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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">x</font>, <font color="Olive" title="b3">p</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="b2">x</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="Olive" title="b1">a</font>,<font color="Olive" title="b3">p</font>) holds <br/> <a href="gtarski1.html#R6" title="GTARSKI1:pred.6">Collinear</a> <font color="Olive" title="b2">x</font>,<font color="Olive" title="b1">a</font>,<font color="Olive" title="b3">p</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">p</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="c3">x</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">a</font>,<font color="Maroon" title="c4">p</font>) implies  <a href="gtarski1.html#R6" title="GTARSKI1:pred.6">Collinear</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">p</font> )</span><br/>







<span class="kw">assume </span><a NAME="E1:3"/>
<font color="Maroon" title="c3">x</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">a</font>,<font color="Maroon" title="c4">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="gtarski1.html#R6" title="GTARSKI1:pred.6">Collinear</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">p</font></span><br/>

<a NAME="E2:3"/><span class="kw">then </span>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">y</font> where <font color="Olive" title="b1">y</font> is   <a href="gtarski1.html#NM1" title="GTARSKI1:NM.1">POINT</a> of <font color="Maroon" title="c1">S</font> :  <a href="gtarski1.html#R6" title="GTARSKI1:pred.6">Collinear</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">p</font>,<font color="Olive" title="b1">y</font> </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="gtarski3.html#D10" onmouseover="rs('gtarski3/D10')" onmouseout="rh()">GTARSKI3:def 10</a></span>;<br/>
<a NAME="E3:3"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">y</font> being   <a href="gtarski1.html#NM1" title="GTARSKI1:NM.1">POINT</a> of <font color="Maroon" title="c1">S</font> st <br/>( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">y</font> &amp;  <a href="gtarski1.html#R6" title="GTARSKI1:pred.6">Collinear</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">p</font>,<font color="Olive" title="b1">y</font> )
 
;<br/>
<span class="kw">hence </span><a NAME="E4:3"/>
 <a href="gtarski1.html#R6" title="GTARSKI1:pred.6">Collinear</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">p</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>; <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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