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<span class="kw">let </span><font color="Maroon" title="c1">S</font> be   <a href="gtarski1.html#V9" title="GTARSKI1:attr.9">satisfying_Tarski-model</a>   <a href="gtarski1.html#L1" title="GTARSKI1:struct.1">TarskiGeometryStruct</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">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  <a href="gtarski4.html#R1" title="GTARSKI4:pred.1">right_angle</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font> &amp;  <a href="gtarski4.html#R1" title="GTARSKI4:pred.1">right_angle</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b3">c</font>,<font color="Olive" title="b2">b</font> holds <br/><font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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"> (  <a href="gtarski4.html#R1" title="GTARSKI4:pred.1">right_angle</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font> &amp;  <a href="gtarski4.html#R1" title="GTARSKI4:pred.1">right_angle</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c3">b</font> implies <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c</font> )</span><br/>







<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:23"/><span class="lab"><font color="Green" title="E21">A1</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="c3">b</font>,<font color="Maroon" title="c4">c</font>
 <span class="kw">and </span><br/><a NAME="E2:23"/><span class="lab"><font color="Green" title="E22">A2</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="c4">c</font>,<font color="Maroon" title="c3">b</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c</font></span><br/>

<span class="kw">assume </span><a NAME="E3:23"/><span class="lab"><font color="Green" title="E23">A3</font></span>: 
<font color="Maroon" title="c3">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</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/>

<span class="kw">set </span><font color="Maroon" title="c5">c9</font> =  <a href="gtarski3.html#K3" title="GTARSKI3:func.3">reflection</a> (<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c6">a9</font> =  <a href="gtarski3.html#K3" title="GTARSKI3:func.3">reflection</a> (<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c2">a</font>);<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="23_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#R1" title="GTARSKI4:pred.1">right_angle</a>  <a href="gtarski3.html#K3" title="GTARSKI3:func.3">reflection</a> (<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c2">a</font>),<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font> &amp;  <a href="gtarski1.html#R1" title="GTARSKI1:pred.1">between</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font>, <a href="gtarski3.html#K3" title="GTARSKI3:func.3">reflection</a> (<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c2">a</font>) )</span></a><div class="add"><a NAME="E1:23_1"/>
 <a href="gtarski3.html#K3" title="GTARSKI3:func.3">reflection</a> (<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c2">a</font>),<font color="Maroon" title="c4">c</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a>  <a href="gtarski3.html#K3" title="GTARSKI3:func.3">reflection</a> (<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c2">a</font>), <a href="gtarski3.html#K3" title="GTARSKI3:func.3">reflection</a> (<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font>)
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="23_1_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="23_1_1_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">a</font>, <a href="gtarski3.html#K3" title="GTARSKI3:func.3">reflection</a> (<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font>) <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a>  <a href="gtarski3.html#K3" title="GTARSKI3:func.3">reflection</a> (<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c2">a</font>),<font color="Maroon" title="c4">c</font> &amp; <font color="Maroon" title="c2">a</font>, <a href="gtarski3.html#K3" title="GTARSKI3:func.3">reflection</a> (<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font>) <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a>  <a href="gtarski3.html#K3" title="GTARSKI3:func.3">reflection</a> (<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c2">a</font>), <a href="gtarski3.html#K3" title="GTARSKI3:func.3">reflection</a> (<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font>) )</span></a><div class="add"><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="23_1_1_1_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">a</font>, <a href="gtarski3.html#K3" title="GTARSKI3:func.3">reflection</a> (<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font>) <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font> &amp; <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a>  <a href="gtarski3.html#K3" title="GTARSKI3:func.3">reflection</a> (<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c2">a</font>),<font color="Maroon" title="c4">c</font> )</span></a><div class="add"><span class="kw">thus </span><a NAME="E1:23_1_1_1_1"/>
<font color="Maroon" title="c2">a</font>, <a href="gtarski3.html#K3" title="GTARSKI3:func.3">reflection</a> (<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font>) <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23"><span class="lab"><font color="Green" title="E21">A1</font></span></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"> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a>  <a href="gtarski3.html#K3" title="GTARSKI3:func.3">reflection</a> (<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c2">a</font>),<font color="Maroon" title="c4">c</font></span><br/><a NAME="E2:23_1_1_1_1"/>
 <a href="gtarski4.html#R1" title="GTARSKI4:pred.1">right_angle</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c2">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="gtarski4.html#T13" target="_self" onmouseover="rs('gtarski4/T13')" onmouseout="rh()">Satz8p2</a>, <a class="ref" href="gtarski4.html#T17" target="_self" onmouseover="rs('gtarski4/T17')" onmouseout="rh()">Satz8p5</a></span>;<br/><span class="kw">hence </span><a NAME="E3:23_1_1_1_1"/>
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a>  <a href="gtarski3.html#K3" title="GTARSKI3:func.3">reflection</a> (<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c2">a</font>),<font color="Maroon" title="c4">c</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="gtarski4.html#T1" target="_self" onmouseover="rs('gtarski4/T1')" onmouseout="rh()">Prelim01</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="E2:23_1_1_1"/>
<font color="Maroon" title="c2">a</font>, <a href="gtarski3.html#K3" title="GTARSKI3:func.3">reflection</a> (<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font>) <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a>  <a href="gtarski3.html#K3" title="GTARSKI3:func.3">reflection</a> (<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c2">a</font>),<font color="Maroon" title="c4">c</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="gtarski3.html#T5" onmouseover="rs('gtarski3/T5')" onmouseout="rh()">GTARSKI3:5</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="gtarski3.html#K3" title="GTARSKI3:func.3">reflection</a> (<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font>) <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a>  <a href="gtarski3.html#K3" title="GTARSKI3:func.3">reflection</a> (<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c2">a</font>), <a href="gtarski3.html#K3" title="GTARSKI3:func.3">reflection</a> (<font color="Maroon" title="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="23_1_1_1_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="gtarski4.html#R1" title="GTARSKI4:pred.1">right_angle</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c2">a</font> &amp; <font color="Maroon" title="c3">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">c</font> &amp;  <a href="gtarski1.html#R6" title="GTARSKI1:pred.6">Collinear</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c3">b</font>, <a href="gtarski3.html#K3" title="GTARSKI3:func.3">reflection</a> (<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font>) )</span></a><div class="add"><span class="kw">thus </span><a NAME="E1:23_1_1_1_2"/>
 <a href="gtarski4.html#R1" title="GTARSKI4:pred.1">right_angle</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c2">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:23"><span class="lab"><font color="Green" title="E22">A2</font></span></a>, <a class="ref" href="gtarski4.html#T13" target="_self" onmouseover="rs('gtarski4/T13')" onmouseout="rh()">Satz8p2</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="c3">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">c</font> &amp;  <a href="gtarski1.html#R6" title="GTARSKI1:pred.6">Collinear</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c3">b</font>, <a href="gtarski3.html#K3" title="GTARSKI3:func.3">reflection</a> (<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font>) )</span><br/><span class="kw">thus </span><a NAME="E2:23_1_1_1_2"/>
<font color="Maroon" title="c3">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:23"><span class="lab"><font color="Green" title="E23">A3</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="c4">c</font>,<font color="Maroon" title="c3">b</font>, <a href="gtarski3.html#K3" title="GTARSKI3:func.3">reflection</a> (<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font>)</span><br/><a NAME="E3:23_1_1_1_2"/>
 <a href="gtarski3.html#R9" title="GTARSKI3:pred.9">Middle</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c3">b</font>, <a href="gtarski3.html#K3" title="GTARSKI3:func.3">reflection</a> (<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="gtarski3.html#D13" onmouseover="rs('gtarski3/D13')" onmouseout="rh()">GTARSKI3:def 13</a></span>;<br/><span class="kw">hence </span><a NAME="E4:23_1_1_1_2"/>
 <a href="gtarski1.html#R6" title="GTARSKI1:pred.6">Collinear</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c3">b</font>, <a href="gtarski3.html#K3" title="GTARSKI3:func.3">reflection</a> (<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="gtarski4.html#T7" target="_self" onmouseover="rs('gtarski4/T7')" onmouseout="rh()">Prelim08a</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="E4:23_1_1_1"/><span class="kw">then </span>
 <a href="gtarski4.html#R1" title="GTARSKI4:pred.1">right_angle</a>  <a href="gtarski3.html#K3" title="GTARSKI3:func.3">reflection</a> (<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font>),<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c2">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="gtarski4.html#T15" target="_self" onmouseover="rs('gtarski4/T15')" onmouseout="rh()">Satz8p3</a></span>;<br/><a NAME="E5:23_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A4</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="c4">c</font>, <a href="gtarski3.html#K3" title="GTARSKI3:func.3">reflection</a> (<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</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="E6:23_1_1_1"/>
<font color="Maroon" title="c2">a</font>, <a href="gtarski3.html#K3" title="GTARSKI3:func.3">reflection</a> (<font color="Maroon" title="c4">c</font>,<span class="p1">(<span class="default"><a href="gtarski3.html#K3" title="GTARSKI3:func.3">reflection</a> (<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font>)</span>)</span>) <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a>  <a href="gtarski3.html#K3" title="GTARSKI3:func.3">reflection</a> (<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c2">a</font>), <a href="gtarski3.html#K3" title="GTARSKI3:func.3">reflection</a> (<font color="Maroon" title="c4">c</font>,<span class="p1">(<span class="default"><a href="gtarski3.html#K3" title="GTARSKI3:func.3">reflection</a> (<font color="Maroon" title="c4">c</font>,<span class="p2">(<span class="default"><a href="gtarski3.html#K3" title="GTARSKI3:func.3">reflection</a> (<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font>)</span>)</span>)</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="gtarski3.html#T105" onmouseover="rs('gtarski3/T105')" onmouseout="rh()">GTARSKI3:105</a></span>;<br/><a NAME="E7:23_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">a</font>, <a href="gtarski3.html#K3" title="GTARSKI3:func.3">reflection</a> (<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font>) <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a>  <a href="gtarski3.html#K3" title="GTARSKI3:func.3">reflection</a> (<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c2">a</font>), <a href="gtarski3.html#K3" title="GTARSKI3:func.3">reflection</a> (<font color="Maroon" title="c4">c</font>,<span class="p1">(<span class="default"><a href="gtarski3.html#K3" title="GTARSKI3:func.3">reflection</a> (<font color="Maroon" title="c4">c</font>,<span class="p2">(<span class="default"><a href="gtarski3.html#K3" title="GTARSKI3:func.3">reflection</a> (<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font>)</span>)</span>)</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:23_1_1_1"><span class="lab"><font color="Green" title="E24">A4</font></span></a>, <a class="ref" href="gtarski3.html#T5" onmouseover="rs('gtarski3/T5')" onmouseout="rh()">GTARSKI3:5</a></span>;<br/><span class="kw">hence </span><a NAME="E8:23_1_1_1"/>
<font color="Maroon" title="c2">a</font>, <a href="gtarski3.html#K3" title="GTARSKI3:func.3">reflection</a> (<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font>) <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a>  <a href="gtarski3.html#K3" title="GTARSKI3:func.3">reflection</a> (<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c2">a</font>), <a href="gtarski3.html#K3" title="GTARSKI3:func.3">reflection</a> (<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="gtarski3.html#T101" onmouseover="rs('gtarski3/T101')" onmouseout="rh()">GTARSKI3:101</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="E2:23_1_1"/>
 <a href="gtarski3.html#K3" title="GTARSKI3:func.3">reflection</a> (<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c2">a</font>),<font color="Maroon" title="c4">c</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a>  <a href="gtarski3.html#K3" title="GTARSKI3:func.3">reflection</a> (<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c2">a</font>), <a href="gtarski3.html#K3" title="GTARSKI3:func.3">reflection</a> (<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="gtarski1.html#D6" onmouseover="rs('gtarski1/D6')" onmouseout="rh()">GTARSKI1:def 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"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E2:23_1"/>
 <a href="gtarski4.html#R1" title="GTARSKI4:pred.1">right_angle</a>  <a href="gtarski3.html#K3" title="GTARSKI3:func.3">reflection</a> (<font color="Maroon" title="c4">c</font>,<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">  <a href="gtarski1.html#R1" title="GTARSKI1:pred.1">between</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font>, <a href="gtarski3.html#K3" title="GTARSKI3:func.3">reflection</a> (<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c2">a</font>)</span><br/><a NAME="E3:23_1"/>
 <a href="gtarski3.html#R9" title="GTARSKI3:pred.9">Middle</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font>, <a href="gtarski3.html#K3" title="GTARSKI3:func.3">reflection</a> (<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c2">a</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="gtarski3.html#D13" onmouseover="rs('gtarski3/D13')" onmouseout="rh()">GTARSKI3:def 13</a></span>;<br/><span class="kw">hence </span><a NAME="E4:23_1"/>
 <a href="gtarski1.html#R1" title="GTARSKI1:pred.1">between</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font>, <a href="gtarski3.html#K3" title="GTARSKI3:func.3">reflection</a> (<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c2">a</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>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E5:23"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23"><span class="lab"><font color="Green" title="E21">A1</font></span></a>, <a class="txt" href="#E3:23"><span class="lab"><font color="Green" title="E23">A3</font></span></a>, <a class="ref" href="gtarski4.html#T18" target="_self" onmouseover="rs('gtarski4/T18')" onmouseout="rh()">Satz8p6</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>
