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<span class="kw">let </span><font color="Maroon" title="c5">CLSP</font> be   <a href="collsp.html#NM2" title="COLLSP:NM.2">CollSp</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">r</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c5">CLSP</font> holds <br/> ( <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">are_collinear</a>  iff <font color="Olive" title="b3">r</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="collsp.html#K1" title="COLLSP:func.1">Line</a> (<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>) )</span><br/><span class="kw">let </span><font color="Maroon" title="c6">a</font>, <font color="Maroon" title="c7">b</font>, <font color="Maroon" title="c8">r</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c5">CLSP</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="c6">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">are_collinear</a>  iff <font color="Maroon" title="c8">r</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="collsp.html#K1" title="COLLSP:func.1">Line</a> (<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>) )</span><br/>







<span class="kw">thus </span><a NAME="E1:18"/>
( <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">are_collinear</a>  implies <font color="Maroon" title="c8">r</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="collsp.html#K1" title="COLLSP:func.1">Line</a> (<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">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"> ( <font color="Maroon" title="c8">r</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="collsp.html#K1" title="COLLSP:func.1">Line</a> (<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>) implies <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">are_collinear</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E2:18"/>
( <font color="Maroon" title="c8">r</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="collsp.html#K1" title="COLLSP:func.1">Line</a> (<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>) implies <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">are_collinear</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"> verum</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="18_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:18_1"/>
<font color="Maroon" title="c8">r</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="collsp.html#K1" title="COLLSP:func.1">Line</a> (<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">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"> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">are_collinear</a> </span><br/>

<a NAME="E2:18_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">p</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c5">CLSP</font> st <br/>( <font color="Maroon" title="c8">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> &amp; <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Olive" title="b1">p</font> <a href="collsp.html#R1" title="COLLSP:pred.1">are_collinear</a>  )
 
;<br/>
<span class="kw">hence </span><a NAME="E3:18_1"/>
<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">are_collinear</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"> verum</span><br/>


</div><span class="kw">end;</span></div>


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