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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="gtarski2.html#NV5" title="GTARSKI2:NV.5">satisfying_Lower_Dimension_Axiom</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  not  <a href="gtarski1.html#R6" title="GTARSKI1:pred.6">Collinear</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font> holds <br/> ex <font color="Olive" title="b4">p</font>, <font color="Olive" title="b5">t</font> being   <a href="gtarski1.html#NM1" title="GTARSKI1:NM.1">POINT</a> of <font color="Maroon" title="c1">S</font> st <br/>(  <a href="gtarski4.html#R6" title="GTARSKI4:pred.6">are_orthogonal</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b4">p</font>,<font color="Olive" title="b1">a</font> &amp;  <a href="gtarski1.html#R6" title="GTARSKI1:pred.6">Collinear</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b5">t</font> &amp;  <a href="gtarski1.html#R1" title="GTARSKI1:pred.1">between</a> <font color="Olive" title="b3">c</font>,<font color="Olive" title="b5">t</font>,<font color="Olive" title="b4">p</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"> (  not  <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> implies  ex <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">t</font> being   <a href="gtarski1.html#NM1" title="GTARSKI1:NM.1">POINT</a> of <font color="Maroon" title="c1">S</font> st <br/>(  <a href="gtarski4.html#R6" title="GTARSKI4:pred.6">are_orthogonal</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c2">a</font> &amp;  <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="Olive" title="b2">t</font> &amp;  <a href="gtarski1.html#R1" title="GTARSKI1:pred.1">between</a> <font color="Maroon" title="c4">c</font>,<font color="Olive" title="b2">t</font>,<font color="Olive" title="b1">p</font> ) )</span><br/>







<span class="kw">assume </span><a NAME="E1:45"/><span class="lab"><font color="Green" title="E35">A1</font></span>: 
 not  <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>
 ; <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">  ex <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">t</font> being   <a href="gtarski1.html#NM1" title="GTARSKI1:NM.1">POINT</a> of <font color="Maroon" title="c1">S</font> st <br/>(  <a href="gtarski4.html#R6" title="GTARSKI4:pred.6">are_orthogonal</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c2">a</font> &amp;  <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="Olive" title="b2">t</font> &amp;  <a href="gtarski1.html#R1" title="GTARSKI1:pred.1">between</a> <font color="Maroon" title="c4">c</font>,<font color="Olive" title="b2">t</font>,<font color="Olive" title="b1">p</font> )</span><br/>

<a NAME="E2:45"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="gtarski3.html#T46" onmouseover="rs('gtarski3/T46')" onmouseout="rh()">GTARSKI3:46</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c5">x</font> being   <a href="gtarski1.html#NM1" title="GTARSKI1:NM.1">POINT</a> of <font color="Maroon" title="c1">S</font><span class="kw"> such that </span><br/><a NAME="E4:45"/><span class="lab"><font color="Green" title="E37">A3</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="gtarski4.html#R8" title="GTARSKI4:pred.8">is_foot</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="#E1:45"><span class="lab"><font color="Green" title="E35">A1</font></span></a>, <a class="ref" href="gtarski4.html#T33" target="_self" onmouseover="rs('gtarski4/T33')" onmouseout="rh()">Satz8p18pExistence</a></span>;<br/>
<a NAME="E5:45"/>
(  <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="c5">x</font> &amp;  <a href="gtarski4.html#R6" title="GTARSKI4:pred.6">are_orthogonal</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">x</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:45"><span class="lab"><font color="Green" title="E37">A3</font></span></a></span>;<br/>
<a NAME="E6:45"/><span class="kw">then </span>
 <a href="gtarski4.html#R5" title="GTARSKI4:pred.5">are_orthogonal</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="gtarski4.html#T29" target="_self" onmouseover="rs('gtarski4/T29')" onmouseout="rh()">Satz8p15</a></span>;<br/>
<a NAME="E7:45"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A4</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> &amp; <font color="Maroon" title="c4">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">x</font> &amp;  <a href="gtarski4.html#R2" title="GTARSKI4:pred.2">are_orthogonal</a>  <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>),<font color="Maroon" title="c5">x</font>, <a href="gtarski4.html#K1" title="GTARSKI4:func.1">Line</a> (<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">x</font>) )
 
;<br/>
<a NAME="E8:45"/><span class="lab"><font color="Green" title="E39">A5</font></span>: 
( <font color="Maroon" title="c2">a</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="c3">b</font>) &amp; <font color="Maroon" title="c4">c</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="c4">c</font>,<font color="Maroon" title="c5">x</font>) )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="gtarski3.html#T83" onmouseover="rs('gtarski3/T83')" onmouseout="rh()">GTARSKI3:83</a></span>;<br/>
<a NAME="E9:45"/><span class="kw">then </span>
 <a href="gtarski4.html#R1" title="GTARSKI4:pred.1">right_angle</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c4">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:45"><span class="lab"><font color="Green" title="E38">A4</font></span></a></span>;<br/>
<a NAME="E10:45"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A6</font></span>: 
<font color="Maroon" title="c2">a</font>, <a href="gtarski3.html#K3" title="GTARSKI3:func.3">reflection</a> (<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c4">c</font>) <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="gtarski3.html#T3" onmouseover="rs('gtarski3/T3')" onmouseout="rh()">GTARSKI3:3</a></span>;<br/>
<a NAME="E11:45"/>
 <a href="gtarski3.html#R9" title="GTARSKI3:pred.9">Middle</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c2">a</font>, <a href="gtarski3.html#K3" title="GTARSKI3:func.3">reflection</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="gtarski3.html#D13" onmouseover="rs('gtarski3/D13')" onmouseout="rh()">GTARSKI3:def 13</a></span>;<br/>
<a NAME="E12:45"/><span class="kw">then </span>
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c2">a</font>, <a href="gtarski3.html#K3" title="GTARSKI3:func.3">reflection</a> (<font color="Maroon" title="c2">a</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/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">p</font> being   <a href="gtarski1.html#NM1" title="GTARSKI1:NM.1">POINT</a> of <font color="Maroon" title="c1">S</font><span class="kw"> such that </span><br/><a NAME="E14:45"/><span class="lab"><font color="Green" title="E41">A7</font></span>: 
 <a href="gtarski3.html#R9" title="GTARSKI3:pred.9">Middle</a>  <a href="gtarski3.html#K3" title="GTARSKI3:func.3">reflection</a> (<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c4">c</font>),<font color="Maroon" title="c6">p</font>, <a href="gtarski3.html#K3" title="GTARSKI3:func.3">reflection</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:45"><span class="lab"><font color="Green" title="E40">A6</font></span></a>, <a class="ref" href="gtarski3.html#T5" onmouseover="rs('gtarski3/T5')" onmouseout="rh()">GTARSKI3:5</a>, <a class="ref" href="gtarski3.html#T117" onmouseover="rs('gtarski3/T117')" onmouseout="rh()">GTARSKI3:117</a></span>;<br/>
<a NAME="E15:45"/><span class="lab"><font color="Green" title="E42">A8</font></span>: 
 <a href="gtarski3.html#R9" title="GTARSKI3:pred.9">Middle</a>  <a href="gtarski3.html#K3" title="GTARSKI3:func.3">reflection</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font>),<font color="Maroon" title="c6">p</font>, <a href="gtarski3.html#K3" title="GTARSKI3:func.3">reflection</a> (<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c4">c</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:45"><span class="lab"><font color="Green" title="E41">A7</font></span></a>, <a class="ref" href="gtarski3.html#T96" onmouseover="rs('gtarski3/T96')" onmouseout="rh()">GTARSKI3:96</a></span>;<br/>
<a NAME="E16:45"/><span class="lab"><font color="Green" title="E43">A9</font></span>: 
 <a href="gtarski4.html#R1" title="GTARSKI4:pred.1">right_angle</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c4">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:45"><span class="lab"><font color="Green" title="E39">A5</font></span></a>, <a class="txt" href="#E7:45"><span class="lab"><font color="Green" title="E38">A4</font></span></a></span>;<br/>
<a NAME="E17:45"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A10</font></span>: 
(  <a href="gtarski4.html#R1" title="GTARSKI4:pred.1">right_angle</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c6">p</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="c6">p</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45"><span class="lab"><font color="Green" title="E35">A1</font></span></a>, <a class="txt" href="#E4:45"><span class="lab"><font color="Green" title="E37">A3</font></span></a>, <a class="txt" href="#E15:45"><span class="lab"><font color="Green" title="E42">A8</font></span></a>, <a class="ref" href="gtarski4.html#T34" target="_self" onmouseover="rs('gtarski4/T34')" onmouseout="rh()">Lemma8p20</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="45_1"><span class="kw">now </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"> (  <a href="gtarski1.html#R1" title="GTARSKI1:pred.1">between</a>  <a href="gtarski3.html#K3" title="GTARSKI3:func.3">reflection</a> (<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c4">c</font>),<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c4">c</font> &amp;  <a href="gtarski1.html#R1" title="GTARSKI1:pred.1">between</a>  <a href="gtarski3.html#K3" title="GTARSKI3:func.3">reflection</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font>),<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font> &amp;  <a href="gtarski1.html#R1" title="GTARSKI1:pred.1">between</a>  <a href="gtarski3.html#K3" title="GTARSKI3:func.3">reflection</a> (<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c4">c</font>),<font color="Maroon" title="c6">p</font>, <a href="gtarski3.html#K3" title="GTARSKI3:func.3">reflection</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font>) )</span></a><div class="add"><a NAME="E1:45_1"/>
 <a href="gtarski3.html#R9" title="GTARSKI3:pred.9">Middle</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">x</font>, <a href="gtarski3.html#K3" title="GTARSKI3:func.3">reflection</a> (<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c4">c</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="gtarski3.html#D13" onmouseover="rs('gtarski3/D13')" onmouseout="rh()">GTARSKI3:def 13</a></span>;<br/><span class="kw">hence </span><a NAME="E2:45_1"/>
 <a href="gtarski1.html#R1" title="GTARSKI1:pred.1">between</a>  <a href="gtarski3.html#K3" title="GTARSKI3:func.3">reflection</a> (<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c4">c</font>),<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c4">c</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="gtarski3.html#T14" onmouseover="rs('gtarski3/T14')" onmouseout="rh()">GTARSKI3:14</a>, <a class="ref" href="gtarski3.html#D12" onmouseover="rs('gtarski3/D12')" onmouseout="rh()">GTARSKI3:def 12</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"> (  <a href="gtarski1.html#R1" title="GTARSKI1:pred.1">between</a>  <a href="gtarski3.html#K3" title="GTARSKI3:func.3">reflection</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font>),<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font> &amp;  <a href="gtarski1.html#R1" title="GTARSKI1:pred.1">between</a>  <a href="gtarski3.html#K3" title="GTARSKI3:func.3">reflection</a> (<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c4">c</font>),<font color="Maroon" title="c6">p</font>, <a href="gtarski3.html#K3" title="GTARSKI3:func.3">reflection</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font>) )</span><br/><a NAME="E3:45_1"/>
 <a href="gtarski3.html#R9" title="GTARSKI3:pred.9">Middle</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c2">a</font>, <a href="gtarski3.html#K3" title="GTARSKI3:func.3">reflection</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="gtarski3.html#D13" onmouseover="rs('gtarski3/D13')" onmouseout="rh()">GTARSKI3:def 13</a></span>;<br/><span class="kw">hence </span><a NAME="E4:45_1"/>
 <a href="gtarski1.html#R1" title="GTARSKI1:pred.1">between</a>  <a href="gtarski3.html#K3" title="GTARSKI3:func.3">reflection</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font>),<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="gtarski3.html#T14" onmouseover="rs('gtarski3/T14')" onmouseout="rh()">GTARSKI3:14</a>, <a class="ref" href="gtarski3.html#D12" onmouseover="rs('gtarski3/D12')" onmouseout="rh()">GTARSKI3:def 12</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">  <a href="gtarski1.html#R1" title="GTARSKI1:pred.1">between</a>  <a href="gtarski3.html#K3" title="GTARSKI3:func.3">reflection</a> (<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c4">c</font>),<font color="Maroon" title="c6">p</font>, <a href="gtarski3.html#K3" title="GTARSKI3:func.3">reflection</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font>)</span><br/><span class="kw">thus </span><a NAME="E5:45_1"/>
 <a href="gtarski1.html#R1" title="GTARSKI1:pred.1">between</a>  <a href="gtarski3.html#K3" title="GTARSKI3:func.3">reflection</a> (<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c4">c</font>),<font color="Maroon" title="c6">p</font>, <a href="gtarski3.html#K3" title="GTARSKI3:func.3">reflection</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:45"><span class="lab"><font color="Green" title="E41">A7</font></span></a>, <a class="ref" href="gtarski3.html#D12" onmouseover="rs('gtarski3/D12')" onmouseout="rh()">GTARSKI3:def 12</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>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">t</font> being   <a href="gtarski1.html#NM1" title="GTARSKI1:NM.1">POINT</a> of <font color="Maroon" title="c1">S</font><span class="kw"> such that </span><br/><a NAME="E20:45"/><span class="lab"><font color="Green" title="E45">A11</font></span>: 
 <a href="gtarski1.html#R1" title="GTARSKI1:pred.1">between</a> <font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c7">t</font>,<font color="Maroon" title="c4">c</font>
 <span class="kw">and </span><br/><a NAME="E21:45"/><span class="lab"><font color="Green" title="E46">A12</font></span>: 
 <a href="gtarski1.html#R1" title="GTARSKI1:pred.1">between</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c7">t</font>,<font color="Maroon" title="c2">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="gtarski3.html#T40" onmouseover="rs('gtarski3/T40')" onmouseout="rh()">GTARSKI3:40</a></span>;<br/>
<a NAME="E22:45"/>
 <a href="gtarski1.html#R6" title="GTARSKI1:pred.6">Collinear</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c7">t</font>,<font color="Maroon" title="c2">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:45"><span class="lab"><font color="Green" title="E46">A12</font></span></a>, <a class="ref" href="gtarski1.html#D17" onmouseover="rs('gtarski1/D17')" onmouseout="rh()">GTARSKI1:def 17</a></span>;<br/>
<a NAME="E23:45"/><span class="kw">then </span>
 <a href="gtarski1.html#R6" title="GTARSKI1:pred.6">Collinear</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c7">t</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>;<br/>
<a NAME="E24:45"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A13</font></span>: 
<font color="Maroon" title="c7">t</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="c5">x</font>,<font color="Maroon" title="c2">a</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="gtarski4.html#E1"><span class="lab"><font color="Green" title="E1">LemmaA1</font></span></a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_2"><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="E1:45_2"/>
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">a</font> or <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_2_1"><span class="kw">suppose </span></a><a NAME="E1:45_2_1"/><span class="lab"><font color="Green" title="E48">A16</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">a</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">  ex <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">t</font> being   <a href="gtarski1.html#NM1" title="GTARSKI1:NM.1">POINT</a> of <font color="Maroon" title="c1">S</font> st <br/>(  <a href="gtarski4.html#R6" title="GTARSKI4:pred.6">are_orthogonal</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c2">a</font> &amp;  <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="Olive" title="b2">t</font> &amp;  <a href="gtarski1.html#R1" title="GTARSKI1:pred.1">between</a> <font color="Maroon" title="c4">c</font>,<font color="Olive" title="b2">t</font>,<font color="Olive" title="b1">p</font> )</span><br/><div class="add"><a NAME="E2:45_2_1"/><span class="kw">then </span>
 <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="c5">x</font>,<font color="Maroon" title="c2">a</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:45"><span class="lab"><font color="Green" title="E36">A2</font></span></a>, <a class="txt" href="#E4:45"><span class="lab"><font color="Green" title="E37">A3</font></span></a>, <a class="txt" href="gtarski4.html#E1"><span class="lab"><font color="Green" title="E1">LemmaA1</font></span></a>, <a class="ref" href="gtarski3.html#T82" onmouseover="rs('gtarski3/T82')" onmouseout="rh()">GTARSKI3:82</a></span>;<br/><a NAME="E3:45_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">a18</font></span>: 
 <a href="gtarski1.html#R6" title="GTARSKI1:pred.6">Collinear</a> <font color="Maroon" title="c7">t</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="gtarski4.html#E2"><span class="lab"><font color="Green" title="E2">LemmaA2</font></span></a>, <a class="txt" href="#E24:45"><span class="lab"><font color="Green" title="E47">A13</font></span></a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c6">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">  ex <font color="Olive" title="b1">t</font> being   <a href="gtarski1.html#NM1" title="GTARSKI1:NM.1">POINT</a> of <font color="Maroon" title="c1">S</font> st <br/>(  <a href="gtarski4.html#R6" title="GTARSKI4:pred.6">are_orthogonal</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c2">a</font> &amp;  <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="Olive" title="b1">t</font> &amp;  <a href="gtarski1.html#R1" title="GTARSKI1:pred.1">between</a> <font color="Maroon" title="c4">c</font>,<font color="Olive" title="b1">t</font>,<font color="Maroon" title="c6">p</font> )</span><br/><span class="kw">take </span>
<font color="Maroon" title="c7">t</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#R6" title="GTARSKI4:pred.6">are_orthogonal</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c2">a</font> &amp;  <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="c7">t</font> &amp;  <a href="gtarski1.html#R1" title="GTARSKI1:pred.1">between</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c7">t</font>,<font color="Maroon" title="c6">p</font> )</span><br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="45_2_1_1"><span class="kw">now </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"> ( <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> &amp; <font color="Maroon" title="c6">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">a</font> &amp;  <a href="gtarski4.html#R3" title="GTARSKI4:pred.3">are_orthogonal</a>  <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="gtarski4.html#K1" title="GTARSKI4:func.1">Line</a> (<font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c2">a</font>) )</span></a><div class="add"><span class="kw">thus </span><a NAME="E1:45_2_1_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45"><span class="lab"><font color="Green" title="E35">A1</font></span></a>, <a class="ref" href="gtarski3.html#T46" onmouseover="rs('gtarski3/T46')" onmouseout="rh()">GTARSKI3:46</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"> ( <font color="Maroon" title="c6">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">a</font> &amp;  <a href="gtarski4.html#R3" title="GTARSKI4:pred.3">are_orthogonal</a>  <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="gtarski4.html#K1" title="GTARSKI4:func.1">Line</a> (<font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c2">a</font>) )</span><br/><span class="kw">thus </span><a NAME="E2:45_2_1_1"/>
<font color="Maroon" title="c6">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:45"><span class="lab"><font color="Green" title="E43">A9</font></span></a>, <a class="txt" href="#E1:45"><span class="lab"><font color="Green" title="E35">A1</font></span></a>, <a class="txt" href="#E4:45"><span class="lab"><font color="Green" title="E37">A3</font></span></a>, <a class="txt" href="#E15:45"><span class="lab"><font color="Green" title="E42">A8</font></span></a>, <a class="ref" href="gtarski4.html#T34" target="_self" onmouseover="rs('gtarski4/T34')" onmouseout="rh()">Lemma8p20</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">  <a href="gtarski4.html#R3" title="GTARSKI4:pred.3">are_orthogonal</a>  <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="gtarski4.html#K1" title="GTARSKI4:func.1">Line</a> (<font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c2">a</font>)</span><br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="45_2_1_1_1"><span class="kw">now </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"> (  <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="gtarski3.html#R8" title="GTARSKI3:pred.8">is_line</a>  &amp;  <a href="gtarski4.html#K1" title="GTARSKI4:func.1">Line</a> (<font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c2">a</font>) <a href="gtarski3.html#R8" title="GTARSKI3:pred.8">is_line</a>  &amp; <font color="Maroon" title="c2">a</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="c3">b</font>) &amp; <font color="Maroon" title="c2">a</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="c6">p</font>,<font color="Maroon" title="c2">a</font>) &amp;  ex <font color="Olive" title="b1">u</font>, <font color="Olive" title="b2">v</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="Olive" title="b1">u</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="c3">b</font>) &amp; <font color="Olive" title="b2">v</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="c6">p</font>,<font color="Maroon" title="c2">a</font>) &amp; <font color="Olive" title="b1">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Olive" title="b2">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">a</font> &amp;  <a href="gtarski4.html#R1" title="GTARSKI4:pred.1">right_angle</a> <font color="Olive" title="b1">u</font>,<font color="Maroon" title="c2">a</font>,<font color="Olive" title="b2">v</font> ) )</span></a><div class="add"><span class="kw">thus </span><a NAME="E1:45_2_1_1_1"/>
 <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="gtarski3.html#R8" title="GTARSKI3:pred.8">is_line</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:45"><span class="lab"><font color="Green" title="E36">A2</font></span></a>, <a class="ref" href="gtarski3.html#D11" onmouseover="rs('gtarski3/D11')" onmouseout="rh()">GTARSKI3:def 11</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"> (  <a href="gtarski4.html#K1" title="GTARSKI4:func.1">Line</a> (<font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c2">a</font>) <a href="gtarski3.html#R8" title="GTARSKI3:pred.8">is_line</a>  &amp; <font color="Maroon" title="c2">a</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="c3">b</font>) &amp; <font color="Maroon" title="c2">a</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="c6">p</font>,<font color="Maroon" title="c2">a</font>) &amp;  ex <font color="Olive" title="b1">u</font>, <font color="Olive" title="b2">v</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="Olive" title="b1">u</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="c3">b</font>) &amp; <font color="Olive" title="b2">v</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="c6">p</font>,<font color="Maroon" title="c2">a</font>) &amp; <font color="Olive" title="b1">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Olive" title="b2">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">a</font> &amp;  <a href="gtarski4.html#R1" title="GTARSKI4:pred.1">right_angle</a> <font color="Olive" title="b1">u</font>,<font color="Maroon" title="c2">a</font>,<font color="Olive" title="b2">v</font> ) )</span><br/><span class="kw">thus </span><a NAME="E2:45_2_1_1_1"/>
 <a href="gtarski4.html#K1" title="GTARSKI4:func.1">Line</a> (<font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c2">a</font>) <a href="gtarski3.html#R8" title="GTARSKI3:pred.8">is_line</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:45"><span class="lab"><font color="Green" title="E44">A10</font></span></a>, <a class="ref" href="gtarski3.html#D11" onmouseover="rs('gtarski3/D11')" onmouseout="rh()">GTARSKI3:def 11</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"> ( <font color="Maroon" title="c2">a</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="c3">b</font>) &amp; <font color="Maroon" title="c2">a</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="c6">p</font>,<font color="Maroon" title="c2">a</font>) &amp;  ex <font color="Olive" title="b1">u</font>, <font color="Olive" title="b2">v</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="Olive" title="b1">u</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="c3">b</font>) &amp; <font color="Olive" title="b2">v</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="c6">p</font>,<font color="Maroon" title="c2">a</font>) &amp; <font color="Olive" title="b1">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Olive" title="b2">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">a</font> &amp;  <a href="gtarski4.html#R1" title="GTARSKI4:pred.1">right_angle</a> <font color="Olive" title="b1">u</font>,<font color="Maroon" title="c2">a</font>,<font color="Olive" title="b2">v</font> ) )</span><br/><span class="kw">thus </span><a NAME="E3:45_2_1_1_1"/>
<font color="Maroon" title="c2">a</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="c3">b</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="gtarski3.html#T83" onmouseover="rs('gtarski3/T83')" onmouseout="rh()">GTARSKI3:83</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"> ( <font color="Maroon" title="c2">a</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="c6">p</font>,<font color="Maroon" title="c2">a</font>) &amp;  ex <font color="Olive" title="b1">u</font>, <font color="Olive" title="b2">v</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="Olive" title="b1">u</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="c3">b</font>) &amp; <font color="Olive" title="b2">v</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="c6">p</font>,<font color="Maroon" title="c2">a</font>) &amp; <font color="Olive" title="b1">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Olive" title="b2">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">a</font> &amp;  <a href="gtarski4.html#R1" title="GTARSKI4:pred.1">right_angle</a> <font color="Olive" title="b1">u</font>,<font color="Maroon" title="c2">a</font>,<font color="Olive" title="b2">v</font> ) )</span><br/><span class="kw">thus </span><a NAME="E4:45_2_1_1_1"/>
<font color="Maroon" title="c2">a</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="c6">p</font>,<font color="Maroon" title="c2">a</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="gtarski3.html#T83" onmouseover="rs('gtarski3/T83')" onmouseout="rh()">GTARSKI3:83</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">  ex <font color="Olive" title="b1">u</font>, <font color="Olive" title="b2">v</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="Olive" title="b1">u</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="c3">b</font>) &amp; <font color="Olive" title="b2">v</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="c6">p</font>,<font color="Maroon" title="c2">a</font>) &amp; <font color="Olive" title="b1">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Olive" title="b2">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">a</font> &amp;  <a href="gtarski4.html#R1" title="GTARSKI4:pred.1">right_angle</a> <font color="Olive" title="b1">u</font>,<font color="Maroon" title="c2">a</font>,<font color="Olive" title="b2">v</font> )</span><br/><span class="kw">thus </span><a NAME="E5:45_2_1_1_1"/>
 ex <font color="Olive" title="b1">u</font>, <font color="Olive" title="b2">v</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="Olive" title="b1">u</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="c3">b</font>) &amp; <font color="Olive" title="b2">v</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="c6">p</font>,<font color="Maroon" title="c2">a</font>) &amp; <font color="Olive" title="b1">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Olive" title="b2">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">a</font> &amp;  <a href="gtarski4.html#R1" title="GTARSKI4:pred.1">right_angle</a> <font color="Olive" title="b1">u</font>,<font color="Maroon" title="c2">a</font>,<font color="Olive" title="b2">v</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"> verum</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="45_2_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">take </span>
<font color="Maroon" title="c5">x</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">  ex <font color="Olive" title="b1">v</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="c5">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="c3">b</font>) &amp; <font color="Olive" title="b1">v</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="c6">p</font>,<font color="Maroon" title="c2">a</font>) &amp; <font color="Maroon" title="c5">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Olive" title="b1">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">a</font> &amp;  <a href="gtarski4.html#R1" title="GTARSKI4:pred.1">right_angle</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c2">a</font>,<font color="Olive" title="b1">v</font> )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c6">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"> ( <font color="Maroon" title="c5">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="c3">b</font>) &amp; <font color="Maroon" title="c6">p</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="c6">p</font>,<font color="Maroon" title="c2">a</font>) &amp; <font color="Maroon" title="c5">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Maroon" title="c6">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">a</font> &amp;  <a href="gtarski4.html#R1" title="GTARSKI4:pred.1">right_angle</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c6">p</font> )</span><br/>

<span class="kw">thus </span><a NAME="E1:45_2_1_1_1_1"/>
<font color="Maroon" title="c5">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="c3">b</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:45"><span class="lab"><font color="Green" title="E37">A3</font></span></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"> ( <font color="Maroon" title="c6">p</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="c6">p</font>,<font color="Maroon" title="c2">a</font>) &amp; <font color="Maroon" title="c5">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Maroon" title="c6">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">a</font> &amp;  <a href="gtarski4.html#R1" title="GTARSKI4:pred.1">right_angle</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c6">p</font> )</span><br/>

<span class="kw">thus </span><a NAME="E2:45_2_1_1_1_1"/>
<font color="Maroon" title="c6">p</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="c6">p</font>,<font color="Maroon" title="c2">a</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="gtarski3.html#T83" onmouseover="rs('gtarski3/T83')" onmouseout="rh()">GTARSKI3:83</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"> ( <font color="Maroon" title="c5">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Maroon" title="c6">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">a</font> &amp;  <a href="gtarski4.html#R1" title="GTARSKI4:pred.1">right_angle</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c6">p</font> )</span><br/>

<span class="kw">thus </span><a NAME="E3:45_2_1_1_1_1"/>
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Maroon" title="c6">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">a</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45_2_1"><span class="lab"><font color="Green" title="E48">A16</font></span></a>, <a class="txt" href="#E16:45"><span class="lab"><font color="Green" title="E43">A9</font></span></a>, <a class="txt" href="#E1:45"><span class="lab"><font color="Green" title="E35">A1</font></span></a>, <a class="txt" href="#E4:45"><span class="lab"><font color="Green" title="E37">A3</font></span></a>, <a class="txt" href="#E15:45"><span class="lab"><font color="Green" title="E42">A8</font></span></a>, <a class="ref" href="gtarski4.html#T34" target="_self" onmouseover="rs('gtarski4/T34')" onmouseout="rh()">Lemma8p20</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">  <a href="gtarski4.html#R1" title="GTARSKI4:pred.1">right_angle</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c6">p</font></span><br/>

<span class="kw">thus </span><a NAME="E4:45_2_1_1_1_1"/>
 <a href="gtarski4.html#R1" title="GTARSKI4:pred.1">right_angle</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c6">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:45"><span class="lab"><font color="Green" title="E43">A9</font></span></a>, <a class="txt" href="#E15:45"><span class="lab"><font color="Green" title="E42">A8</font></span></a>, <a class="ref" href="gtarski4.html#T34" target="_self" onmouseover="rs('gtarski4/T34')" onmouseout="rh()">Lemma8p20</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><span class="kw">hence </span><a NAME="E4:45_2_1_1"/>
 <a href="gtarski4.html#R3" title="GTARSKI4:pred.3">are_orthogonal</a>  <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="gtarski4.html#K1" title="GTARSKI4:func.1">Line</a> (<font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c2">a</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="gtarski4.html#T24" target="_self" onmouseover="rs('gtarski4/T24')" onmouseout="rh()">Satz8p13</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><span class="kw">hence </span><a NAME="E5:45_2_1"/>
(  <a href="gtarski4.html#R6" title="GTARSKI4:pred.6">are_orthogonal</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c2">a</font> &amp;  <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="c7">t</font> &amp;  <a href="gtarski1.html#R1" title="GTARSKI1:pred.1">between</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c7">t</font>,<font color="Maroon" title="c6">p</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:45_2_1"><span class="lab"><font color="Green" title="E49">a18</font></span></a>, <a class="txt" href="#E20:45"><span class="lab"><font color="Green" title="E45">A11</font></span></a>, <a class="ref" href="gtarski3.html#T14" onmouseover="rs('gtarski3/T14')" onmouseout="rh()">GTARSKI3:14</a>, <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/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_2_2"><span class="kw">suppose </span></a><a NAME="E1:45_2_2"/><span class="lab"><font color="Green" title="E48">A19</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</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">  ex <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">t</font> being   <a href="gtarski1.html#NM1" title="GTARSKI1:NM.1">POINT</a> of <font color="Maroon" title="c1">S</font> st <br/>(  <a href="gtarski4.html#R6" title="GTARSKI4:pred.6">are_orthogonal</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c2">a</font> &amp;  <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="Olive" title="b2">t</font> &amp;  <a href="gtarski1.html#R1" title="GTARSKI1:pred.1">between</a> <font color="Maroon" title="c4">c</font>,<font color="Olive" title="b2">t</font>,<font color="Olive" title="b1">p</font> )</span><br/><div class="add"><a NAME="E2:45_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A20</font></span>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">t</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:45"><span class="lab"><font color="Green" title="E46">A12</font></span></a>, <a class="ref" href="gtarski1.html#D10" onmouseover="rs('gtarski1/D10')" onmouseout="rh()">GTARSKI1:def 10</a></span>;<br/><div><span class="lab"><font color="Green" title="E50">a21</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="45_2_2_1"><span class="kw">now </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"> ( <font color="Maroon" title="c6">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Maroon" title="c4">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Maroon" title="c4">c</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="c6">p</font>) )</span></a><div class="add"><span class="kw">thus </span><a NAME="E1:45_2_2_1"/>
<font color="Maroon" title="c6">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:45"><span class="lab"><font color="Green" title="E43">A9</font></span></a>, <a class="txt" href="#E1:45"><span class="lab"><font color="Green" title="E35">A1</font></span></a>, <a class="txt" href="#E4:45"><span class="lab"><font color="Green" title="E37">A3</font></span></a>, <a class="txt" href="#E15:45"><span class="lab"><font color="Green" title="E42">A8</font></span></a>, <a class="ref" href="gtarski4.html#T34" target="_self" onmouseover="rs('gtarski4/T34')" onmouseout="rh()">Lemma8p20</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"> ( <font color="Maroon" title="c4">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Maroon" title="c4">c</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="c6">p</font>) )</span><br/><span class="kw">thus </span><a NAME="E2:45_2_2_1"/>
<font color="Maroon" title="c4">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">a</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="c4">c</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="c6">p</font>)</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="45_2_2_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:45_2_2_1_1"/>
<font color="Maroon" title="c4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</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"> contradiction</span><br/>

<a NAME="E2:45_2_2_1_1"/><span class="kw">then </span>
 <a href="gtarski1.html#R6" title="GTARSKI1:pred.6">Collinear</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c3">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="gtarski3.html#T46" onmouseover="rs('gtarski3/T46')" onmouseout="rh()">GTARSKI3:46</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:45_2_2_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45"><span class="lab"><font color="Green" title="E35">A1</font></span></a>, <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/>


</div><span class="kw">end;</span></div><a NAME="E3:45_2_2_1"/>
 <a href="gtarski1.html#R6" title="GTARSKI1:pred.6">Collinear</a> <font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:45_2_2"><span class="lab"><font color="Green" title="E49">A20</font></span></a>, <a class="txt" href="#E20:45"><span class="lab"><font color="Green" title="E45">A11</font></span></a>, <a class="ref" href="gtarski1.html#D17" onmouseover="rs('gtarski1/D17')" onmouseout="rh()">GTARSKI1:def 17</a></span>;<br/><span class="kw">hence </span><a NAME="E4:45_2_2_1"/>
<font color="Maroon" title="c4">c</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="c6">p</font>)
 <span class="kw">by</span> <span class="lab"><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><span class="kw">take </span>
<font color="Maroon" title="c6">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">  ex <font color="Olive" title="b1">t</font> being   <a href="gtarski1.html#NM1" title="GTARSKI1:NM.1">POINT</a> of <font color="Maroon" title="c1">S</font> st <br/>(  <a href="gtarski4.html#R6" title="GTARSKI4:pred.6">are_orthogonal</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c2">a</font> &amp;  <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="Olive" title="b1">t</font> &amp;  <a href="gtarski1.html#R1" title="GTARSKI1:pred.1">between</a> <font color="Maroon" title="c4">c</font>,<font color="Olive" title="b1">t</font>,<font color="Maroon" title="c6">p</font> )</span><br/><span class="kw">take </span>
<font color="Maroon" title="c7">t</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#R6" title="GTARSKI4:pred.6">are_orthogonal</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c2">a</font> &amp;  <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="c7">t</font> &amp;  <a href="gtarski1.html#R1" title="GTARSKI1:pred.1">between</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c7">t</font>,<font color="Maroon" title="c6">p</font> )</span><br/><a NAME="E4:45_2_2"/>
(  <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="c2">a</font> &amp;  <a href="gtarski4.html#R6" title="GTARSKI4:pred.6">are_orthogonal</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c2">a</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:45"><span class="lab"><font color="Green" title="E37">A3</font></span></a>, <a class="txt" href="#E1:45_2_2"><span class="lab"><font color="Green" title="E48">A19</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E5:45_2_2"/>
(  <a href="gtarski4.html#R6" title="GTARSKI4:pred.6">are_orthogonal</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c2">a</font> &amp;  <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="c7">t</font> &amp;  <a href="gtarski1.html#R1" title="GTARSKI1:pred.1">between</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c7">t</font>,<font color="Maroon" title="c6">p</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:45_2_2"><span class="lab"><font color="Green" title="E50">a21</font></span></a>, <a class="txt" href="#E20:45"><span class="lab"><font color="Green" title="E45">A11</font></span></a>, <a class="ref" href="gtarski3.html#T14" onmouseover="rs('gtarski3/T14')" onmouseout="rh()">GTARSKI3:14</a>, <a class="ref" href="gtarski3.html#T82" onmouseover="rs('gtarski3/T82')" onmouseout="rh()">GTARSKI3:82</a>, <a class="txt" href="#E1:45_2_2"><span class="lab"><font color="Green" title="E48">A19</font></span></a>, <a class="txt" href="#E21:45"><span class="lab"><font color="Green" title="E46">A12</font></span></a>, <a class="ref" href="gtarski1.html#D10" onmouseover="rs('gtarski1/D10')" onmouseout="rh()">GTARSKI1:def 10</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>
