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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> being   <a href="gtarski1.html#NM1" title="GTARSKI1:NM.1">POINT</a> of <font color="Maroon" title="c1">S</font>  ex <font color="Olive" title="b3">x</font> being   <a href="gtarski1.html#NM1" title="GTARSKI1:NM.1">POINT</a> of <font color="Maroon" title="c1">S</font> st  <a href="gtarski3.html#R9" title="GTARSKI3:pred.9">Middle</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b3">x</font>,<font color="Olive" title="b2">b</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</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">  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  <a href="gtarski3.html#R9" title="GTARSKI3:pred.9">Middle</a> <font color="Maroon" title="c2">a</font>,<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c3">b</font></span><br/>



<span class="kw">set </span><font color="Maroon" title="c4">c</font> =  the   <a href="gtarski1.html#NM1" title="GTARSKI1:NM.1">POINT</a> of <font color="Maroon" title="c1">S</font>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:50_1"/>
( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> or <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/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_1"><span class="kw">suppose </span></a><a NAME="E1:50_1_1"/>
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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">  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  <a href="gtarski3.html#R9" title="GTARSKI3:pred.9">Middle</a> <font color="Maroon" title="c2">a</font>,<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c3">b</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:50_1_1"/>
 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  <a href="gtarski3.html#R9" title="GTARSKI3:pred.9">Middle</a> <font color="Maroon" title="c2">a</font>,<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c3">b</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>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_2"><span class="kw">suppose </span></a><a NAME="E1:50_1_2"/><span class="lab"><font color="Green" title="E40">A1</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>
 ; <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  <a href="gtarski3.html#R9" title="GTARSKI3:pred.9">Middle</a> <font color="Maroon" title="c2">a</font>,<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c3">b</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">q</font>, <font color="Maroon" title="c6">t</font> being   <a href="gtarski1.html#NM1" title="GTARSKI1:NM.1">POINT</a> of <font color="Maroon" title="c1">S</font><span class="kw"> such that </span><br/><a NAME="E3:50_1_2"/><span class="lab"><font color="Green" title="E41">A2</font></span>: 
 <a href="gtarski4.html#R6" title="GTARSKI4:pred.6">are_orthogonal</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c3">b</font>
 <span class="kw">and </span><br/><a NAME="E4:50_1_2"/>
 <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="c6">t</font>
 <span class="kw">and </span><br/><a NAME="E5:50_1_2"/>
 <a href="gtarski1.html#R1" title="GTARSKI1:pred.1">between</a>  the   <a href="gtarski1.html#NM1" title="GTARSKI1:NM.1">POINT</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c6">t</font>,<font color="Maroon" title="c5">q</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="gtarski4.html#T36" target="_self" onmouseover="rs('gtarski4/T36')" onmouseout="rh()">Satz8p21</a></span>;<br/><a NAME="E6:50_1_2"/><span class="lab"><font color="Green" title="E42">A3</font></span>: 
 <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="c5">q</font>,<font color="Maroon" title="c3">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:50_1_2"><span class="lab"><font color="Green" title="E41">A2</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c7">p</font>, <font color="Maroon" title="c8">t9</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="E8:50_1_2"/><span class="lab"><font color="Green" title="E43">A4</font></span>: 
 <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="c7">p</font>,<font color="Maroon" title="c2">a</font>
 <span class="kw">and </span><br/><a NAME="E9:50_1_2"/><span class="lab"><font color="Green" title="E44">A5</font></span>: 
 <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="c8">t9</font>
 <span class="kw">and </span><br/><a NAME="E10:50_1_2"/><span class="lab"><font color="Green" title="E45">A6</font></span>: 
 <a href="gtarski1.html#R1" title="GTARSKI1:pred.1">between</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c8">t9</font>,<font color="Maroon" title="c7">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:50_1_2"><span class="lab"><font color="Green" title="E40">A1</font></span></a>, <a class="ref" href="gtarski4.html#T36" target="_self" onmouseover="rs('gtarski4/T36')" onmouseout="rh()">Satz8p21</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_2_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:50_1_2_1"/>
( <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c7">p</font> <a href="gtarski3.html#R6" title="GTARSKI3:pred.6">&lt;=</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">q</font> or <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">q</font> <a href="gtarski3.html#R6" title="GTARSKI3:pred.6">&lt;=</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c7">p</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="gtarski3.html#T64" onmouseover="rs('gtarski3/T64')" onmouseout="rh()">GTARSKI3:64</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:50_1_2_1_1"/>
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c7">p</font> <a href="gtarski3.html#R6" title="GTARSKI3:pred.6">&lt;=</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">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">  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  <a href="gtarski3.html#R9" title="GTARSKI3:pred.9">Middle</a> <font color="Maroon" title="c2">a</font>,<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c3">b</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:50_1_2_1_1"/>
 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  <a href="gtarski3.html#R9" title="GTARSKI3:pred.9">Middle</a> <font color="Maroon" title="c2">a</font>,<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c3">b</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="gtarski4.html#T38" target="_self" onmouseover="rs('gtarski4/T38')" onmouseout="rh()">Satz8p22lemma</a>, <a class="txt" href="#E6:50_1_2"><span class="lab"><font color="Green" title="E42">A3</font></span></a>, <a class="txt" href="#E8:50_1_2"><span class="lab"><font color="Green" title="E43">A4</font></span></a>, <a class="txt" href="#E9:50_1_2"><span class="lab"><font color="Green" title="E44">A5</font></span></a>, <a class="txt" href="#E10:50_1_2"><span class="lab"><font color="Green" title="E45">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"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:50_1_2_1_2"/><span class="lab"><font color="Green" title="E46">A7</font></span>: 
<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">q</font> <a href="gtarski3.html#R6" title="GTARSKI3:pred.6">&lt;=</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c7">p</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">x</font> being   <a href="gtarski1.html#NM1" title="GTARSKI1:NM.1">POINT</a> of <font color="Maroon" title="c1">S</font> st  <a href="gtarski3.html#R9" title="GTARSKI3:pred.9">Middle</a> <font color="Maroon" title="c2">a</font>,<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c3">b</font></span><br/><div class="add"><a NAME="E2:50_1_2_1_2"/>
(  <a href="gtarski4.html#R6" title="GTARSKI4:pred.6">are_orthogonal</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c7">p</font>,<font color="Maroon" title="c2">a</font> &amp;  <a href="gtarski4.html#R6" title="GTARSKI4:pred.6">are_orthogonal</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c3">b</font> &amp;  <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="c8">t9</font> &amp;  <a href="gtarski1.html#R1" title="GTARSKI1:pred.1">between</a> <font color="Maroon" title="c7">p</font>,<font color="Maroon" title="c8">t9</font>,<font color="Maroon" title="c5">q</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:50_1_2"><span class="lab"><font color="Green" title="E45">A6</font></span></a>, <a class="txt" href="#E8:50_1_2"><span class="lab"><font color="Green" title="E43">A4</font></span></a>, <a class="txt" href="#E3:50_1_2"><span class="lab"><font color="Green" title="E41">A2</font></span></a>, <a class="txt" href="#E9:50_1_2"><span class="lab"><font color="Green" title="E44">A5</font></span></a>, <a class="ref" href="gtarski3.html#T14" onmouseover="rs('gtarski3/T14')" onmouseout="rh()">GTARSKI3:14</a>, <a class="ref" href="gtarski3.html#T45" onmouseover="rs('gtarski3/T45')" onmouseout="rh()">GTARSKI3:45</a></span>;<br/><a NAME="E3:50_1_2_1_2"/><span class="kw">then </span>
 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  <a href="gtarski3.html#R9" title="GTARSKI3:pred.9">Middle</a> <font color="Maroon" title="c3">b</font>,<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c2">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="gtarski4.html#T38" target="_self" onmouseover="rs('gtarski4/T38')" onmouseout="rh()">Satz8p22lemma</a>, <a class="txt" href="#E1:50_1_2_1_2"><span class="lab"><font color="Green" title="E46">A7</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:50_1_2_1_2"/>
 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  <a href="gtarski3.html#R9" title="GTARSKI3:pred.9">Middle</a> <font color="Maroon" title="c2">a</font>,<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c3">b</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="gtarski3.html#T96" onmouseover="rs('gtarski3/T96')" onmouseout="rh()">GTARSKI3:96</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

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