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<span class="kw">let </span><font color="Maroon" title="c1">S</font> be   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="gtarski1.html#V9" title="GTARSKI1:attr.9">satisfying_Tarski-model</a>   <a href="gtarski1.html#L1" title="GTARSKI1:struct.1">TarskiGeometryStruct</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">z</font>, <font color="Olive" title="b4">a</font>, <font color="Olive" title="b5">b</font>, <font color="Olive" title="b6">c</font>, <font color="Olive" title="b7">c9</font>, <font color="Olive" title="b8">p</font>, <font color="Olive" title="b9">q</font>, <font color="Olive" title="b10">q9</font> being   <a href="gtarski1.html#NM1" title="GTARSKI1:NM.1">POINT</a> of <font color="Maroon" title="c1">S</font>  st  not  <a href="gtarski1.html#R6" title="GTARSKI1:pred.6">Collinear</a> <font color="Olive" title="b4">a</font>,<font color="Olive" title="b5">b</font>,<font color="Olive" title="b6">c</font> &amp;  <a href="gtarski1.html#R1" title="GTARSKI1:pred.1">between</a> <font color="Olive" title="b5">b</font>,<font color="Olive" title="b4">a</font>,<font color="Olive" title="b2">y</font> &amp; <font color="Olive" title="b4">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">y</font> &amp;  <a href="gtarski1.html#R1" title="GTARSKI1:pred.1">between</a> <font color="Olive" title="b4">a</font>,<font color="Olive" title="b2">y</font>,<font color="Olive" title="b3">z</font> &amp; <font color="Olive" title="b2">y</font>,<font color="Olive" title="b3">z</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Olive" title="b2">y</font>,<font color="Olive" title="b8">p</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b8">p</font> &amp; <font color="Olive" title="b10">q9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="gtarski3.html#K3" title="GTARSKI3:func.3">reflection</a> (<font color="Olive" title="b3">z</font>,<font color="Olive" title="b9">q</font>) &amp;  <a href="gtarski3.html#R9" title="GTARSKI3:pred.9">Middle</a> <font color="Olive" title="b6">c</font>,<font color="Olive" title="b1">x</font>,<font color="Olive" title="b7">c9</font> &amp; <font color="Olive" title="b6">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">y</font> &amp;  <a href="gtarski1.html#R1" title="GTARSKI1:pred.1">between</a> <font color="Olive" title="b10">q9</font>,<font color="Olive" title="b2">y</font>,<font color="Olive" title="b7">c9</font> &amp;  <a href="gtarski3.html#R9" title="GTARSKI3:pred.9">Middle</a> <font color="Olive" title="b2">y</font>,<font color="Olive" title="b8">p</font>,<font color="Olive" title="b6">c</font> &amp;  <a href="gtarski1.html#R1" title="GTARSKI1:pred.1">between</a> <font color="Olive" title="b8">p</font>,<font color="Olive" title="b2">y</font>,<font color="Olive" title="b9">q</font> &amp; <font color="Olive" title="b9">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b10">q9</font> holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">y</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">y</font>, <font color="Maroon" title="c4">z</font>, <font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c6">b</font>, <font color="Maroon" title="c7">c</font>, <font color="Maroon" title="c8">c9</font>, <font color="Maroon" title="c9">p</font>, <font color="Maroon" title="c10">q</font>, <font color="Maroon" title="c11">q9</font> be   <a href="gtarski1.html#NM1" title="GTARSKI1:NM.1">POINT</a> of <font color="Maroon" title="c1">S</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not  <a href="gtarski1.html#R6" title="GTARSKI1:pred.6">Collinear</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">c</font> &amp;  <a href="gtarski1.html#R1" title="GTARSKI1:pred.1">between</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c3">y</font> &amp; <font color="Maroon" title="c5">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">y</font> &amp;  <a href="gtarski1.html#R1" title="GTARSKI1:pred.1">between</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c4">z</font> &amp; <font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c4">z</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c9">p</font> &amp; <font color="Maroon" title="c3">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">p</font> &amp; <font color="Maroon" title="c11">q9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="gtarski3.html#K3" title="GTARSKI3:func.3">reflection</a> (<font color="Maroon" title="c4">z</font>,<font color="Maroon" title="c10">q</font>) &amp;  <a href="gtarski3.html#R9" title="GTARSKI3:pred.9">Middle</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c8">c9</font> &amp; <font color="Maroon" title="c7">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">y</font> &amp;  <a href="gtarski1.html#R1" title="GTARSKI1:pred.1">between</a> <font color="Maroon" title="c11">q9</font>,<font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c8">c9</font> &amp;  <a href="gtarski3.html#R9" title="GTARSKI3:pred.9">Middle</a> <font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c9">p</font>,<font color="Maroon" title="c7">c</font> &amp;  <a href="gtarski1.html#R1" title="GTARSKI1:pred.1">between</a> <font color="Maroon" title="c9">p</font>,<font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c10">q</font> &amp; <font color="Maroon" title="c10">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">q9</font> implies <font color="Maroon" title="c2">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">y</font> )</span><br/>





















<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:42"/><span class="lab"><font color="Green" title="E32">A1</font></span>: 
 not  <a href="gtarski1.html#R6" title="GTARSKI1:pred.6">Collinear</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">c</font>
 <span class="kw">and </span><br/><a NAME="E2:42"/><span class="lab"><font color="Green" title="E33">A2</font></span>: 
 <a href="gtarski1.html#R1" title="GTARSKI1:pred.1">between</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c3">y</font>
 <span class="kw">and </span><br/><a NAME="E3:42"/><span class="lab"><font color="Green" title="E34">A3</font></span>: 
<font color="Maroon" title="c5">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">y</font>
 <span class="kw">and </span><br/><a NAME="E4:42"/><span class="lab"><font color="Green" title="E35">A4</font></span>: 
 <a href="gtarski1.html#R1" title="GTARSKI1:pred.1">between</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c4">z</font>
 <span class="kw">and </span><br/><a NAME="E5:42"/><span class="lab"><font color="Green" title="E36">A5</font></span>: 
<font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c4">z</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c9">p</font>
 <span class="kw">and </span><br/><a NAME="E6:42"/><span class="lab"><font color="Green" title="E37">A6</font></span>: 
<font color="Maroon" title="c3">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">p</font>
 <span class="kw">and </span><br/><a NAME="E7:42"/><span class="lab"><font color="Green" title="E38">A7</font></span>: 
<font color="Maroon" title="c11">q9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="gtarski3.html#K3" title="GTARSKI3:func.3">reflection</a> (<font color="Maroon" title="c4">z</font>,<font color="Maroon" title="c10">q</font>)
 <span class="kw">and </span><br/><a NAME="E8:42"/><span class="lab"><font color="Green" title="E39">A8</font></span>: 
 <a href="gtarski3.html#R9" title="GTARSKI3:pred.9">Middle</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c8">c9</font>
 <span class="kw">and </span><br/><a NAME="E9:42"/><span class="lab"><font color="Green" title="E40">A9</font></span>: 
<font color="Maroon" title="c7">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">y</font>
 <span class="kw">and </span><br/><a NAME="E10:42"/><span class="lab"><font color="Green" title="E41">A10</font></span>: 
 <a href="gtarski1.html#R1" title="GTARSKI1:pred.1">between</a> <font color="Maroon" title="c11">q9</font>,<font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c8">c9</font>
 <span class="kw">and </span><br/><a NAME="E11:42"/><span class="lab"><font color="Green" title="E42">A11</font></span>: 
 <a href="gtarski3.html#R9" title="GTARSKI3:pred.9">Middle</a> <font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c9">p</font>,<font color="Maroon" title="c7">c</font>
 <span class="kw">and </span><br/><a NAME="E12:42"/><span class="lab"><font color="Green" title="E43">A12</font></span>: 
 <a href="gtarski1.html#R1" title="GTARSKI1:pred.1">between</a> <font color="Maroon" title="c9">p</font>,<font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c10">q</font>
 <span class="kw">and </span><br/><a NAME="E13:42"/><span class="lab"><font color="Green" title="E44">A13</font></span>: 
<font color="Maroon" title="c10">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">q9</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">y</font></span><br/>

<span class="kw">assume </span><a NAME="E14:42"/><span class="lab"><font color="Green" title="E45">A14</font></span>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">y</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="42_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="gtarski4.html#K1" title="GTARSKI4:func.1">Line</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>) &amp; <font color="Maroon" title="c5">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">y</font> &amp; <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> )</span></a><div class="add"><a NAME="E1:42_1"/>
 <a href="gtarski1.html#R6" title="GTARSKI1:pred.6">Collinear</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c3">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:42"><span class="lab"><font color="Green" title="E33">A2</font></span></a>, <a class="ref" href="gtarski1.html#D17" onmouseover="rs('gtarski1/D17')" onmouseout="rh()">GTARSKI1:def 17</a></span>;<br/><span class="kw">hence </span><a NAME="E2:42_1"/>
<font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="gtarski4.html#K1" title="GTARSKI4:func.1">Line</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="gtarski4.html#E1"><span class="lab"><font color="Green" title="E1">LemmaA1</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">y</font> &amp; <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> )</span><br/><span class="kw">thus </span><a NAME="E3:42_1"/>
<font color="Maroon" title="c5">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:42"><span class="lab"><font color="Green" title="E34">A3</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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></span><br/><span class="kw">thus </span><a NAME="E4:42_1"/>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="ref" href="gtarski3.html#T46" onmouseover="rs('gtarski3/T46')" onmouseout="rh()">GTARSKI3:46</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E16:42"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A15</font></span>: 
 <a href="gtarski4.html#K1" title="GTARSKI4:func.1">Line</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c3">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="gtarski4.html#K1" title="GTARSKI4:func.1">Line</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="gtarski3.html#T82" onmouseover="rs('gtarski3/T82')" onmouseout="rh()">GTARSKI3:82</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="42_2"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="gtarski4.html#K1" title="GTARSKI4:func.1">Line</a> (<font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c5">a</font>) &amp; <font color="Maroon" title="c3">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">a</font> &amp; <font color="Maroon" title="c3">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">z</font> )</span></a><div class="add"><a NAME="E1:42_2"/>
 <a href="gtarski1.html#R6" title="GTARSKI1:pred.6">Collinear</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c4">z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:42"><span class="lab"><font color="Green" title="E35">A4</font></span></a>, <a class="ref" href="gtarski1.html#D17" onmouseover="rs('gtarski1/D17')" onmouseout="rh()">GTARSKI1:def 17</a></span>;<br/><span class="kw">hence </span><a NAME="E2:42_2"/>
<font color="Maroon" title="c4">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="gtarski4.html#K1" title="GTARSKI4:func.1">Line</a> (<font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c5">a</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="gtarski4.html#E1"><span class="lab"><font color="Green" title="E1">LemmaA1</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">a</font> &amp; <font color="Maroon" title="c3">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">z</font> )</span><br/><span class="kw">thus </span><a NAME="E3:42_2"/>
<font color="Maroon" title="c3">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:42"><span class="lab"><font color="Green" title="E34">A3</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">z</font></span><br/><span class="kw">thus </span><a NAME="E4:42_2"/>
<font color="Maroon" title="c3">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:42"><span class="lab"><font color="Green" title="E36">A5</font></span></a>, <a class="txt" href="#E6:42"><span class="lab"><font color="Green" title="E37">A6</font></span></a>, <a class="ref" href="gtarski1.html#D7" onmouseover="rs('gtarski1/D7')" onmouseout="rh()">GTARSKI1:def 7</a>, <a class="ref" href="gtarski3.html#T4" onmouseover="rs('gtarski3/T4')" onmouseout="rh()">GTARSKI3:4</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E18:42"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A16</font></span>: 
 <a href="gtarski4.html#K1" title="GTARSKI4:func.1">Line</a> (<font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c4">z</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="gtarski4.html#K1" title="GTARSKI4:func.1">Line</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:42"><span class="lab"><font color="Green" title="E46">A15</font></span></a>, <a class="ref" href="gtarski3.html#T82" onmouseover="rs('gtarski3/T82')" onmouseout="rh()">GTARSKI3:82</a></span>;<br/>
<a NAME="E19:42"/><span class="lab"><font color="Green" title="E48">A17</font></span>: 
 <a href="gtarski4.html#K1" title="GTARSKI4:func.1">Line</a> (<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c7">c</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="gtarski4.html#K1" title="GTARSKI4:func.1">Line</a> (<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c8">c9</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:42"><span class="lab"><font color="Green" title="E39">A8</font></span></a>, <a class="txt" href="#E9:42"><span class="lab"><font color="Green" title="E40">A9</font></span></a>, <a class="txt" href="#E14:42"><span class="lab"><font color="Green" title="E45">A14</font></span></a>, <a class="ref" href="gtarski4.html#T10" target="_self" onmouseover="rs('gtarski4/T10')" onmouseout="rh()">Prelim10</a></span>;<br/>
<a NAME="E20:42"/>
 <a href="gtarski1.html#R6" title="GTARSKI1:pred.6">Collinear</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c8">c9</font>,<font color="Maroon" title="c11">q9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:42"><span class="lab"><font color="Green" title="E41">A10</font></span></a>, <a class="txt" href="#E14:42"><span class="lab"><font color="Green" title="E45">A14</font></span></a>, <a class="ref" href="gtarski1.html#D17" onmouseover="rs('gtarski1/D17')" onmouseout="rh()">GTARSKI1:def 17</a></span>;<br/>
<a NAME="E21:42"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A18</font></span>: 
<font color="Maroon" title="c11">q9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="gtarski4.html#K1" title="GTARSKI4:func.1">Line</a> (<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c7">c</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:42"><span class="lab"><font color="Green" title="E48">A17</font></span></a>, <a class="txt" href="gtarski4.html#E1"><span class="lab"><font color="Green" title="E1">LemmaA1</font></span></a></span>;<br/>
<a NAME="E22:42"/>
(  <a href="gtarski1.html#R1" title="GTARSKI1:pred.1">between</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c9">p</font>,<font color="Maroon" title="c2">x</font> &amp;  <a href="gtarski1.html#R1" title="GTARSKI1:pred.1">between</a> <font color="Maroon" title="c9">p</font>,<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c10">q</font> &amp; <font color="Maroon" title="c9">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">x</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:42"><span class="lab"><font color="Green" title="E45">A14</font></span></a>, <a class="txt" href="#E6:42"><span class="lab"><font color="Green" title="E37">A6</font></span></a>, <a class="txt" href="#E11:42"><span class="lab"><font color="Green" title="E42">A11</font></span></a>, <a class="txt" href="#E12:42"><span class="lab"><font color="Green" title="E43">A12</font></span></a>, <a class="ref" href="gtarski3.html#D12" onmouseover="rs('gtarski3/D12')" onmouseout="rh()">GTARSKI3:def 12</a>, <a class="ref" href="gtarski3.html#T14" onmouseover="rs('gtarski3/T14')" onmouseout="rh()">GTARSKI3:14</a></span>;<br/>
<a NAME="E23:42"/><span class="kw">then </span>
 <a href="gtarski1.html#R1" title="GTARSKI1:pred.1">between</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c10">q</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="gtarski3.html#T19" onmouseover="rs('gtarski3/T19')" onmouseout="rh()">GTARSKI3:19</a></span>;<br/>
<a NAME="E24:42"/><span class="kw">then </span>
 <a href="gtarski1.html#R6" title="GTARSKI1:pred.6">Collinear</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c10">q</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="gtarski1.html#D17" onmouseover="rs('gtarski1/D17')" onmouseout="rh()">GTARSKI1:def 17</a></span>;<br/>
<a NAME="E25:42"/><span class="kw">then </span>
<font color="Maroon" title="c10">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="gtarski4.html#K1" title="GTARSKI4:func.1">Line</a> (<font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c2">x</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="gtarski4.html#E1"><span class="lab"><font color="Green" title="E1">LemmaA1</font></span></a></span>;<br/>
<a NAME="E26:42"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A19</font></span>: 
 <a href="gtarski4.html#K1" title="GTARSKI4:func.1">Line</a> (<font color="Maroon" title="c10">q</font>,<font color="Maroon" title="c11">q9</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="gtarski4.html#K1" title="GTARSKI4:func.1">Line</a> (<font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c2">x</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:42"><span class="lab"><font color="Green" title="E44">A13</font></span></a>, <a class="txt" href="#E9:42"><span class="lab"><font color="Green" title="E40">A9</font></span></a>, <a class="txt" href="#E14:42"><span class="lab"><font color="Green" title="E45">A14</font></span></a>, <a class="txt" href="#E21:42"><span class="lab"><font color="Green" title="E49">A18</font></span></a>, <a class="ref" href="gtarski4.html#T11" target="_self" onmouseover="rs('gtarski4/T11')" onmouseout="rh()">Prelim11</a></span>;<br/>
<a NAME="E27:42"/>
 <a href="gtarski3.html#R9" title="GTARSKI3:pred.9">Middle</a> <font color="Maroon" title="c10">q</font>,<font color="Maroon" title="c4">z</font>,<font color="Maroon" title="c11">q9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:42"><span class="lab"><font color="Green" title="E38">A7</font></span></a>, <a class="ref" href="gtarski3.html#D13" onmouseover="rs('gtarski3/D13')" onmouseout="rh()">GTARSKI3:def 13</a></span>;<br/>
<a NAME="E28:42"/><span class="kw">then </span>
 <a href="gtarski1.html#R1" title="GTARSKI1:pred.1">between</a> <font color="Maroon" title="c10">q</font>,<font color="Maroon" title="c4">z</font>,<font color="Maroon" title="c11">q9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="gtarski3.html#D12" onmouseover="rs('gtarski3/D12')" onmouseout="rh()">GTARSKI3:def 12</a></span>;<br/>
<a NAME="E29:42"/><span class="kw">then </span>
 <a href="gtarski1.html#R6" title="GTARSKI1:pred.6">Collinear</a> <font color="Maroon" title="c10">q</font>,<font color="Maroon" title="c4">z</font>,<font color="Maroon" title="c11">q9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="gtarski1.html#D17" onmouseover="rs('gtarski1/D17')" onmouseout="rh()">GTARSKI1:def 17</a></span>;<br/>
<a NAME="E30:42"/><span class="kw">then </span>
 <a href="gtarski1.html#R6" title="GTARSKI1:pred.6">Collinear</a> <font color="Maroon" title="c10">q</font>,<font color="Maroon" title="c11">q9</font>,<font color="Maroon" title="c4">z</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="gtarski3.html#T45" onmouseover="rs('gtarski3/T45')" onmouseout="rh()">GTARSKI3:45</a></span>;<br/>
<a NAME="E31:42"/><span class="kw">then </span>
<font color="Maroon" title="c4">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="gtarski4.html#K1" title="GTARSKI4:func.1">Line</a> (<font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c2">x</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:42"><span class="lab"><font color="Green" title="E50">A19</font></span></a>, <a class="txt" href="gtarski4.html#E1"><span class="lab"><font color="Green" title="E1">LemmaA1</font></span></a></span>;<br/>
<a NAME="E32:42"/><span class="kw">then </span>
 <a href="gtarski1.html#R6" title="GTARSKI1:pred.6">Collinear</a> <font color="Maroon" title="c4">z</font>,<font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c2">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="gtarski4.html#E2"><span class="lab"><font color="Green" title="E2">LemmaA2</font></span></a></span>;<br/>
<a NAME="E33:42"/><span class="kw">then </span>
 <a href="gtarski1.html#R6" title="GTARSKI1:pred.6">Collinear</a> <font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c4">z</font>,<font color="Maroon" title="c7">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:42"><span class="lab"><font color="Green" title="E45">A14</font></span></a>, <a class="ref" href="gtarski3.html#T45" onmouseover="rs('gtarski3/T45')" onmouseout="rh()">GTARSKI3:45</a></span>;<br/>
<a NAME="E34:42"/><span class="kw">then </span>
<font color="Maroon" title="c7">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="gtarski4.html#K1" title="GTARSKI4:func.1">Line</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:42"><span class="lab"><font color="Green" title="E47">A16</font></span></a>, <a class="txt" href="gtarski4.html#E1"><span class="lab"><font color="Green" title="E1">LemmaA1</font></span></a></span>;<br/>
<a NAME="E35:42"/><span class="kw">then </span>
 <a href="gtarski1.html#R6" title="GTARSKI1:pred.6">Collinear</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="gtarski4.html#E2"><span class="lab"><font color="Green" title="E2">LemmaA2</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E36:42"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="ref" href="gtarski3.html#T45" onmouseover="rs('gtarski3/T45')" onmouseout="rh()">GTARSKI3:45</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
