24_1_2_2"/><i><font color="Green" title="E33">A21</font></i>: 
not <font color="Maroon" title="c5">P</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c6">Q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">x</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="c4">K</font>,<font color="Maroon" title="c5">P</font></span><br/><a NAME="E4:24_1_2_2"/><i><font color="Green" title="E34">A22</font></i>: 
<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>
 <b>by </b><i><a class="txt" href="#E1:24"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="#E3:24_1_2_2"><i><font color="Green" title="E33">A21</font></i></a>, <a class="ref" href="aff_1.html#T55" title="AFF_1:th.55">AFF_1:55</a></i>;<br/><div><i><font color="Green" title="E35">A23</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_2_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:24_1_2_2_1"/><i><font color="Green" title="E36">A24</font></i>: 
<font color="Maroon" title="c13">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">a</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">x</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="c4">K</font>,<font color="Maroon" title="c5">P</font></span><br/><a NAME="E2:24_1_2_2_1"/><b>then </b><i><font color="Green" title="E37">A25</font></i>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">a'</font>
 <b>by </b><i><a class="txt" href="#E1:24"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="#E4:24_1"><i><font color="Green" title="E22">A3</font></i></a>, <a class="txt" href="#E2:24_1_2_2"><i><font color="Green" title="E32">A20</font></i></a>, <a class="txt" href="#E4:24_1_2_2"><i><font color="Green" title="E34">A22</font></i></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></i>;<br/><div><i><font color="Green" title="E38">A26</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_2_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:24_1_2_2_1_1"/><i><font color="Green" title="E39">A27</font></i>: 
<font color="Maroon" title="c13">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">a'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">x</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="c4">K</font>,<font color="Maroon" title="c5">P</font></span><br/><a NAME="E2:24_1_2_2_1_1"/>
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c13">q</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c6">Q</font>
 <b>by </b><i><a class="txt" href="#E1:24"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="#E2:24_1_2_2"><i><font color="Green" title="E32">A20</font></i></a>, <a class="ref" href="aff_1.html#T37" title="AFF_1:th.37">AFF_1:37</a></i>;<br/><a NAME="E3:24_1_2_2_1_1"/><b>then </b>
<font color="Maroon" title="c6">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="c4">K</font>,<font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="#E1:24"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="#E4:24_1"><i><font color="Green" title="E22">A3</font></i></a>, <a class="txt" href="#E1:24_1_2_2_1"><i><font color="Green" title="E36">A24</font></i></a>, <a class="txt" href="#E1:24_1_2_2_1_1"><i><font color="Green" title="E39">A27</font></i></a>, <a class="txt" href="aff_4.html#E24"><i><font color="Green" title="E18">Lm4</font></i></a>, <a class="ref" href="aff_1.html#T67" title="AFF_1:th.67">AFF_1:67</a></i>;<br/><b>hence </b><a NAME="E4:24_1_2_2_1_1"/>
<font color="Maroon" title="c7">x</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="c4">K</font>,<font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="#E1:24_1"><i><font color="Green" title="E21">A2</font></i></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><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_2_2_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:24_1_2_2_1_2"/><i><font color="Green" title="E39">A28</font></i>: 
<font color="Maroon" title="c13">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">a'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">x</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="c4">K</font>,<font color="Maroon" title="c5">P</font></span><br/><a NAME="E2:24_1_2_2_1_2"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c13">q</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c8">c</font>
 <b>by </b><i><a class="txt" href="#E1:24"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="#E1:24_1"><i><font color="Green" title="E21">A2</font></i></a>, <a class="txt" href="#E2:24_1_2_2"><i><font color="Green" title="E32">A20</font></i></a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></i>;<br/><b>then consider </b><font color="Maroon" title="c14">c'</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="E4:24_1_2_2_1_2"/><i><font color="Green" title="E40">A29</font></i>: 
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c13">q</font>,<font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c14">c'</font> &amp; <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c9">a'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c14">c'</font> )
 <b>by </b><i><a class="txt" href="#E1:24_1_2_2_1"><i><font color="Green" title="E36">A24</font></i></a>, <a class="ref" href="aff_4.html#T1" target="_self" title="AFF_4:th.1">Th1</a></i>;<br/><a NAME="E5:24_1_2_2_1_2"/><i><font color="Green" title="E41">A30</font></i>: 
<font color="Maroon" title="c13">q</font>,<font color="Maroon" title="c9">a'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c13">q</font>,<font color="Maroon" title="c14">c'</font>
 <b>by </b><i><a class="txt" href="#E4:24_1_2_2_1_2"><i><font color="Green" title="E40">A29</font></i></a>, <a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></i>;<br/><a NAME="E6:24_1_2_2_1_2"/>
<font color="Maroon" title="c13">q</font>,<font color="Maroon" title="c9">a'</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="#E1:24"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="#E4:24_1"><i><font color="Green" title="E22">A3</font></i></a>, <a class="txt" href="#E2:24_1_2_2"><i><font color="Green" title="E32">A20</font></i></a>, <a class="ref" href="aff_1.html#T37" title="AFF_1:th.37">AFF_1:37</a></i>;<br/><a NAME="E7:24_1_2_2_1_2"/><b>then </b>
<font color="Maroon" title="c13">q</font>,<font color="Maroon" title="c14">c'</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="#E1:24_1_2_2_1_2"><i><font color="Green" title="E39">A28</font></i></a>, <a class="txt" href="#E5:24_1_2_2_1_2"><i><font color="Green" title="E41">A30</font></i></a>, <a class="ref" href="aff_1.html#T46" title="AFF_1:th.46">AFF_1:46</a></i>;<br/><a NAME="E8:24_1_2_2_1_2"/><b>then </b><i><font color="Green" title="E42">A31</font></i>: 
<font color="Maroon" title="c14">c'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="#E2:24_1_2_2"><i><font color="Green" title="E32">A20</font></i></a>, <a class="ref" href="aff_4.html#T2" target="_self" title="AFF_4:th.2">Th2</a></i>;<br/><a NAME="E9:24_1_2_2_1_2"/>
<font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c14">c'</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c4">K</font>
 <b>by </b><i><a class="txt" href="#E4:24_1"><i><font color="Green" title="E22">A3</font></i></a>, <a class="txt" href="#E2:24_1_2_2_1"><i><font color="Green" title="E37">A25</font></i></a>, <a class="txt" href="#E4:24_1_2_2_1_2"><i><font color="Green" title="E40">A29</font></i></a>, <a class="ref" href="aff_1.html#T46" title="AFF_1:th.46">AFF_1:46</a></i>;<br/><b>hence </b><a NAME="E10:24_1_2_2_1_2"/>
<font color="Maroon" title="c7">x</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="c4">K</font>,<font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="#E8:24_1_2_2_1_2"><i><font color="Green" title="E42">A31</font></i></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><b>hence </b><a NAME="E5:24_1_2_2_1"/>
<font color="Maroon" title="c7">x</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="c4">K</font>,<font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="#E3:24_1_2_2_1"><i><font color="Green" title="E38">A26</font></i></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><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_2_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:24_1_2_2_2"/><i><font color="Green" title="E36">A32</font></i>: 
<font color="Maroon" title="c13">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">x</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="c4">K</font>,<font color="Maroon" title="c5">P</font></span><br/><a NAME="E2:24_1_2_2_2"/><b>then </b><i><font color="Green" title="E37">A33</font></i>: 
<font color="Maroon" title="c3">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">b'</font>
 <b>by </b><i><a class="txt" href="#E1:24"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="#E6:24_1"><i><font color="Green" title="E23">A4</font></i></a>, <a class="txt" href="#E2:24_1_2_2"><i><font color="Green" title="E32">A20</font></i></a>, <a class="txt" href="#E4:24_1_2_2"><i><font color="Green" title="E34">A22</font></i></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></i>;<br/><div><i><font color="Green" title="E38">A34</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_2_2_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:24_1_2_2_2_1"/><i><font color="Green" title="E39">A35</font></i>: 
<font color="Maroon" title="c13">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">x</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="c4">K</font>,<font color="Maroon" title="c5">P</font></span><br/><a NAME="E2:24_1_2_2_2_1"/>
<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c13">q</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c6">Q</font>
 <b>by </b><i><a class="txt" href="#E1:24"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="#E2:24_1_2_2"><i><font color="Green" title="E32">A20</font></i></a>, <a class="ref" href="aff_1.html#T37" title="AFF_1:th.37">AFF_1:37</a></i>;<br/><a NAME="E3:24_1_2_2_2_1"/><b>then </b>
<font color="Maroon" title="c6">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="c4">K</font>,<font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="#E1:24"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="#E6:24_1"><i><font color="Green" title="E23">A4</font></i></a>, <a class="txt" href="#E1:24_1_2_2_2"><i><font color="Green" title="E36">A32</font></i></a>, <a class="txt" href="#E1:24_1_2_2_2_1"><i><font color="Green" title="E39">A35</font></i></a>, <a class="txt" href="aff_4.html#E24"><i><font color="Green" title="E18">Lm4</font></i></a>, <a class="ref" href="aff_1.html#T67" title="AFF_1:th.67">AFF_1:67</a></i>;<br/><b>hence </b><a NAME="E4:24_1_2_2_2_1"/>
<font color="Maroon" title="c7">x</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="c4">K</font>,<font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="#E1:24_1"><i><font color="Green" title="E21">A2</font></i></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><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_2_2_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:24_1_2_2_2_2"/><i><font color="Green" title="E39">A36</font></i>: 
<font color="Maroon" title="c13">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">b'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">x</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="c4">K</font>,<font color="Maroon" title="c5">P</font></span><br/><a NAME="E2:24_1_2_2_2_2"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c13">q</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c8">c</font>
 <b>by </b><i><a class="txt" href="#E1:24"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="#E1:24_1"><i><font color="Green" title="E21">A2</font></i></a>, <a class="txt" href="#E2:24_1_2_2"><i><font color="Green" title="E32">A20</font></i></a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></i>;<br/><b>then consider </b><font color="Maroon" title="c14">c'</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="E4:24_1_2_2_2_2"/><i><font color="Green" title="E40">A37</font></i>: 
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c13">q</font>,<font color="Maroon" title="c10">b'</font>,<font color="Maroon" title="c14">c'</font> &amp; <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c10">b'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c14">c'</font> )
 <b>by </b><i><a class="txt" href="#E1:24_1_2_2_2"><i><font color="Green" title="E36">A32</font></i></a>, <a class="ref" href="aff_4.html#T1" target="_self" title="AFF_4:th.1">Th1</a></i>;<br/><a NAME="E5:24_1_2_2_2_2"/><i><font color="Green" title="E41">A38</font></i>: 
<font color="Maroon" title="c13">q</font>,<font color="Maroon" title="c10">b'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c13">q</font>,<font color="Maroon" title="c14">c'</font>
 <b>by </b><i><a class="txt" href="#E4:24_1_2_2_2_2"><i><font color="Green" title="E40">A37</font></i></a>, <a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></i>;<br/><a NAME="E6:24_1_2_2_2_2"/>
<font color="Maroon" title="c13">q</font>,<font color="Maroon" title="c10">b'</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="#E1:24"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="#E6:24_1"><i><font color="Green" title="E23">A4</font></i></a>, <a class="txt" href="#E2:24_1_2_2"><i><font color="Green" title="E32">A20</font></i></a>, <a class="ref" href="aff_1.html#T37" title="AFF_1:th.37">AFF_1:37</a></i>;<br/><a NAME="E7:24_1_2_2_2_2"/><b>then </b>
<font color="Maroon" title="c13">q</font>,<font color="Maroon" title="c14">c'</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="#E1:24_1_2_2_2_2"><i><font color="Green" title="E39">A36</font></i></a>, <a class="txt" href="#E5:24_1_2_2_2_2"><i><font color="Green" title="E41">A38</font></i></a>, <a class="ref" href="aff_1.html#T46" title="AFF_1:th.46">AFF_1:46</a></i>;<br/><a NAME="E8:24_1_2_2_2_2"/><b>then </b><i><font color="Green" title="E42">A39</font></i>: 
<font color="Maroon" title="c14">c'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="#E2:24_1_2_2"><i><font color="Green" title="E32">A20</font></i></a>, <a class="ref" href="aff_4.html#T2" target="_self" title="AFF_4:th.2">Th2</a></i>;<br/><a NAME="E9:24_1_2_2_2_2"/>
<font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c14">c'</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c4">K</font>
 <b>by </b><i><a class="txt" href="#E6:24_1"><i><font color="Green" title="E23">A4</font></i></a>, <a class="txt" href="#E2:24_1_2_2_2"><i><font color="Green" title="E37">A33</font></i></a>, <a class="txt" href="#E4:24_1_2_2_2_2"><i><font color="Green" title="E40">A37</font></i></a>, <a class="ref" href="aff_1.html#T46" title="AFF_1:th.46">AFF_1:46</a></i>;<br/><b>hence </b><a NAME="E10:24_1_2_2_2_2"/>
<font color="Maroon" title="c7">x</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="c4">K</font>,<font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="#E8:24_1_2_2_2_2"><i><font color="Green" title="E42">A39</font></i></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><b>hence </b><a NAME="E5:24_1_2_2_2"/>
<font color="Maroon" title="c7">x</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="c4">K</font>,<font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="#E3:24_1_2_2_2"><i><font color="Green" title="E38">A34</font></i></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><b>hence </b><a NAME="E7:24_1_2_2"/>
<font color="Maroon" title="c7">x</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="c4">K</font>,<font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="#E1:24"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="#E5:24_1_2_2"><i><font color="Green" title="E35">A23</font></i></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><b>hence </b><a NAME="E5:24_1_2"/>
<font color="Maroon" title="c7">x</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="c4">K</font>,<font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="#E2:24_1_2"><i><font color="Green" title="E30">A11</font></i></a>, <a class="txt" href="#E3:24_1_2"><i><font color="Green" title="E31">A12</font></i></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><b>hence </b><a NAME="E17:24_1"/>
<font color="Maroon" title="c7">x</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="c4">K</font>,<font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="#E15:24_1"><i><font color="Green" title="E29">A10</font></i></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>
<b>hence </b><a NAME="E3:24"/>
<font color="Maroon" title="c6">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="c4">K</font>,<font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</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>
