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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="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">A9</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">S</font>  st  <a href="gtarski4.html#R3" title="GTARSKI4:pred.3">are_orthogonal</a> <font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">A9</font> holds <br/><font color="Olive" title="b1">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">A9</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">A</font>, <font color="Maroon" title="c3">A9</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</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#R3" title="GTARSKI4:pred.3">are_orthogonal</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">A9</font> implies <font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">A9</font> )</span><br/>





<span class="kw">assume </span><a NAME="E1:33"/>
 <a href="gtarski4.html#R3" title="GTARSKI4:pred.3">are_orthogonal</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">A9</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="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">A9</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">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="E3:33"/><span class="lab"><font color="Green" title="E27">A1</font></span>: 
 <a href="gtarski4.html#R2" title="GTARSKI4:pred.2">are_orthogonal</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">A9</font>
 ;<br/>
<a NAME="E4:33"/><span class="lab"><font color="Green" title="E28">A2</font></span>: 
( <font color="Maroon" title="c2">A</font> <a href="gtarski3.html#R8" title="GTARSKI3:pred.8">is_line</a>  &amp; <font color="Maroon" title="c3">A9</font> <a href="gtarski3.html#R8" title="GTARSKI3:pred.8">is_line</a>  &amp; <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A9</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> <font color="Maroon" title="c2">A</font> &amp; <font color="Olive" title="b2">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A9</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="c4">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="c4">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="c4">x</font>,<font color="Olive" title="b2">v</font> ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:33"><span class="lab"><font color="Green" title="E27">A1</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">consider </span><font color="Maroon" title="c5">u</font>, <font color="Maroon" title="c6">v</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="E6:33"/><span class="lab"><font color="Green" title="E29">A3</font></span>: 
<font color="Maroon" title="c5">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 <span class="kw">and </span><br/><a NAME="E7:33"/><span class="lab"><font color="Green" title="E30">A4</font></span>: 
<font color="Maroon" title="c6">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A9</font>
 <span class="kw">and </span><br/><a NAME="E8:33"/><span class="lab"><font color="Green" title="E31">A5</font></span>: 
<font color="Maroon" title="c5">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">x</font>
 <span class="kw">and </span><br/><a NAME="E9:33"/><span class="lab"><font color="Green" title="E32">A6</font></span>: 
<font color="Maroon" title="c6">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">x</font>
 <span class="kw">and </span><br/><a NAME="E10:33"/><span class="lab"><font color="Green" title="E33">A7</font></span>: 
 <a href="gtarski4.html#R1" title="GTARSKI4:pred.1">right_angle</a> <font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c6">v</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:33"><span class="lab"><font color="Green" title="E27">A1</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">assume </span><a NAME="E11:33"/>
<font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A9</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="E12:33"/><span class="kw">then </span>
 <a href="gtarski1.html#R6" title="GTARSKI1:pred.6">Collinear</a> <font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c6">v</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:33"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="txt" href="#E6:33"><span class="lab"><font color="Green" title="E29">A3</font></span></a>, <a class="txt" href="#E7:33"><span class="lab"><font color="Green" title="E30">A4</font></span></a>, <a class="ref" href="gtarski3.html#T90" onmouseover="rs('gtarski3/T90')" onmouseout="rh()">GTARSKI3:90</a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:33"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:33"><span class="lab"><font color="Green" title="E31">A5</font></span></a>, <a class="txt" href="#E9:33"><span class="lab"><font color="Green" title="E32">A6</font></span></a>, <a class="txt" href="#E10:33"><span class="lab"><font color="Green" title="E33">A7</font></span></a>, <a class="ref" href="gtarski4.html#T21" target="_self" onmouseover="rs('gtarski4/T21')" onmouseout="rh()">Satz8p9</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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