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<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">a</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="b2">Q</font>, <font color="Olive" title="b3">K</font>, <font color="Olive" title="b4">P</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">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">Q</font> &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="b3">K</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Olive" title="b2">Q</font> holds <br/><font color="Olive" title="b2">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/><span class="kw">let </span><font color="Maroon" title="c2">a</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">Q</font>, <font color="Olive" title="b2">K</font>, <font color="Olive" title="b3">P</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="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Q</font> &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="b2">K</font>,<font color="Olive" title="b3">P</font> &amp; <font color="Olive" title="b2">K</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Olive" title="b1">Q</font> holds <br/><font color="Olive" title="b1">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="b2">K</font>,<font color="Olive" title="b3">P</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">Q</font>, <font color="Maroon" title="c4">K</font>, <font color="Maroon" title="c5">P</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Q</font> &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="c4">K</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c3">Q</font> implies <font color="Maroon" title="c3">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/>









<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:23"/><span class="lab"><font color="Green" title="E19">A1</font></span>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Q</font>
 <span class="kw">and </span><br/><a NAME="E2:23"/><span class="lab"><font color="Green" title="E20">A2</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E3:23"/><span class="lab"><font color="Green" title="E21">A3</font></span>: 
<font color="Maroon" title="c4">K</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c3">Q</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="c3">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="23_1"><span class="kw">now </span></a><div class="add"><a NAME="E1:23_1"/><span class="lab"><font color="Green" title="E22">A4</font></span>: 
 <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> <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="c3">Q</font>,<font color="Maroon" title="c5">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:23"><span class="lab"><font color="Green" title="E21">A3</font></span></a>, <a class="ref" href="aff_4.html#T16" target="_self" title="AFF_4:th.16">Th16</a></span>;<br/><span class="kw">let </span><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <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="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Q</font> implies <font color="Maroon" title="c6">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/><span class="kw">assume </span><a NAME="E2:23_1"/><span class="lab"><font color="Green" title="E23">A5</font></span>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Q</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="c6">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/><span class="kw">reconsider </span><font color="Maroon" title="c7">c</font> = <font color="Maroon" title="c6">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:23_1"><span class="lab"><font color="Green" title="E23">A5</font></span></a></span>;<br/><a NAME="E4:23_1"/><span class="lab"><font color="Green" title="E24">A6</font></span>: 
<font color="Maroon" title="c3">Q</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:23"><span class="lab"><font color="Green" title="E21">A3</font></span></a>, <a class="ref" href="aff_1.html#T50" title="AFF_1:th.50">AFF_1:50</a></span>;<br/><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E6:23_1"/><span class="lab"><font color="Green" title="E25">A7</font></span>: 
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c8">b</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c4">K</font>
 <span class="kw">and </span><br/><a NAME="E7:23_1"/><span class="lab"><font color="Green" title="E26">A8</font></span>: 
<font color="Maroon" title="c8">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:23"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="txt" href="aff_4.html#E17"><span class="lab"><font color="Green" title="E14">Lm3</font></span></a></span>;<br/><a NAME="E8:23_1"/>
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c8">b</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c3">Q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:23"><span class="lab"><font color="Green" title="E21">A3</font></span></a>, <a class="txt" href="#E6:23_1"><span class="lab"><font color="Green" title="E25">A7</font></span></a>, <a class="ref" href="aff_1.html#T57" title="AFF_1:th.57">AFF_1:57</a></span>;<br/><a NAME="E9:23_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="txt" href="#E4:23_1"><span class="lab"><font color="Green" title="E24">A6</font></span></a>, <a class="ref" href="aff_1.html#T37" title="AFF_1:th.37">AFF_1:37</a></span>;<br/><a NAME="E10:23_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">b</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c3">Q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:23_1"><span class="lab"><font color="Green" title="E23">A5</font></span></a>, <a class="txt" href="#E4:23_1"><span class="lab"><font color="Green" title="E24">A6</font></span></a>, <a class="ref" href="aff_1.html#T37" title="AFF_1:th.37">AFF_1:37</a></span>;<br/><span class="kw">hence </span><a NAME="E11:23_1"/>
<font color="Maroon" title="c6">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E7:23_1"><span class="lab"><font color="Green" title="E26">A8</font></span></a>, <a class="txt" href="#E1:23_1"><span class="lab"><font color="Green" title="E22">A4</font></span></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>
<span class="kw">hence </span><a NAME="E5:23"/>
<font color="Maroon" title="c3">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 class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</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/>


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