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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">p</font>, <font color="Olive" title="b2">q</font>, <font color="Olive" title="b3">r</font> being   <a href="gtarski1.html#NM1" title="GTARSKI1:NM.1">POINT</a> of <font color="Maroon" title="c1">S</font>  st <font color="Olive" title="b1">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">q</font> &amp; <font color="Olive" title="b1">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">r</font> &amp; (  <a href="gtarski1.html#R6" title="GTARSKI1:pred.6">Collinear</a> <font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font>,<font color="Olive" title="b3">r</font> or  <a href="gtarski1.html#R6" title="GTARSKI1:pred.6">Collinear</a> <font color="Olive" title="b2">q</font>,<font color="Olive" title="b3">r</font>,<font color="Olive" title="b1">p</font> or  <a href="gtarski1.html#R6" title="GTARSKI1:pred.6">Collinear</a> <font color="Olive" title="b3">r</font>,<font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font> or  <a href="gtarski1.html#R6" title="GTARSKI1:pred.6">Collinear</a> <font color="Olive" title="b1">p</font>,<font color="Olive" title="b3">r</font>,<font color="Olive" title="b2">q</font> or  <a href="gtarski1.html#R6" title="GTARSKI1:pred.6">Collinear</a> <font color="Olive" title="b2">q</font>,<font color="Olive" title="b1">p</font>,<font color="Olive" title="b3">r</font> or  <a href="gtarski1.html#R6" title="GTARSKI1:pred.6">Collinear</a> <font color="Olive" title="b3">r</font>,<font color="Olive" title="b2">q</font>,<font color="Olive" title="b1">p</font> ) holds <br/>(  <a href="gtarski4.html#K1" title="GTARSKI4:func.1">Line</a> (<font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</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="b1">p</font>,<font color="Olive" title="b3">r</font>) &amp;  <a href="gtarski4.html#K1" title="GTARSKI4:func.1">Line</a> (<font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</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">r</font>,<font color="Olive" title="b1">p</font>) &amp;  <a href="gtarski4.html#K1" title="GTARSKI4:func.1">Line</a> (<font color="Olive" title="b2">q</font>,<font color="Olive" title="b1">p</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="b1">p</font>,<font color="Olive" title="b3">r</font>) &amp;  <a href="gtarski4.html#K1" title="GTARSKI4:func.1">Line</a> (<font color="Olive" title="b2">q</font>,<font color="Olive" title="b1">p</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">r</font>,<font color="Olive" title="b1">p</font>) )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">p</font>, <font color="Maroon" title="c3">q</font>, <font color="Maroon" title="c4">r</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"> ( <font color="Maroon" title="c2">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">q</font> &amp; <font color="Maroon" title="c2">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">r</font> &amp; (  <a href="gtarski1.html#R6" title="GTARSKI1:pred.6">Collinear</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c4">r</font> or  <a href="gtarski1.html#R6" title="GTARSKI1:pred.6">Collinear</a> <font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c4">r</font>,<font color="Maroon" title="c2">p</font> or  <a href="gtarski1.html#R6" title="GTARSKI1:pred.6">Collinear</a> <font color="Maroon" title="c4">r</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font> or  <a href="gtarski1.html#R6" title="GTARSKI1:pred.6">Collinear</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">r</font>,<font color="Maroon" title="c3">q</font> or  <a href="gtarski1.html#R6" title="GTARSKI1:pred.6">Collinear</a> <font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">r</font> or  <a href="gtarski1.html#R6" title="GTARSKI1:pred.6">Collinear</a> <font color="Maroon" title="c4">r</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c2">p</font> ) implies (  <a href="gtarski4.html#K1" title="GTARSKI4:func.1">Line</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</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">p</font>,<font color="Maroon" title="c4">r</font>) &amp;  <a href="gtarski4.html#K1" title="GTARSKI4:func.1">Line</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</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">r</font>,<font color="Maroon" title="c2">p</font>) &amp;  <a href="gtarski4.html#K1" title="GTARSKI4:func.1">Line</a> (<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c2">p</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">p</font>,<font color="Maroon" title="c4">r</font>) &amp;  <a href="gtarski4.html#K1" title="GTARSKI4:func.1">Line</a> (<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c2">p</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">r</font>,<font color="Maroon" title="c2">p</font>) ) )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:8"/><span class="lab"><font color="Green" title="E8">A1</font></span>: 
<font color="Maroon" title="c2">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">q</font>
 <span class="kw">and </span><br/><a NAME="E2:8"/><span class="lab"><font color="Green" title="E9">A2</font></span>: 
<font color="Maroon" title="c2">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">r</font>
 <span class="kw">and </span><br/><a NAME="E3:8"/><span class="lab"><font color="Green" title="E10">A3</font></span>: 
(  <a href="gtarski1.html#R6" title="GTARSKI1:pred.6">Collinear</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c4">r</font> or  <a href="gtarski1.html#R6" title="GTARSKI1:pred.6">Collinear</a> <font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c4">r</font>,<font color="Maroon" title="c2">p</font> or  <a href="gtarski1.html#R6" title="GTARSKI1:pred.6">Collinear</a> <font color="Maroon" title="c4">r</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font> or  <a href="gtarski1.html#R6" title="GTARSKI1:pred.6">Collinear</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">r</font>,<font color="Maroon" title="c3">q</font> or  <a href="gtarski1.html#R6" title="GTARSKI1:pred.6">Collinear</a> <font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">r</font> or  <a href="gtarski1.html#R6" title="GTARSKI1:pred.6">Collinear</a> <font color="Maroon" title="c4">r</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c2">p</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="c2">p</font>,<font color="Maroon" title="c3">q</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">p</font>,<font color="Maroon" title="c4">r</font>) &amp;  <a href="gtarski4.html#K1" title="GTARSKI4:func.1">Line</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</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">r</font>,<font color="Maroon" title="c2">p</font>) &amp;  <a href="gtarski4.html#K1" title="GTARSKI4:func.1">Line</a> (<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c2">p</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">p</font>,<font color="Maroon" title="c4">r</font>) &amp;  <a href="gtarski4.html#K1" title="GTARSKI4:func.1">Line</a> (<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c2">p</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">r</font>,<font color="Maroon" title="c2">p</font>) )</span><br/>

<a NAME="E4:8"/>
 <a href="gtarski1.html#R6" title="GTARSKI1:pred.6">Collinear</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c4">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:8"><span class="lab"><font color="Green" title="E10">A3</font></span></a>, <a class="ref" href="gtarski3.html#T45" onmouseover="rs('gtarski3/T45')" onmouseout="rh()">GTARSKI3:45</a></span>;<br/>
<a NAME="E5:8"/><span class="kw">then </span>
<font color="Maroon" title="c4">r</font> <a href="gtarski1.html#R7" title="GTARSKI1:pred.7">on_line</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8"><span class="lab"><font color="Green" title="E8">A1</font></span></a>, <a class="ref" href="gtarski3.html#T84" onmouseover="rs('gtarski3/T84')" onmouseout="rh()">GTARSKI3:84</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:8"/>
(  <a href="gtarski4.html#K1" title="GTARSKI4:func.1">Line</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</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">p</font>,<font color="Maroon" title="c4">r</font>) &amp;  <a href="gtarski4.html#K1" title="GTARSKI4:func.1">Line</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</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">r</font>,<font color="Maroon" title="c2">p</font>) &amp;  <a href="gtarski4.html#K1" title="GTARSKI4:func.1">Line</a> (<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c2">p</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">p</font>,<font color="Maroon" title="c4">r</font>) &amp;  <a href="gtarski4.html#K1" title="GTARSKI4:func.1">Line</a> (<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c2">p</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">r</font>,<font color="Maroon" title="c2">p</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8"><span class="lab"><font color="Green" title="E8">A1</font></span></a>, <a class="txt" href="#E2:8"><span class="lab"><font color="Green" title="E9">A2</font></span></a>, <a class="ref" href="gtarski1.html#T39" onmouseover="rs('gtarski1/T39')" onmouseout="rh()">GTARSKI1:39</a>, <a class="ref" href="gtarski3.html#T85" onmouseover="rs('gtarski3/T85')" onmouseout="rh()">GTARSKI3:85</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/>


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