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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">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font> being   <a href="gtarski1.html#NM1" title="GTARSKI1:NM.1">POINT</a> of <font color="Maroon" title="c1">S</font>  st (  <a href="gtarski3.html#R9" title="GTARSKI3:pred.9">Middle</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font> or  <a href="gtarski3.html#R9" title="GTARSKI3:pred.9">Middle</a> <font color="Olive" title="b3">c</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b1">a</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> or <font color="Olive" title="b2">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">c</font> ) holds <br/>(  <a href="gtarski4.html#K1" title="GTARSKI4:func.1">Line</a> (<font color="Olive" title="b2">b</font>,<font color="Olive" title="b1">a</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="gtarski4.html#K1" title="GTARSKI4:func.1">Line</a> (<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font>) &amp;  <a href="gtarski4.html#K1" title="GTARSKI4:func.1">Line</a> (<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="gtarski4.html#K1" title="GTARSKI4:func.1">Line</a> (<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font>) &amp;  <a href="gtarski4.html#K1" title="GTARSKI4:func.1">Line</a> (<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="gtarski4.html#K1" title="GTARSKI4:func.1">Line</a> (<font color="Olive" title="b3">c</font>,<font color="Olive" title="b2">b</font>) &amp;  <a href="gtarski4.html#K1" title="GTARSKI4:func.1">Line</a> (<font color="Olive" title="b2">b</font>,<font color="Olive" title="b1">a</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="gtarski4.html#K1" title="GTARSKI4:func.1">Line</a> (<font color="Olive" title="b3">c</font>,<font color="Olive" title="b2">b</font>) )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font>, <font color="Maroon" title="c4">c</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"> ( (  <a href="gtarski3.html#R9" title="GTARSKI3:pred.9">Middle</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font> or  <a href="gtarski3.html#R9" title="GTARSKI3:pred.9">Middle</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">a</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> or <font color="Maroon" title="c3">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">c</font> ) implies (  <a href="gtarski4.html#K1" title="GTARSKI4:func.1">Line</a> (<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">a</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="c3">b</font>,<font color="Maroon" title="c4">c</font>) &amp;  <a href="gtarski4.html#K1" title="GTARSKI4:func.1">Line</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</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="c3">b</font>,<font color="Maroon" title="c4">c</font>) &amp;  <a href="gtarski4.html#K1" title="GTARSKI4:func.1">Line</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</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="c4">c</font>,<font color="Maroon" title="c3">b</font>) &amp;  <a href="gtarski4.html#K1" title="GTARSKI4:func.1">Line</a> (<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">a</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="c4">c</font>,<font color="Maroon" title="c3">b</font>) ) )</span><br/>







<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:13"/><span class="lab"><font color="Green" title="E13">A1</font></span>: 
(  <a href="gtarski3.html#R9" title="GTARSKI3:pred.9">Middle</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font> or  <a href="gtarski3.html#R9" title="GTARSKI3:pred.9">Middle</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">a</font> )
 <span class="kw">and </span><br/><a NAME="E2:13"/><span class="lab"><font color="Green" title="E14">A2</font></span>: 
( <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> or <font color="Maroon" title="c3">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">c</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"> (  <a href="gtarski4.html#K1" title="GTARSKI4:func.1">Line</a> (<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">a</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="c3">b</font>,<font color="Maroon" title="c4">c</font>) &amp;  <a href="gtarski4.html#K1" title="GTARSKI4:func.1">Line</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</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="c3">b</font>,<font color="Maroon" title="c4">c</font>) &amp;  <a href="gtarski4.html#K1" title="GTARSKI4:func.1">Line</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</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="c4">c</font>,<font color="Maroon" title="c3">b</font>) &amp;  <a href="gtarski4.html#K1" title="GTARSKI4:func.1">Line</a> (<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">a</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="c4">c</font>,<font color="Maroon" title="c3">b</font>) )</span><br/>

<a NAME="E3:13"/><span class="lab"><font color="Green" title="E15">A3</font></span>: 
 <a href="gtarski3.html#R9" title="GTARSKI3:pred.9">Middle</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:13"><span class="lab"><font color="Green" title="E13">A1</font></span></a>, <a class="ref" href="gtarski3.html#T96" onmouseover="rs('gtarski3/T96')" onmouseout="rh()">GTARSKI3:96</a></span>;<br/>
<a NAME="E4:13"/><span class="kw">then </span>
 <a href="gtarski1.html#R1" title="GTARSKI1:pred.1">between</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</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="E5:13"/><span class="kw">then </span><span class="lab"><font color="Green" title="E16">a4</font></span>: 
 <a href="gtarski1.html#R6" title="GTARSKI1:pred.6">Collinear</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</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/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:13_1"/>
( <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> or <font color="Maroon" title="c3">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">c</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:13"><span class="lab"><font color="Green" title="E14">A2</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_1_1"><span class="kw">suppose </span></a><a NAME="E1:13_1_1"/><span class="lab"><font color="Green" title="E17">A5</font></span>: 
<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>
 ; <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"> (  <a href="gtarski4.html#K1" title="GTARSKI4:func.1">Line</a> (<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">a</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="c3">b</font>,<font color="Maroon" title="c4">c</font>) &amp;  <a href="gtarski4.html#K1" title="GTARSKI4:func.1">Line</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</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="c3">b</font>,<font color="Maroon" title="c4">c</font>) &amp;  <a href="gtarski4.html#K1" title="GTARSKI4:func.1">Line</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</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="c4">c</font>,<font color="Maroon" title="c3">b</font>) &amp;  <a href="gtarski4.html#K1" title="GTARSKI4:func.1">Line</a> (<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">a</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="c4">c</font>,<font color="Maroon" title="c3">b</font>) )</span><br/><div class="add"><a NAME="E2:13_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:13"><span class="lab"><font color="Green" title="E15">A3</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#D12" onmouseover="rs('gtarski3/D12')" onmouseout="rh()">GTARSKI3:def 12</a></span>;<br/><span class="kw">hence </span><a NAME="E3:13_1_1"/>
(  <a href="gtarski4.html#K1" title="GTARSKI4:func.1">Line</a> (<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">a</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="c3">b</font>,<font color="Maroon" title="c4">c</font>) &amp;  <a href="gtarski4.html#K1" title="GTARSKI4:func.1">Line</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</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="c3">b</font>,<font color="Maroon" title="c4">c</font>) &amp;  <a href="gtarski4.html#K1" title="GTARSKI4:func.1">Line</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</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="c4">c</font>,<font color="Maroon" title="c3">b</font>) &amp;  <a href="gtarski4.html#K1" title="GTARSKI4:func.1">Line</a> (<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">a</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="c4">c</font>,<font color="Maroon" title="c3">b</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:13"><span class="lab"><font color="Green" title="E16">a4</font></span></a>, <a class="txt" href="#E1:13_1_1"><span class="lab"><font color="Green" title="E17">A5</font></span></a>, <a class="ref" href="gtarski3.html#T82" onmouseover="rs('gtarski3/T82')" onmouseout="rh()">GTARSKI3:82</a>, <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"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_1_2"><span class="kw">suppose </span></a><a NAME="E1:13_1_2"/><span class="lab"><font color="Green" title="E17">A6</font></span>: 
<font color="Maroon" title="c3">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">c</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"> (  <a href="gtarski4.html#K1" title="GTARSKI4:func.1">Line</a> (<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">a</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="c3">b</font>,<font color="Maroon" title="c4">c</font>) &amp;  <a href="gtarski4.html#K1" title="GTARSKI4:func.1">Line</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</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="c3">b</font>,<font color="Maroon" title="c4">c</font>) &amp;  <a href="gtarski4.html#K1" title="GTARSKI4:func.1">Line</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</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="c4">c</font>,<font color="Maroon" title="c3">b</font>) &amp;  <a href="gtarski4.html#K1" title="GTARSKI4:func.1">Line</a> (<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">a</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="c4">c</font>,<font color="Maroon" title="c3">b</font>) )</span><br/><div class="add"><a NAME="E2:13_1_2"/><span class="kw">then </span>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:13"><span class="lab"><font color="Green" title="E13">A1</font></span></a>, <a class="ref" href="gtarski4.html#T9" target="_self" onmouseover="rs('gtarski4/T9')" onmouseout="rh()">Prelim09</a></span>;<br/><span class="kw">hence </span><a NAME="E3:13_1_2"/>
(  <a href="gtarski4.html#K1" title="GTARSKI4:func.1">Line</a> (<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">a</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="c3">b</font>,<font color="Maroon" title="c4">c</font>) &amp;  <a href="gtarski4.html#K1" title="GTARSKI4:func.1">Line</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</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="c3">b</font>,<font color="Maroon" title="c4">c</font>) &amp;  <a href="gtarski4.html#K1" title="GTARSKI4:func.1">Line</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</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="c4">c</font>,<font color="Maroon" title="c3">b</font>) &amp;  <a href="gtarski4.html#K1" title="GTARSKI4:func.1">Line</a> (<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">a</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="c4">c</font>,<font color="Maroon" title="c3">b</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:13"><span class="lab"><font color="Green" title="E16">a4</font></span></a>, <a class="txt" href="#E1:13_1_2"><span class="lab"><font color="Green" title="E17">A6</font></span></a>, <a class="ref" href="gtarski3.html#T82" onmouseover="rs('gtarski3/T82')" onmouseout="rh()">GTARSKI3:82</a>, <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"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div>
