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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">x</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="b4">x</font> <a href="gtarski4.html#R8" title="GTARSKI4:pred.8">is_foot</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</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">x</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">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><br/>







<span class="kw">assume </span><a NAME="E1:43"/><span class="lab"><font color="Green" title="E33">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">x</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">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><br/>

<span class="kw">consider </span><font color="Maroon" title="c5">y</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="E3:43"/><span class="lab"><font color="Green" title="E34">A2</font></span>: 
 <a href="gtarski1.html#R1" title="GTARSKI1:pred.1">between</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">y</font>
 <span class="kw">and </span><br/><a NAME="E4:43"/><span class="lab"><font color="Green" title="E35">A3</font></span>: 
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">y</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="gtarski1.html#D8" onmouseover="rs('gtarski1/D8')" onmouseout="rh()">GTARSKI1:def 8</a></span>;<br/>
<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="E6:43"/><span class="lab"><font color="Green" title="E36">A4</font></span>: 
 <a href="gtarski3.html#R9" title="GTARSKI3:pred.9">Middle</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c4">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:43"><span class="lab"><font color="Green" title="E35">A3</font></span></a>, <a class="ref" href="gtarski3.html#T117" onmouseover="rs('gtarski3/T117')" onmouseout="rh()">GTARSKI3:117</a></span>;<br/>
<a NAME="E7:43"/><span class="lab"><font color="Green" title="E37">A5</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="c6">p</font>,<font color="Maroon" title="c5">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:43"><span class="lab"><font color="Green" title="E35">A3</font></span></a>, <a class="txt" href="#E6:43"><span class="lab"><font color="Green" title="E36">A4</font></span></a>, <a class="ref" href="gtarski3.html#D13" onmouseover="rs('gtarski3/D13')" onmouseout="rh()">GTARSKI3:def 13</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">z</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="E9:43"/><span class="lab"><font color="Green" title="E38">A6</font></span>: 
 <a href="gtarski1.html#R1" title="GTARSKI1:pred.1">between</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c7">z</font>
 <span class="kw">and </span><br/><a NAME="E10:43"/><span class="lab"><font color="Green" title="E39">A7</font></span>: 
<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c7">z</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">p</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="gtarski1.html#D8" onmouseover="rs('gtarski1/D8')" onmouseout="rh()">GTARSKI1:def 8</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c8">q</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="E12:43"/><span class="lab"><font color="Green" title="E40">A8</font></span>: 
 <a href="gtarski1.html#R1" title="GTARSKI1:pred.1">between</a> <font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c8">q</font>
 <span class="kw">and </span><br/><a NAME="E13:43"/><span class="lab"><font color="Green" title="E41">A9</font></span>: 
<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c8">q</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c2">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="gtarski1.html#D8" onmouseover="rs('gtarski1/D8')" onmouseout="rh()">GTARSKI1:def 8</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c9">q9</font> =  <a href="gtarski3.html#K3" title="GTARSKI3:func.3">reflection</a> (<font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c8">q</font>);<br/>
<span class="kw">consider </span><font color="Maroon" title="c10">c9</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="E15:43"/><span class="lab"><font color="Green" title="E42">A10</font></span>: 
 <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="c7">z</font>,<font color="Maroon" title="c8">q</font>),<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c10">c9</font>
 <span class="kw">and </span><br/><a NAME="E16:43"/><span class="lab"><font color="Green" title="E43">A11</font></span>: 
<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c10">c9</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c4">c</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="gtarski1.html#D8" onmouseover="rs('gtarski1/D8')" onmouseout="rh()">GTARSKI1:def 8</a></span>;<br/>
<a NAME="E17:43"/><span class="lab"><font color="Green" title="E44">A12</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="txt" href="#E1:43"><span class="lab"><font color="Green" title="E33">A1</font></span></a>, <a class="ref" href="gtarski3.html#T46" onmouseover="rs('gtarski3/T46')" onmouseout="rh()">GTARSKI3:46</a></span>;<br/>
<a NAME="E18:43"/><span class="lab"><font color="Green" title="E45">A13</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="c5">y</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="43_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:43_1"/>
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">y</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="E2:43_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:43"><span class="lab"><font color="Green" title="E35">A3</font></span></a>, <a class="ref" href="gtarski1.html#D7" onmouseover="rs('gtarski1/D7')" onmouseout="rh()">GTARSKI1:def 7</a>, <a class="ref" href="gtarski3.html#T4" onmouseover="rs('gtarski3/T4')" onmouseout="rh()">GTARSKI3:4</a></span>;<br/>
<a NAME="E3:43_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="E4:43_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43"><span class="lab"><font color="Green" title="E33">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>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="43_2"><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> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c7">z</font> &amp;  <a href="gtarski1.html#R1" title="GTARSKI1:pred.1">between</a> <font color="Maroon" title="c8">q</font>,<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">p</font> &amp; <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">y</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c8">q</font>,<font color="Maroon" title="c5">y</font> &amp; <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c7">z</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">p</font> &amp; <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c8">q</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c8">q</font>,<font color="Maroon" title="c2">a</font> &amp; <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c8">q</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c2">a</font> )</span></a><div class="add"><span class="kw">thus </span><a NAME="E1:43_2"/>
 <a href="gtarski1.html#R1" title="GTARSKI1:pred.1">between</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c7">z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:43"><span class="lab"><font color="Green" title="E38">A6</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"> (  <a href="gtarski1.html#R1" title="GTARSKI1:pred.1">between</a> <font color="Maroon" title="c8">q</font>,<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">p</font> &amp; <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">y</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c8">q</font>,<font color="Maroon" title="c5">y</font> &amp; <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c7">z</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">p</font> &amp; <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c8">q</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c8">q</font>,<font color="Maroon" title="c2">a</font> &amp; <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c8">q</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c2">a</font> )</span><br/><span class="kw">thus </span><a NAME="E2:43_2"/>
 <a href="gtarski1.html#R1" title="GTARSKI1:pred.1">between</a> <font color="Maroon" title="c8">q</font>,<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:43"><span class="lab"><font color="Green" title="E40">A8</font></span></a>, <a class="ref" href="gtarski3.html#T14" onmouseover="rs('gtarski3/T14')" onmouseout="rh()">GTARSKI3:14</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>,<font color="Maroon" title="c5">y</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c8">q</font>,<font color="Maroon" title="c5">y</font> &amp; <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c7">z</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">p</font> &amp; <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c8">q</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c8">q</font>,<font color="Maroon" title="c2">a</font> &amp; <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c8">q</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c2">a</font> )</span><br/><a NAME="E3:43_2"/>
<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c2">a</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c8">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:43"><span class="lab"><font color="Green" title="E41">A9</font></span></a>, <a class="ref" href="gtarski3.html#T3" onmouseover="rs('gtarski3/T3')" onmouseout="rh()">GTARSKI3:3</a></span>;<br/><a NAME="E4:43_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">y</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c8">q</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="gtarski3.html#T6" onmouseover="rs('gtarski3/T6')" onmouseout="rh()">GTARSKI3:6</a></span>;<br/><span class="kw">hence </span><a NAME="E5:43_2"/>
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">y</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c8">q</font>,<font color="Maroon" title="c5">y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="gtarski3.html#T7" onmouseover="rs('gtarski3/T7')" onmouseout="rh()">GTARSKI3:7</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">y</font>,<font color="Maroon" title="c7">z</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">p</font> &amp; <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c8">q</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c8">q</font>,<font color="Maroon" title="c2">a</font> &amp; <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c8">q</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c2">a</font> )</span><br/><span class="kw">thus </span><a NAME="E6:43_2"/>
<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c7">z</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:43"><span class="lab"><font color="Green" title="E39">A7</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="c2">a</font>,<font color="Maroon" title="c8">q</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c8">q</font>,<font color="Maroon" title="c2">a</font> &amp; <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c8">q</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c2">a</font> )</span><br/><span class="kw">thus </span><a NAME="E7:43_2"/>
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c8">q</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c8">q</font>,<font color="Maroon" title="c2">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="gtarski3.html#T2" onmouseover="rs('gtarski3/T2')" onmouseout="rh()">GTARSKI3:2</a>, <a class="ref" href="gtarski3.html#T6" onmouseover="rs('gtarski3/T6')" onmouseout="rh()">GTARSKI3:6</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">y</font>,<font color="Maroon" title="c8">q</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c2">a</font></span><br/><span class="kw">thus </span><a NAME="E8:43_2"/>
<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c8">q</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c2">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:43"><span class="lab"><font color="Green" title="E41">A9</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>
<a NAME="E20:43"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A14</font></span>: 
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c8">q</font> <a href="gtarski3.html#R1" title="GTARSKI3:pred.1">AFS</a> <font color="Maroon" title="c8">q</font>,<font color="Maroon" title="c5">y</font>,<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#D2" onmouseover="rs('gtarski3/D2')" onmouseout="rh()">GTARSKI3:def 2</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="43_3"><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>,<font color="Maroon" title="c6">p</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c8">q</font>,<font color="Maroon" title="c7">z</font> &amp; <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">y</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c8">q</font>,<font color="Maroon" title="c5">y</font> &amp; <font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c5">y</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c5">y</font> )</span></a><div class="add"><a NAME="E1:43_3"/>
<font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c2">a</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c8">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:43"><span class="lab"><font color="Green" title="E46">A14</font></span></a>, <a class="txt" href="#E18:43"><span class="lab"><font color="Green" title="E45">A13</font></span></a>, <a class="ref" href="gtarski3.html#T3" onmouseover="rs('gtarski3/T3')" onmouseout="rh()">GTARSKI3:3</a>, <a class="ref" href="gtarski3.html#T10" onmouseover="rs('gtarski3/T10')" onmouseout="rh()">GTARSKI3:10</a></span>;<br/><a NAME="E2:43_3"/><span class="kw">then </span>
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c6">p</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c8">q</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="gtarski3.html#T6" onmouseover="rs('gtarski3/T6')" onmouseout="rh()">GTARSKI3:6</a></span>;<br/><span class="kw">hence </span><a NAME="E3:43_3"/>
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c6">p</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c8">q</font>,<font color="Maroon" title="c7">z</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="gtarski3.html#T7" onmouseover="rs('gtarski3/T7')" onmouseout="rh()">GTARSKI3:7</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>,<font color="Maroon" title="c5">y</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c8">q</font>,<font color="Maroon" title="c5">y</font> &amp; <font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c5">y</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c5">y</font> )</span><br/><a NAME="E4:43_3"/>
<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c2">a</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c8">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:43"><span class="lab"><font color="Green" title="E41">A9</font></span></a>, <a class="ref" href="gtarski3.html#T3" onmouseover="rs('gtarski3/T3')" onmouseout="rh()">GTARSKI3:3</a></span>;<br/><a NAME="E5:43_3"/><span class="kw">then </span>
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">y</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c8">q</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="gtarski3.html#T6" onmouseover="rs('gtarski3/T6')" onmouseout="rh()">GTARSKI3:6</a></span>;<br/><span class="kw">hence </span><a NAME="E6:43_3"/>
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">y</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c8">q</font>,<font color="Maroon" title="c5">y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="gtarski3.html#T7" onmouseover="rs('gtarski3/T7')" onmouseout="rh()">GTARSKI3:7</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>,<font color="Maroon" title="c5">y</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c5">y</font></span><br/><a NAME="E7:43_3"/>
<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">p</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c7">z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:43"><span class="lab"><font color="Green" title="E39">A7</font></span></a>, <a class="ref" href="gtarski3.html#T4" onmouseover="rs('gtarski3/T4')" onmouseout="rh()">GTARSKI3:4</a></span>;<br/><a NAME="E8:43_3"/><span class="kw">then </span>
<font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c5">y</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c7">z</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="gtarski3.html#T6" onmouseover="rs('gtarski3/T6')" onmouseout="rh()">GTARSKI3:6</a></span>;<br/><span class="kw">hence </span><a NAME="E9:43_3"/>
<font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c5">y</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c5">y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="gtarski3.html#T7" onmouseover="rs('gtarski3/T7')" onmouseout="rh()">GTARSKI3:7</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="E22:43"/><span class="kw">then </span>
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c5">y</font> <a href="gtarski1.html#R3" title="GTARSKI1:pred.3">cong</a> <font color="Maroon" title="c8">q</font>,<font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c5">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="gtarski1.html#D3" onmouseover="rs('gtarski1/D3')" onmouseout="rh()">GTARSKI1:def 3</a></span>;<br/>
<a NAME="E23:43"/><span class="kw">then </span>
 <a href="gtarski4.html#R1" title="GTARSKI4:pred.1">right_angle</a> <font color="Maroon" title="c8">q</font>,<font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c5">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:43"><span class="lab"><font color="Green" title="E37">A5</font></span></a>, <a class="ref" href="gtarski4.html#T22" target="_self" onmouseover="rs('gtarski4/T22')" onmouseout="rh()">Satz8p10</a></span>;<br/>
<a NAME="E24:43"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A15</font></span>: 
 <a href="gtarski4.html#R1" title="GTARSKI4:pred.1">right_angle</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c8">q</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="gtarski4.html#T13" target="_self" onmouseover="rs('gtarski4/T13')" onmouseout="rh()">Satz8p2</a></span>;<br/>
<a NAME="E25:43"/><span class="lab"><font color="Green" title="E48">A16</font></span>: 
<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c4">c</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c10">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:43"><span class="lab"><font color="Green" title="E43">A11</font></span></a>, <a class="ref" href="gtarski3.html#T3" onmouseover="rs('gtarski3/T3')" onmouseout="rh()">GTARSKI3:3</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c11">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="E27:43"/><span class="lab"><font color="Green" title="E49">A17</font></span>: 
 <a href="gtarski3.html#R9" title="GTARSKI3:pred.9">Middle</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c10">c9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:43"><span class="lab"><font color="Green" title="E43">A11</font></span></a>, <a class="ref" href="gtarski3.html#T3" onmouseover="rs('gtarski3/T3')" onmouseout="rh()">GTARSKI3:3</a>, <a class="ref" href="gtarski3.html#T117" onmouseover="rs('gtarski3/T117')" onmouseout="rh()">GTARSKI3:117</a></span>;<br/>
<a NAME="E28:43"/><span class="lab"><font color="Green" title="E50">A18</font></span>: 
<font color="Maroon" title="c5">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">p</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="43_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:43_4"/>
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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"> contradiction</span><br/>

<a NAME="E2:43_4"/><span class="kw">then </span>
<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c4">c</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c5">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:43"><span class="lab"><font color="Green" title="E36">A4</font></span></a>, <a class="ref" href="gtarski3.html#T4" onmouseover="rs('gtarski3/T4')" onmouseout="rh()">GTARSKI3:4</a>, <a class="ref" href="gtarski3.html#D12" onmouseover="rs('gtarski3/D12')" onmouseout="rh()">GTARSKI3:def 12</a></span>;<br/>
<a NAME="E3:43_4"/><span class="kw">then </span>
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="gtarski1.html#D7" onmouseover="rs('gtarski1/D7')" onmouseout="rh()">GTARSKI1:def 7</a></span>;<br/>
<a NAME="E4:43_4"/><span class="kw">then </span>
 <a href="gtarski1.html#R6" title="GTARSKI1:pred.6">Collinear</a> <font color="Maroon" title="c3">b</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="#E3:43"><span class="lab"><font color="Green" title="E34">A2</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="E5:43_4"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43"><span class="lab"><font color="Green" title="E33">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>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="43_5"><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> <font color="Maroon" title="c8">q</font>,<font color="Maroon" title="c5">y</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="c7">z</font>,<font color="Maroon" title="c8">q</font>),<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c10">c9</font> &amp; <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c8">q</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c5">y</font>, <a href="gtarski3.html#K3" title="GTARSKI3:func.3">reflection</a> (<font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c8">q</font>) &amp; <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c4">c</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c10">c9</font> &amp;  <a href="gtarski3.html#R9" title="GTARSKI3:pred.9">Middle</a> <font color="Maroon" title="c8">q</font>,<font color="Maroon" title="c7">z</font>, <a href="gtarski3.html#K3" title="GTARSKI3:func.3">reflection</a> (<font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c8">q</font>) &amp;  <a href="gtarski3.html#R9" title="GTARSKI3:pred.9">Middle</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c10">c9</font> )</span></a><div class="add"><a NAME="E1:43_5"/>
(  <a href="gtarski1.html#R1" title="GTARSKI1:pred.1">between</a> <font color="Maroon" title="c8">q</font>,<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">p</font> &amp;  <a href="gtarski1.html#R1" title="GTARSKI1:pred.1">between</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c4">c</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:43"><span class="lab"><font color="Green" title="E36">A4</font></span></a>, <a class="txt" href="#E12:43"><span class="lab"><font color="Green" title="E40">A8</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#D12" onmouseover="rs('gtarski3/D12')" onmouseout="rh()">GTARSKI3:def 12</a></span>;<br/><span class="kw">hence </span><a NAME="E2:43_5"/>
 <a href="gtarski1.html#R1" title="GTARSKI1:pred.1">between</a> <font color="Maroon" title="c8">q</font>,<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c4">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:43"><span class="lab"><font color="Green" title="E50">A18</font></span></a>, <a class="ref" href="gtarski3.html#T19" onmouseover="rs('gtarski3/T19')" onmouseout="rh()">GTARSKI3:19</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="c7">z</font>,<font color="Maroon" title="c8">q</font>),<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c10">c9</font> &amp; <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c8">q</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c5">y</font>, <a href="gtarski3.html#K3" title="GTARSKI3:func.3">reflection</a> (<font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c8">q</font>) &amp; <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c4">c</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c10">c9</font> &amp;  <a href="gtarski3.html#R9" title="GTARSKI3:pred.9">Middle</a> <font color="Maroon" title="c8">q</font>,<font color="Maroon" title="c7">z</font>, <a href="gtarski3.html#K3" title="GTARSKI3:func.3">reflection</a> (<font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c8">q</font>) &amp;  <a href="gtarski3.html#R9" title="GTARSKI3:pred.9">Middle</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c10">c9</font> )</span><br/><span class="kw">thus </span><a NAME="E3:43_5"/>
 <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="c7">z</font>,<font color="Maroon" title="c8">q</font>),<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c10">c9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:43"><span class="lab"><font color="Green" title="E42">A10</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="c5">y</font>,<font color="Maroon" title="c8">q</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c5">y</font>, <a href="gtarski3.html#K3" title="GTARSKI3:func.3">reflection</a> (<font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c8">q</font>) &amp; <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c4">c</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c10">c9</font> &amp;  <a href="gtarski3.html#R9" title="GTARSKI3:pred.9">Middle</a> <font color="Maroon" title="c8">q</font>,<font color="Maroon" title="c7">z</font>, <a href="gtarski3.html#K3" title="GTARSKI3:func.3">reflection</a> (<font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c8">q</font>) &amp;  <a href="gtarski3.html#R9" title="GTARSKI3:pred.9">Middle</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c10">c9</font> )</span><br/><span class="kw">thus </span><a NAME="E4:43_5"/>
<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c8">q</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c5">y</font>, <a href="gtarski3.html#K3" title="GTARSKI3:func.3">reflection</a> (<font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c8">q</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:43"><span class="lab"><font color="Green" title="E47">A15</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="c5">y</font>,<font color="Maroon" title="c4">c</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c10">c9</font> &amp;  <a href="gtarski3.html#R9" title="GTARSKI3:pred.9">Middle</a> <font color="Maroon" title="c8">q</font>,<font color="Maroon" title="c7">z</font>, <a href="gtarski3.html#K3" title="GTARSKI3:func.3">reflection</a> (<font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c8">q</font>) &amp;  <a href="gtarski3.html#R9" title="GTARSKI3:pred.9">Middle</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c10">c9</font> )</span><br/><span class="kw">thus </span><a NAME="E5:43_5"/>
<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c4">c</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c10">c9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:43"><span class="lab"><font color="Green" title="E43">A11</font></span></a>, <a class="ref" href="gtarski3.html#T3" onmouseover="rs('gtarski3/T3')" onmouseout="rh()">GTARSKI3:3</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="gtarski3.html#R9" title="GTARSKI3:pred.9">Middle</a> <font color="Maroon" title="c8">q</font>,<font color="Maroon" title="c7">z</font>, <a href="gtarski3.html#K3" title="GTARSKI3:func.3">reflection</a> (<font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c8">q</font>) &amp;  <a href="gtarski3.html#R9" title="GTARSKI3:pred.9">Middle</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c10">c9</font> )</span><br/><span class="kw">thus </span><a NAME="E6:43_5"/>
 <a href="gtarski3.html#R9" title="GTARSKI3:pred.9">Middle</a> <font color="Maroon" title="c8">q</font>,<font color="Maroon" title="c7">z</font>, <a href="gtarski3.html#K3" title="GTARSKI3:func.3">reflection</a> (<font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c8">q</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>; <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="gtarski3.html#R9" title="GTARSKI3:pred.9">Middle</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c10">c9</font></span><br/><span class="kw">thus </span><a NAME="E7:43_5"/>
 <a href="gtarski3.html#R9" title="GTARSKI3:pred.9">Middle</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c10">c9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:43"><span class="lab"><font color="Green" title="E49">A17</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>
<a NAME="E30:43"/><span class="kw">then </span>
 <a href="gtarski3.html#R10" title="GTARSKI3:pred.10">Krippenfigur</a> <font color="Maroon" title="c8">q</font>,<font color="Maroon" title="c7">z</font>, <a href="gtarski3.html#K3" title="GTARSKI3:func.3">reflection</a> (<font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c8">q</font>),<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c10">c9</font>,<font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c4">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="gtarski3.html#D14" onmouseover="rs('gtarski3/D14')" onmouseout="rh()">GTARSKI3:def 14</a></span>;<br/>
<a NAME="E31:43"/><span class="kw">then </span>
 <a href="gtarski1.html#R1" title="GTARSKI1:pred.1">between</a> <font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c11">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="gtarski3.html#T116" onmouseover="rs('gtarski3/T116')" onmouseout="rh()">GTARSKI3:116</a></span>;<br/>
<a NAME="E32:43"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A19</font></span>: 
 <a href="gtarski1.html#R6" title="GTARSKI1:pred.6">Collinear</a> <font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c11">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="gtarski1.html#D17" onmouseover="rs('gtarski1/D17')" onmouseout="rh()">GTARSKI1:def 17</a></span>;<br/>
<a NAME="E33:43"/><span class="lab"><font color="Green" title="E52">A20</font></span>: 
<font color="Maroon" title="c4">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">y</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="43_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:43_6"/>
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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"> contradiction</span><br/>

<a NAME="E2:43_6"/><span class="kw">then </span>
 <a href="gtarski1.html#R6" title="GTARSKI1:pred.6">Collinear</a> <font color="Maroon" title="c3">b</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="#E3:43"><span class="lab"><font color="Green" title="E34">A2</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="E3:43_6"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43"><span class="lab"><font color="Green" title="E33">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="E34:43"/><span class="lab"><font color="Green" title="E53">A21</font></span>: 
<font color="Maroon" title="c5">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">z</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:43"><span class="lab"><font color="Green" title="E39">A7</font></span></a>, <a class="txt" href="#E28:43"><span class="lab"><font color="Green" title="E50">A18</font></span></a>, <a class="ref" href="gtarski1.html#D7" onmouseover="rs('gtarski1/D7')" onmouseout="rh()">GTARSKI1:def 7</a>, <a class="ref" href="gtarski3.html#T4" onmouseover="rs('gtarski3/T4')" onmouseout="rh()">GTARSKI3:4</a></span>;<br/>
<a NAME="E35:43"/><span class="lab"><font color="Green" title="E54">A22</font></span>: 
<font color="Maroon" title="c8">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="gtarski3.html#K3" title="GTARSKI3:func.3">reflection</a> (<font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c8">q</font>)
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="43_7"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:43_7"/>
<font color="Maroon" title="c8">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="gtarski3.html#K3" title="GTARSKI3:func.3">reflection</a> (<font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c8">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"> contradiction</span><br/>

<a NAME="E2:43_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A23</font></span>: 
 <a href="gtarski3.html#R9" title="GTARSKI3:pred.9">Middle</a> <font color="Maroon" title="c8">q</font>,<font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c8">q</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="E3:43_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A24</font></span>: 
<font color="Maroon" title="c8">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">z</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="gtarski3.html#T97" onmouseover="rs('gtarski3/T97')" onmouseout="rh()">GTARSKI3:97</a></span>;<br/>
<a NAME="E4:43_7"/><span class="lab"><font color="Green" title="E57">A25</font></span>: 
<font color="Maroon" title="c5">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">q</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43_7"><span class="lab"><font color="Green" title="E55">A23</font></span></a>, <a class="ref" href="gtarski3.html#T97" onmouseover="rs('gtarski3/T97')" onmouseout="rh()">GTARSKI3:97</a>, <a class="txt" href="#E34:43"><span class="lab"><font color="Green" title="E53">A21</font></span></a></span>;<br/>
<a NAME="E5:43_7"/>
 <a href="gtarski1.html#R6" title="GTARSKI1:pred.6">Collinear</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43"><span class="lab"><font color="Green" title="E34">A2</font></span></a>, <a class="ref" href="gtarski1.html#D17" onmouseover="rs('gtarski1/D17')" onmouseout="rh()">GTARSKI1:def 17</a></span>;<br/>
<a NAME="E6:43_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A26</font></span>: 
 <a href="gtarski4.html#K1" title="GTARSKI4:func.1">Line</a> (<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c2">a</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">a</font>,<font color="Maroon" title="c3">b</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:43"><span class="lab"><font color="Green" title="E44">A12</font></span></a>, <a class="txt" href="#E18:43"><span class="lab"><font color="Green" title="E45">A13</font></span></a>, <a class="ref" href="gtarski3.html#T82" onmouseover="rs('gtarski3/T82')" onmouseout="rh()">GTARSKI3:82</a>, <a class="txt" href="gtarski4.html#E1"><span class="lab"><font color="Green" title="E1">LemmaA1</font></span></a></span>;<br/>
<a NAME="E7:43_7"/>
 <a href="gtarski1.html#R6" title="GTARSKI1:pred.6">Collinear</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c7">z</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:43"><span class="lab"><font color="Green" title="E38">A6</font></span></a>, <a class="ref" href="gtarski1.html#D17" onmouseover="rs('gtarski1/D17')" onmouseout="rh()">GTARSKI1:def 17</a></span>;<br/>
<a NAME="E8:43_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A27</font></span>: 
 <a href="gtarski4.html#K1" title="GTARSKI4:func.1">Line</a> (<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c8">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">a</font>,<font color="Maroon" title="c3">b</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:43"><span class="lab"><font color="Green" title="E45">A13</font></span></a>, <a class="txt" href="#E34:43"><span class="lab"><font color="Green" title="E53">A21</font></span></a>, <a class="txt" href="#E6:43_7"><span class="lab"><font color="Green" title="E58">A26</font></span></a>, <a class="ref" href="gtarski3.html#T82" onmouseover="rs('gtarski3/T82')" onmouseout="rh()">GTARSKI3:82</a>, <a class="txt" href="#E3:43_7"><span class="lab"><font color="Green" title="E56">A24</font></span></a>, <a class="txt" href="gtarski4.html#E1"><span class="lab"><font color="Green" title="E1">LemmaA1</font></span></a></span>;<br/>
<a NAME="E9:43_7"/>
 <a href="gtarski1.html#R6" title="GTARSKI1:pred.6">Collinear</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c8">q</font>,<font color="Maroon" title="c6">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:43"><span class="lab"><font color="Green" title="E40">A8</font></span></a>, <a class="ref" href="gtarski1.html#D17" onmouseover="rs('gtarski1/D17')" onmouseout="rh()">GTARSKI1:def 17</a></span>;<br/>
<a NAME="E10:43_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A28</font></span>: 
 <a href="gtarski4.html#K1" title="GTARSKI4:func.1">Line</a> (<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">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">a</font>,<font color="Maroon" title="c3">b</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:43_7"><span class="lab"><font color="Green" title="E57">A25</font></span></a>, <a class="txt" href="#E28:43"><span class="lab"><font color="Green" title="E50">A18</font></span></a>, <a class="ref" href="gtarski3.html#T82" onmouseover="rs('gtarski3/T82')" onmouseout="rh()">GTARSKI3:82</a>, <a class="txt" href="#E8:43_7"><span class="lab"><font color="Green" title="E59">A27</font></span></a>, <a class="txt" href="gtarski4.html#E1"><span class="lab"><font color="Green" title="E1">LemmaA1</font></span></a></span>;<br/>
<a NAME="E11:43_7"/>
 <a href="gtarski1.html#R1" title="GTARSKI1:pred.1">between</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c4">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:43"><span class="lab"><font color="Green" title="E36">A4</font></span></a>, <a class="ref" href="gtarski3.html#D12" onmouseover="rs('gtarski3/D12')" onmouseout="rh()">GTARSKI3:def 12</a></span>;<br/>
<a NAME="E12:43_7"/><span class="kw">then </span>
 <a href="gtarski1.html#R6" title="GTARSKI1:pred.6">Collinear</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c4">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="gtarski1.html#D17" onmouseover="rs('gtarski1/D17')" onmouseout="rh()">GTARSKI1:def 17</a></span>;<br/>
<a NAME="E13:43_7"/><span class="kw">then </span>
<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="c3">b</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:43_7"><span class="lab"><font color="Green" title="E60">A28</font></span></a>, <a class="txt" href="gtarski4.html#E1"><span class="lab"><font color="Green" title="E1">LemmaA1</font></span></a></span>;<br/>
<a NAME="E14:43_7"/><span class="kw">then </span>
 <a href="gtarski1.html#R6" title="GTARSKI1:pred.6">Collinear</a> <font color="Maroon" title="c4">c</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></span>;<br/>
<span class="kw">hence </span><a NAME="E15:43_7"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43"><span class="lab"><font color="Green" title="E33">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>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="43_8"><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="c5">y</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#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">y</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="c3">b</font> )</span></a><div class="add"><a NAME="E1:43_8"/>
 <a href="gtarski1.html#R6" title="GTARSKI1:pred.6">Collinear</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43"><span class="lab"><font color="Green" title="E34">A2</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="E2:43_8"/>
<font color="Maroon" title="c5">y</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="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="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">y</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="c3">b</font> )</span><br/><span class="kw">thus </span><a NAME="E3:43_8"/>
<font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:43"><span class="lab"><font color="Green" title="E45">A13</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="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b</font></span><br/><span class="kw">thus </span><a NAME="E4:43_8"/>
<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="#E1:43"><span class="lab"><font color="Green" title="E33">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"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E37:43"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A29</font></span>: 
 <a href="gtarski4.html#K1" title="GTARSKI4:func.1">Line</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">y</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">a</font>,<font color="Maroon" title="c3">b</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="gtarski3.html#T82" onmouseover="rs('gtarski3/T82')" onmouseout="rh()">GTARSKI3:82</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="43_9"><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="c7">z</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">y</font>,<font color="Maroon" title="c2">a</font>) &amp; <font color="Maroon" title="c5">y</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="c5">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">z</font> )</span></a><div class="add"><a NAME="E1:43_9"/>
 <a href="gtarski1.html#R6" title="GTARSKI1:pred.6">Collinear</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c7">z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:43"><span class="lab"><font color="Green" title="E38">A6</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="E2:43_9"/>
<font color="Maroon" title="c7">z</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">y</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>; <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">y</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="c5">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">z</font> )</span><br/><span class="kw">thus </span><a NAME="E3:43_9"/>
<font color="Maroon" title="c5">y</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="#E18:43"><span class="lab"><font color="Green" title="E45">A13</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="c5">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">z</font></span><br/><span class="kw">thus </span><a NAME="E4:43_9"/>
<font color="Maroon" title="c5">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:43"><span class="lab"><font color="Green" title="E39">A7</font></span></a>, <a class="txt" href="#E28:43"><span class="lab"><font color="Green" title="E50">A18</font></span></a>, <a class="ref" href="gtarski1.html#D7" onmouseover="rs('gtarski1/D7')" onmouseout="rh()">GTARSKI1:def 7</a>, <a class="ref" href="gtarski3.html#T4" onmouseover="rs('gtarski3/T4')" onmouseout="rh()">GTARSKI3:4</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="E39:43"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A30</font></span>: 
 <a href="gtarski4.html#K1" title="GTARSKI4:func.1">Line</a> (<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c7">z</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">a</font>,<font color="Maroon" title="c3">b</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:43"><span class="lab"><font color="Green" title="E55">A29</font></span></a>, <a class="ref" href="gtarski3.html#T82" onmouseover="rs('gtarski3/T82')" onmouseout="rh()">GTARSKI3:82</a></span>;<br/>
<a NAME="E40:43"/><span class="lab"><font color="Green" title="E57">A31</font></span>: 
<font color="Maroon" title="c4">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">x</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="43_10"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:43_10"/>
<font color="Maroon" title="c4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">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"> contradiction</span><br/>

<a NAME="E2:43_10"/><span class="kw">then </span>
<font color="Maroon" title="c10">c9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="gtarski3.html#K3" title="GTARSKI3:func.3">reflection</a> (<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c4">c</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:43"><span class="lab"><font color="Green" title="E49">A17</font></span></a>, <a class="ref" href="gtarski3.html#D13" onmouseover="rs('gtarski3/D13')" onmouseout="rh()">GTARSKI3:def 13</a></span>;<br/>
<a NAME="E3:43_10"/><span class="kw">then </span>
 <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="c7">z</font>,<font color="Maroon" title="c8">q</font>),<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c4">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:43"><span class="lab"><font color="Green" title="E42">A10</font></span></a>, <a class="ref" href="gtarski3.html#T104" onmouseover="rs('gtarski3/T104')" onmouseout="rh()">GTARSKI3:104</a></span>;<br/>
<a NAME="E4:43_10"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">a32</font></span>: 
 <a href="gtarski1.html#R6" title="GTARSKI1:pred.6">Collinear</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c4">c</font>, <a href="gtarski3.html#K3" title="GTARSKI3:func.3">reflection</a> (<font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c8">q</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="gtarski1.html#D17" onmouseover="rs('gtarski1/D17')" onmouseout="rh()">GTARSKI1:def 17</a></span>;<br/>
<a NAME="E5:43_10"/>
 <a href="gtarski1.html#R1" title="GTARSKI1:pred.1">between</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c4">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:43"><span class="lab"><font color="Green" title="E36">A4</font></span></a>, <a class="ref" href="gtarski3.html#D12" onmouseover="rs('gtarski3/D12')" onmouseout="rh()">GTARSKI3:def 12</a></span>;<br/>
<a NAME="E6:43_10"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A33</font></span>: 
 <a href="gtarski1.html#R6" title="GTARSKI1:pred.6">Collinear</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c4">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="gtarski1.html#D17" onmouseover="rs('gtarski1/D17')" onmouseout="rh()">GTARSKI1:def 17</a></span>;<br/>
<a NAME="E7:43_10"/><span class="lab"><font color="Green" title="E60">A34</font></span>: 
<font color="Maroon" title="c8">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:43"><span class="lab"><font color="Green" title="E45">A13</font></span></a>, <a class="txt" href="#E13:43"><span class="lab"><font color="Green" title="E41">A9</font></span></a>, <a class="ref" href="gtarski3.html#T4" onmouseover="rs('gtarski3/T4')" onmouseout="rh()">GTARSKI3:4</a>, <a class="ref" href="gtarski1.html#D7" onmouseover="rs('gtarski1/D7')" onmouseout="rh()">GTARSKI1:def 7</a></span>;<br/>
<a NAME="E8:43_10"/>
<font color="Maroon" title="c5">y</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="#E6:43"><span class="lab"><font color="Green" title="E36">A4</font></span></a>, <a class="txt" href="#E33:43"><span class="lab"><font color="Green" title="E52">A20</font></span></a>, <a class="ref" href="gtarski4.html#T9" target="_self" onmouseover="rs('gtarski4/T9')" onmouseout="rh()">Prelim09</a></span>;<br/>
<a NAME="E9:43_10"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A35</font></span>: 
 <a href="gtarski4.html#K1" title="GTARSKI4:func.1">Line</a> (<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">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="c5">y</font>,<font color="Maroon" title="c4">c</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:43"><span class="lab"><font color="Green" title="E52">A20</font></span></a>, <a class="txt" href="#E6:43_10"><span class="lab"><font color="Green" title="E59">A33</font></span></a>, <a class="ref" href="gtarski3.html#T82" onmouseover="rs('gtarski3/T82')" onmouseout="rh()">GTARSKI3:82</a>, <a class="txt" href="gtarski4.html#E1"><span class="lab"><font color="Green" title="E1">LemmaA1</font></span></a></span>;<br/>
<a NAME="E10:43_10"/>
 <a href="gtarski1.html#R6" title="GTARSKI1:pred.6">Collinear</a> <font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c8">q</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:43"><span class="lab"><font color="Green" title="E40">A8</font></span></a>, <a class="ref" href="gtarski1.html#D17" onmouseover="rs('gtarski1/D17')" onmouseout="rh()">GTARSKI1:def 17</a></span>;<br/>
<a NAME="E11:43_10"/><span class="kw">then </span>
 <a href="gtarski4.html#K1" title="GTARSKI4:func.1">Line</a> (<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c8">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="c5">y</font>,<font color="Maroon" title="c4">c</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:43_10"><span class="lab"><font color="Green" title="E60">A34</font></span></a>, <a class="txt" href="#E33:43"><span class="lab"><font color="Green" title="E52">A20</font></span></a>, <a class="ref" href="gtarski3.html#T82" onmouseover="rs('gtarski3/T82')" onmouseout="rh()">GTARSKI3:82</a>, <a class="txt" href="#E9:43_10"><span class="lab"><font color="Green" title="E61">A35</font></span></a>, <a class="txt" href="gtarski4.html#E1"><span class="lab"><font color="Green" title="E1">LemmaA1</font></span></a></span>;<br/>
<a NAME="E12:43_10"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A37</font></span>: 
 <a href="gtarski4.html#K1" title="GTARSKI4:func.1">Line</a> (<font color="Maroon" title="c8">q</font>,<span class="p1">(<span class="default"><a href="gtarski3.html#K3" title="GTARSKI3:func.3">reflection</a> (<font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c8">q</font>)</span>)</span>) <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">y</font>,<font color="Maroon" title="c4">c</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:43"><span class="lab"><font color="Green" title="E54">A22</font></span></a>, <a class="txt" href="#E7:43_10"><span class="lab"><font color="Green" title="E60">A34</font></span></a>, <a class="ref" href="gtarski3.html#T82" onmouseover="rs('gtarski3/T82')" onmouseout="rh()">GTARSKI3:82</a>, <a class="txt" href="#E4:43_10"><span class="lab"><font color="Green" title="E58">a32</font></span></a>, <a class="txt" href="gtarski4.html#E1"><span class="lab"><font color="Green" title="E1">LemmaA1</font></span></a></span>;<br/>
<a NAME="E13:43_10"/>
 <a href="gtarski3.html#R9" title="GTARSKI3:pred.9">Middle</a> <font color="Maroon" title="c8">q</font>,<font color="Maroon" title="c7">z</font>, <a href="gtarski3.html#K3" title="GTARSKI3:func.3">reflection</a> (<font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c8">q</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="E14:43_10"/><span class="kw">then </span>
 <a href="gtarski1.html#R1" title="GTARSKI1:pred.1">between</a> <font color="Maroon" title="c8">q</font>,<font color="Maroon" title="c7">z</font>, <a href="gtarski3.html#K3" title="GTARSKI3:func.3">reflection</a> (<font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c8">q</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/>
<a NAME="E15:43_10"/><span class="kw">then </span>
 <a href="gtarski1.html#R6" title="GTARSKI1:pred.6">Collinear</a> <font color="Maroon" title="c8">q</font>,<font color="Maroon" title="c7">z</font>, <a href="gtarski3.html#K3" title="GTARSKI3:func.3">reflection</a> (<font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c8">q</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="gtarski1.html#D17" onmouseover="rs('gtarski1/D17')" onmouseout="rh()">GTARSKI1:def 17</a></span>;<br/>
<a NAME="E16:43_10"/><span class="kw">then </span>
 <a href="gtarski1.html#R6" title="GTARSKI1:pred.6">Collinear</a> <font color="Maroon" title="c8">q</font>, <a href="gtarski3.html#K3" title="GTARSKI3:func.3">reflection</a> (<font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c8">q</font>),<font color="Maroon" title="c7">z</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="E17:43_10"/><span class="kw">then </span>
<font color="Maroon" title="c7">z</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">y</font>,<font color="Maroon" title="c4">c</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:43_10"><span class="lab"><font color="Green" title="E62">A37</font></span></a>, <a class="txt" href="gtarski4.html#E1"><span class="lab"><font color="Green" title="E1">LemmaA1</font></span></a></span>;<br/>
<a NAME="E18:43_10"/><span class="kw">then </span>
 <a href="gtarski1.html#R6" title="GTARSKI1:pred.6">Collinear</a> <font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c4">c</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></span>;<br/>
<a NAME="E19:43_10"/><span class="kw">then </span>
<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="c5">y</font>,<font color="Maroon" title="c7">z</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/>
<a NAME="E20:43_10"/><span class="kw">then </span>
 <a href="gtarski1.html#R6" title="GTARSKI1:pred.6">Collinear</a> <font color="Maroon" title="c4">c</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="#E39:43"><span class="lab"><font color="Green" title="E56">A30</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E21:43_10"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43"><span class="lab"><font color="Green" title="E33">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>
<span class="kw">take </span>
<font color="Maroon" title="c11">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"> <font color="Maroon" title="c11">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><br/>

<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="43_11"><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#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="c11">x</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="c11">x</font> )</span></a><div class="add"><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="43_11_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="c5">y</font>,<font color="Maroon" title="c7">z</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="c4">c</font>,<font color="Maroon" title="c11">x</font>) <a href="gtarski3.html#R8" title="GTARSKI3:pred.8">is_line</a>  &amp; <font color="Maroon" title="c11">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="c5">y</font>,<font color="Maroon" title="c7">z</font>) &amp; <font color="Maroon" title="c11">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="c4">c</font>,<font color="Maroon" title="c11">x</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="c5">y</font>,<font color="Maroon" title="c7">z</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="c4">c</font>,<font color="Maroon" title="c11">x</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="c11">x</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="c11">x</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="c11">x</font>,<font color="Olive" title="b2">v</font> ) )</span></a><div class="add"><span class="kw">thus </span><a NAME="E1:43_11_1"/>
 <a href="gtarski4.html#K1" title="GTARSKI4:func.1">Line</a> (<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c7">z</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="#E34:43"><span class="lab"><font color="Green" title="E53">A21</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="c4">c</font>,<font color="Maroon" title="c11">x</font>) <a href="gtarski3.html#R8" title="GTARSKI3:pred.8">is_line</a>  &amp; <font color="Maroon" title="c11">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="c5">y</font>,<font color="Maroon" title="c7">z</font>) &amp; <font color="Maroon" title="c11">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="c4">c</font>,<font color="Maroon" title="c11">x</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="c5">y</font>,<font color="Maroon" title="c7">z</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="c4">c</font>,<font color="Maroon" title="c11">x</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="c11">x</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="c11">x</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="c11">x</font>,<font color="Olive" title="b2">v</font> ) )</span><br/><span class="kw">thus </span><a NAME="E2:43_11_1"/>
 <a href="gtarski4.html#K1" title="GTARSKI4:func.1">Line</a> (<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c11">x</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="#E40:43"><span class="lab"><font color="Green" title="E57">A31</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="c11">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="c5">y</font>,<font color="Maroon" title="c7">z</font>) &amp; <font color="Maroon" title="c11">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="c4">c</font>,<font color="Maroon" title="c11">x</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="c5">y</font>,<font color="Maroon" title="c7">z</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="c4">c</font>,<font color="Maroon" title="c11">x</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="c11">x</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="c11">x</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="c11">x</font>,<font color="Olive" title="b2">v</font> ) )</span><br/><span class="kw">thus </span><a NAME="E3:43_11_1"/>
<font color="Maroon" title="c11">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="c5">y</font>,<font color="Maroon" title="c7">z</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>, <a class="txt" href="#E32:43"><span class="lab"><font color="Green" title="E51">A19</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="c11">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="c4">c</font>,<font color="Maroon" title="c11">x</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="c5">y</font>,<font color="Maroon" title="c7">z</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="c4">c</font>,<font color="Maroon" title="c11">x</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="c11">x</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="c11">x</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="c11">x</font>,<font color="Olive" title="b2">v</font> ) )</span><br/><span class="kw">thus </span><a NAME="E4:43_11_1"/>
<font color="Maroon" title="c11">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="c4">c</font>,<font color="Maroon" title="c11">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>; <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="c5">y</font>,<font color="Maroon" title="c7">z</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="c4">c</font>,<font color="Maroon" title="c11">x</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="c11">x</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="c11">x</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="c11">x</font>,<font color="Olive" title="b2">v</font> )</span><br/><span class="kw">thus </span><a NAME="E5:43_11_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="c5">y</font>,<font color="Maroon" title="c7">z</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="c4">c</font>,<font color="Maroon" title="c11">x</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="c11">x</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="c11">x</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="c11">x</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="43_11_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">take </span>
<font color="Maroon" title="c5">y</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">y</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">y</font>,<font color="Maroon" title="c7">z</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="c4">c</font>,<font color="Maroon" title="c11">x</font>) &amp; <font color="Maroon" title="c5">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">x</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="c11">x</font> &amp;  <a href="gtarski4.html#R1" title="GTARSKI4:pred.1">right_angle</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c11">x</font>,<font color="Olive" title="b1">v</font> )</span><br/>

<span class="kw">take </span>
<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"> ( <font color="Maroon" title="c5">y</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">y</font>,<font color="Maroon" title="c7">z</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="c11">x</font>) &amp; <font color="Maroon" title="c5">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">x</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="c11">x</font> &amp;  <a href="gtarski4.html#R1" title="GTARSKI4:pred.1">right_angle</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c4">c</font> )</span><br/>

<span class="kw">thus </span><a NAME="E1:43_11_1_1"/>
<font color="Maroon" title="c5">y</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">y</font>,<font color="Maroon" title="c7">z</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="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="c11">x</font>) &amp; <font color="Maroon" title="c5">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">x</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="c11">x</font> &amp;  <a href="gtarski4.html#R1" title="GTARSKI4:pred.1">right_angle</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c4">c</font> )</span><br/>

<span class="kw">thus </span><a NAME="E2:43_11_1_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="c4">c</font>,<font color="Maroon" title="c11">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>; <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">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">x</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="c11">x</font> &amp;  <a href="gtarski4.html#R1" title="GTARSKI4:pred.1">right_angle</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c4">c</font> )</span><br/>

<span class="kw">thus </span><a NAME="E3:43_11_1_1"/>
<font color="Maroon" title="c5">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="gtarski4.html#T32" target="_self" onmouseover="rs('gtarski4/T32')" onmouseout="rh()">Satz8p18Lemma</a>, <a class="txt" href="#E1:43"><span class="lab"><font color="Green" title="E33">A1</font></span></a>, <a class="txt" href="#E3:43"><span class="lab"><font color="Green" title="E34">A2</font></span></a>, <a class="txt" href="#E18:43"><span class="lab"><font color="Green" title="E45">A13</font></span></a>, <a class="txt" href="#E10:43"><span class="lab"><font color="Green" title="E39">A7</font></span></a>, <a class="txt" href="#E28:43"><span class="lab"><font color="Green" title="E50">A18</font></span></a>, <a class="txt" href="#E27:43"><span class="lab"><font color="Green" title="E49">A17</font></span></a>, <a class="txt" href="#E33:43"><span class="lab"><font color="Green" title="E52">A20</font></span></a>, <a class="txt" href="#E15:43"><span class="lab"><font color="Green" title="E42">A10</font></span></a>, <a class="txt" href="#E6:43"><span class="lab"><font color="Green" title="E36">A4</font></span></a>, <a class="txt" href="#E12:43"><span class="lab"><font color="Green" title="E40">A8</font></span></a>, <a class="txt" href="#E35:43"><span class="lab"><font color="Green" title="E54">A22</font></span></a>, <a class="txt" href="#E9:43"><span class="lab"><font color="Green" title="E38">A6</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="c4">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">x</font> &amp;  <a href="gtarski4.html#R1" title="GTARSKI4:pred.1">right_angle</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c4">c</font> )</span><br/>

<span class="kw">thus </span><a NAME="E4:43_11_1_1"/>
<font color="Maroon" title="c4">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:43"><span class="lab"><font color="Green" title="E57">A31</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">  <a href="gtarski4.html#R1" title="GTARSKI4:pred.1">right_angle</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c4">c</font></span><br/>

<span class="kw">thus </span><a NAME="E5:43_11_1_1"/>
 <a href="gtarski4.html#R1" title="GTARSKI4:pred.1">right_angle</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c4">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:43"><span class="lab"><font color="Green" title="E48">A16</font></span></a>, <a class="txt" href="#E27:43"><span class="lab"><font color="Green" title="E49">A17</font></span></a>, <a class="ref" href="gtarski3.html#D13" onmouseover="rs('gtarski3/D13')" onmouseout="rh()">GTARSKI3:def 13</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><a NAME="E2:43_11"/><span class="kw">then </span>
 <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="c4">c</font>,<font color="Maroon" title="c11">x</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:43"><span class="lab"><font color="Green" title="E56">A30</font></span></a>, <a class="ref" href="gtarski4.html#T24" target="_self" onmouseover="rs('gtarski4/T24')" onmouseout="rh()">Satz8p13</a></span>;<br/><span class="kw">hence </span><a NAME="E3:43_11"/>
 <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="c11">x</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>, <a class="txt" href="#E1:43"><span class="lab"><font color="Green" title="E33">A1</font></span></a>, <a class="txt" href="#E40:43"><span class="lab"><font color="Green" title="E57">A31</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">  <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="c11">x</font></span><br/><a NAME="E4:43_11"/>
<font color="Maroon" title="c11">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="c5">y</font>,<font color="Maroon" title="c7">z</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>, <a class="txt" href="#E32:43"><span class="lab"><font color="Green" title="E51">A19</font></span></a></span>;<br/><a NAME="E5:43_11"/><span class="kw">then </span>
 <a href="gtarski1.html#R6" title="GTARSKI1:pred.6">Collinear</a> <font color="Maroon" title="c11">x</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="#E39:43"><span class="lab"><font color="Green" title="E56">A30</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E6:43_11"/>
 <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="c11">x</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/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E42:43"/>
<font color="Maroon" title="c11">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>
 ; <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>
