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<span class="kw">let </span><font color="Maroon" title="c10">CLSP</font> be   <a href="collsp.html#V4" title="COLLSP:attr.4">proper</a>  <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">p</font>, <font color="Olive" title="b2">q</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c10">CLSP</font><br/>  for <font color="Olive" title="b3">P</font> being    <a href="collsp.html#M2" title="COLLSP:mode.2">LINE</a> of <font color="Maroon" title="c10">CLSP</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="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b3">P</font> &amp; <font color="Olive" title="b2">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b3">P</font> holds <br/> <a href="collsp.html#K1" title="COLLSP:func.1">Line</a> (<font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b3">P</font></span><br/><span class="kw">let </span><font color="Maroon" title="c11">p</font>, <font color="Maroon" title="c12">q</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c10">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">  for <font color="Olive" title="b1">P</font> being    <a href="collsp.html#M2" title="COLLSP:mode.2">LINE</a> of <font color="Maroon" title="c10">CLSP</font>  st <font color="Maroon" title="c11">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">q</font> &amp; <font color="Maroon" title="c11">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">P</font> &amp; <font color="Maroon" title="c12">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">P</font> holds <br/> <a href="collsp.html#K1" title="COLLSP:func.1">Line</a> (<font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">P</font></span><br/><span class="kw">let </span><font color="Maroon" title="c13">P</font> be    <a href="collsp.html#M2" title="COLLSP:mode.2">LINE</a> of <font color="Maroon" title="c10">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="c11">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">q</font> &amp; <font color="Maroon" title="c11">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c13">P</font> &amp; <font color="Maroon" title="c12">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c13">P</font> implies  <a href="collsp.html#K1" title="COLLSP:func.1">Line</a> (<font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">P</font> )</span><br/>







<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:34"/><span class="lab"><font color="Green" title="E28">A1</font></span>: 
<font color="Maroon" title="c11">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">q</font>
 <span class="kw">and </span><br/><a NAME="E2:34"/><span class="lab"><font color="Green" title="E29">A2</font></span>: 
( <font color="Maroon" title="c11">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c13">P</font> &amp; <font color="Maroon" title="c12">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c13">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="collsp.html#K1" title="COLLSP:func.1">Line</a> (<font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">P</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c14">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</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"> (  not <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="collsp.html#K1" title="COLLSP:func.1">Line</a> (<font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font>) or <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">P</font> )</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c15">a</font>, <font color="Maroon" title="c16">b</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c10">CLSP</font><span class="kw"> such that </span><br/><a NAME="E4:34"/>
<font color="Maroon" title="c15">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c16">b</font>
 <span class="kw">and </span><br/><a NAME="E5:34"/><span class="lab"><font color="Green" title="E30">A3</font></span>: 
<font color="Maroon" title="c13">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="collsp.html#K1" title="COLLSP:func.1">Line</a> (<font color="Maroon" title="c15">a</font>,<font color="Maroon" title="c16">b</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="collsp.html#D7" target="_self" onmouseover="rs('collsp/D7')" onmouseout="rh()">Def7</a></span>;<br/>
<span class="kw">assume </span><a NAME="E6:34"/>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="collsp.html#K1" title="COLLSP:func.1">Line</a> (<font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</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="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">P</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c17">r</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c10">CLSP</font><span class="kw"> such that </span><br/><a NAME="E8:34"/><span class="lab"><font color="Green" title="E31">A4</font></span>: 
<font color="Maroon" title="c17">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">x</font>
 <span class="kw">and </span><br/><a NAME="E9:34"/><span class="lab"><font color="Green" title="E32">A5</font></span>: 
<font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c17">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">are_collinear</a> 
 ;<br/>
<a NAME="E10:34"/>
( <font color="Maroon" title="c15">a</font>,<font color="Maroon" title="c16">b</font>,<font color="Maroon" title="c11">p</font> <a href="collsp.html#R1" title="COLLSP:pred.1">are_collinear</a>  &amp; <font color="Maroon" title="c15">a</font>,<font color="Maroon" title="c16">b</font>,<font color="Maroon" title="c12">q</font> <a href="collsp.html#R1" title="COLLSP:pred.1">are_collinear</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:34"><span class="lab"><font color="Green" title="E29">A2</font></span></a>, <a class="txt" href="#E5:34"><span class="lab"><font color="Green" title="E30">A3</font></span></a>, <a class="ref" href="collsp.html#T11" target="_self" onmouseover="rs('collsp/T11')" onmouseout="rh()">Th11</a></span>;<br/>
<a NAME="E11:34"/><span class="kw">then </span>
<font color="Maroon" title="c15">a</font>,<font color="Maroon" title="c16">b</font>,<font color="Maroon" title="c17">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">are_collinear</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34"><span class="lab"><font color="Green" title="E28">A1</font></span></a>, <a class="txt" href="#E9:34"><span class="lab"><font color="Green" title="E32">A5</font></span></a>, <a class="ref" href="collsp.html#T9" target="_self" onmouseover="rs('collsp/T9')" onmouseout="rh()">Th9</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:34"/>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:34"><span class="lab"><font color="Green" title="E30">A3</font></span></a>, <a class="txt" href="#E8:34"><span class="lab"><font color="Green" title="E31">A4</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/>


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