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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><br/>  for <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> holds <br/> (  <a href="gtarski4.html#R2" title="GTARSKI4:pred.2">are_orthogonal</a> <font color="Olive" title="b1">A</font>,<font color="Olive" title="b3">x</font>,<font color="Olive" title="b2">A9</font> iff ( <font color="Olive" title="b1">A</font> <a href="gtarski3.html#R8" title="GTARSKI3:pred.8">is_line</a>  &amp; <font color="Olive" title="b2">A9</font> <a href="gtarski3.html#R8" title="GTARSKI3:pred.8">is_line</a>  &amp; <font color="Olive" title="b3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">A</font> &amp; <font color="Olive" title="b3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">A9</font> &amp;  ex <font color="Olive" title="b4">u</font>, <font color="Olive" title="b5">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="b4">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">A</font> &amp; <font color="Olive" title="b5">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">A9</font> &amp; <font color="Olive" title="b4">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">x</font> &amp; <font color="Olive" title="b5">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">x</font> &amp;  <a href="gtarski4.html#R1" title="GTARSKI4:pred.1">right_angle</a> <font color="Olive" title="b4">u</font>,<font color="Olive" title="b3">x</font>,<font color="Olive" title="b5">v</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">  for <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> holds <br/> (  <a href="gtarski4.html#R2" title="GTARSKI4:pred.2">are_orthogonal</a> <font color="Maroon" title="c2">A</font>,<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c3">A9</font> iff ( <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="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Olive" title="b1">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="b2">u</font>, <font color="Olive" title="b3">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="b2">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="b3">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="b2">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b3">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">x</font> &amp;  <a href="gtarski4.html#R1" title="GTARSKI4:pred.1">right_angle</a> <font color="Olive" title="b2">u</font>,<font color="Olive" title="b1">x</font>,<font color="Olive" title="b3">v</font> ) ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">x</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#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> iff ( <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><br/>







<div><span class="kw">hereby </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#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> ) implies  <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> )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:32_1"/><span class="lab"><font color="Green" title="E26">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>
 ; <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#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><br/><span class="kw">hence </span><a NAME="E2:32_1"/>
( <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> )
 ; <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> <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><br/><span class="kw">consider </span><font color="Maroon" title="c5">p</font>, <font color="Maroon" title="c6">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="E4:32_1"/><span class="lab"><font color="Green" title="E27">A2</font></span>: 
<font color="Maroon" title="c5">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">q</font>
 <span class="kw">and </span><br/><a NAME="E5:32_1"/><span class="lab"><font color="Green" title="E28">A3</font></span>: 
<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="c5">p</font>,<font color="Maroon" title="c6">q</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32_1"><span class="lab"><font color="Green" title="E26">A1</font></span></a>, <a class="ref" href="gtarski3.html#D11" onmouseover="rs('gtarski3/D11')" onmouseout="rh()">GTARSKI3:def 11</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c7">p9</font>, <font color="Maroon" title="c8">q9</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="E7:32_1"/><span class="lab"><font color="Green" title="E29">A4</font></span>: 
<font color="Maroon" title="c7">p9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">q9</font>
 <span class="kw">and </span><br/><a NAME="E8:32_1"/><span class="lab"><font color="Green" title="E30">A5</font></span>: 
<font color="Maroon" title="c3">A9</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="c7">p9</font>,<font color="Maroon" title="c8">q9</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32_1"><span class="lab"><font color="Green" title="E26">A1</font></span></a>, <a class="ref" href="gtarski3.html#D11" onmouseover="rs('gtarski3/D11')" onmouseout="rh()">GTARSKI3:def 11</a></span>;<br/><span class="kw">thus </span><a NAME="E9:32_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> <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> )
  <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="32_1_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_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="E1:32_1_1_1"/>
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p</font> or <font color="Maroon" title="c4">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">p</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:32_1_1_1_1"/><span class="lab"><font color="Green" title="E31">A6</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">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">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><br/><div class="add"><span class="kw">take </span>
<font color="Maroon" title="c6">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">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="c6">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A9</font> &amp; <font color="Maroon" title="c6">q</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="b1">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="Maroon" title="c6">q</font>,<font color="Maroon" title="c4">x</font>,<font color="Olive" title="b1">v</font> )</span><br/><a NAME="E2:32_1_1_1_1"/><span class="lab"><font color="Green" title="E32">A7</font></span>: 
<font color="Maroon" title="c6">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:32_1"><span class="lab"><font color="Green" title="E28">A3</font></span></a>, <a class="ref" href="gtarski3.html#T83" onmouseover="rs('gtarski3/T83')" onmouseout="rh()">GTARSKI3:83</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1_1_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="E1:32_1_1_1_1_1"/>
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p9</font> or <font color="Maroon" title="c4">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">p9</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:32_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E33">A8</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p9</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="c6">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A9</font> &amp; <font color="Maroon" title="c6">q</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="b1">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="Maroon" title="c6">q</font>,<font color="Maroon" title="c4">x</font>,<font color="Olive" title="b1">v</font> )</span><br/><div class="add"><span class="kw">take </span>
<font color="Maroon" title="c8">q9</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="c6">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c8">q9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A9</font> &amp; <font color="Maroon" title="c6">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">x</font> &amp; <font color="Maroon" title="c8">q9</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="Maroon" title="c6">q</font>,<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c8">q9</font> )</span><br/><span class="kw">thus </span><a NAME="E2:32_1_1_1_1_1_1"/>
( <font color="Maroon" title="c6">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c8">q9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A9</font> &amp; <font color="Maroon" title="c6">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">x</font> &amp; <font color="Maroon" title="c8">q9</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="Maroon" title="c6">q</font>,<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c8">q9</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E33">A8</font></span></a>, <a class="txt" href="#E7:32_1"><span class="lab"><font color="Green" title="E29">A4</font></span></a>, <a class="txt" href="#E1:32_1_1_1_1"><span class="lab"><font color="Green" title="E31">A6</font></span></a>, <a class="txt" href="#E4:32_1"><span class="lab"><font color="Green" title="E27">A2</font></span></a>, <a class="txt" href="#E1:32_1"><span class="lab"><font color="Green" title="E26">A1</font></span></a>, <a class="txt" href="#E8:32_1"><span class="lab"><font color="Green" title="E30">A5</font></span></a>, <a class="txt" href="#E2:32_1_1_1_1"><span class="lab"><font color="Green" title="E32">A7</font></span></a>, <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"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:32_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E33">A9</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">p9</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="c6">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A9</font> &amp; <font color="Maroon" title="c6">q</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="b1">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="Maroon" title="c6">q</font>,<font color="Maroon" title="c4">x</font>,<font color="Olive" title="b1">v</font> )</span><br/><div class="add"><span class="kw">take </span>
<font color="Maroon" title="c7">p9</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="c6">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c7">p9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A9</font> &amp; <font color="Maroon" title="c6">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">x</font> &amp; <font color="Maroon" title="c7">p9</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="Maroon" title="c6">q</font>,<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c7">p9</font> )</span><br/><a NAME="E2:32_1_1_1_1_1_2"/>
<font color="Maroon" title="c7">p9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:32_1"><span class="lab"><font color="Green" title="E30">A5</font></span></a>, <a class="ref" href="gtarski3.html#T83" onmouseover="rs('gtarski3/T83')" onmouseout="rh()">GTARSKI3:83</a></span>;<br/><span class="kw">hence </span><a NAME="E3:32_1_1_1_1_1_2"/>
( <font color="Maroon" title="c6">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c7">p9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A9</font> &amp; <font color="Maroon" title="c6">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">x</font> &amp; <font color="Maroon" title="c7">p9</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="Maroon" title="c6">q</font>,<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c7">p9</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E33">A9</font></span></a>, <a class="txt" href="#E1:32_1_1_1_1"><span class="lab"><font color="Green" title="E31">A6</font></span></a>, <a class="txt" href="#E4:32_1"><span class="lab"><font color="Green" title="E27">A2</font></span></a>, <a class="txt" href="#E5:32_1"><span class="lab"><font color="Green" title="E28">A3</font></span></a>, <a class="txt" href="#E1:32_1"><span class="lab"><font color="Green" title="E26">A1</font></span></a>, <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"> 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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:32_1_1_1_2"/><span class="lab"><font color="Green" title="E31">A10</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">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">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><br/><div class="add"><span class="kw">take </span>
<font color="Maroon" title="c5">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">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">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A9</font> &amp; <font color="Maroon" title="c5">p</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="b1">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="Maroon" title="c5">p</font>,<font color="Maroon" title="c4">x</font>,<font color="Olive" title="b1">v</font> )</span><br/><a NAME="E2:32_1_1_1_2"/><span class="lab"><font color="Green" title="E32">A11</font></span>: 
<font color="Maroon" title="c5">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:32_1"><span class="lab"><font color="Green" title="E28">A3</font></span></a>, <a class="ref" href="gtarski3.html#T83" onmouseover="rs('gtarski3/T83')" onmouseout="rh()">GTARSKI3:83</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_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="E1:32_1_1_1_2_1"/>
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p9</font> or <font color="Maroon" title="c4">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">p9</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:32_1_1_1_2_1_1"/><span class="lab"><font color="Green" title="E33">A12</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p9</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">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A9</font> &amp; <font color="Maroon" title="c5">p</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="b1">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="Maroon" title="c5">p</font>,<font color="Maroon" title="c4">x</font>,<font color="Olive" title="b1">v</font> )</span><br/><div class="add"><span class="kw">take </span>
<font color="Maroon" title="c8">q9</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">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c8">q9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A9</font> &amp; <font color="Maroon" title="c5">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">x</font> &amp; <font color="Maroon" title="c8">q9</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="Maroon" title="c5">p</font>,<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c8">q9</font> )</span><br/><a NAME="E2:32_1_1_1_2_1_1"/>
<font color="Maroon" title="c8">q9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:32_1"><span class="lab"><font color="Green" title="E30">A5</font></span></a>, <a class="ref" href="gtarski3.html#T83" onmouseover="rs('gtarski3/T83')" onmouseout="rh()">GTARSKI3:83</a></span>;<br/><span class="kw">hence </span><a NAME="E3:32_1_1_1_2_1_1"/>
( <font color="Maroon" title="c5">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c8">q9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A9</font> &amp; <font color="Maroon" title="c5">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">x</font> &amp; <font color="Maroon" title="c8">q9</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="Maroon" title="c5">p</font>,<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c8">q9</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E33">A12</font></span></a>, <a class="txt" href="#E7:32_1"><span class="lab"><font color="Green" title="E29">A4</font></span></a>, <a class="txt" href="#E1:32_1_1_1_2"><span class="lab"><font color="Green" title="E31">A10</font></span></a>, <a class="txt" href="#E5:32_1"><span class="lab"><font color="Green" title="E28">A3</font></span></a>, <a class="ref" href="gtarski3.html#T83" onmouseover="rs('gtarski3/T83')" onmouseout="rh()">GTARSKI3:83</a>, <a class="txt" href="#E1:32_1"><span class="lab"><font color="Green" title="E26">A1</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="32_1_1_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:32_1_1_1_2_1_2"/><span class="lab"><font color="Green" title="E33">A13</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">p9</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">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A9</font> &amp; <font color="Maroon" title="c5">p</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="b1">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="Maroon" title="c5">p</font>,<font color="Maroon" title="c4">x</font>,<font color="Olive" title="b1">v</font> )</span><br/><div class="add"><span class="kw">take </span>
<font color="Maroon" title="c7">p9</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">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c7">p9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A9</font> &amp; <font color="Maroon" title="c5">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">x</font> &amp; <font color="Maroon" title="c7">p9</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="Maroon" title="c5">p</font>,<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c7">p9</font> )</span><br/><span class="kw">thus </span><a NAME="E2:32_1_1_1_2_1_2"/>
( <font color="Maroon" title="c5">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c7">p9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A9</font> &amp; <font color="Maroon" title="c5">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">x</font> &amp; <font color="Maroon" title="c7">p9</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="Maroon" title="c5">p</font>,<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c7">p9</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:32_1"><span class="lab"><font color="Green" title="E30">A5</font></span></a>, <a class="ref" href="gtarski3.html#T83" onmouseover="rs('gtarski3/T83')" onmouseout="rh()">GTARSKI3:83</a>, <a class="txt" href="#E1:32_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E33">A13</font></span></a>, <a class="txt" href="#E2:32_1_1_1_2"><span class="lab"><font color="Green" title="E32">A11</font></span></a>, <a class="txt" href="#E1:32_1_1_1_2"><span class="lab"><font color="Green" title="E31">A10</font></span></a>, <a class="txt" href="#E1:32_1"><span class="lab"><font color="Green" title="E26">A1</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><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></div>
<span class="kw">end;</span></div>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:32"/><span class="lab"><font color="Green" title="E26">A14</font></span>: 
<font color="Maroon" title="c2">A</font> <a href="gtarski3.html#R8" title="GTARSKI3:pred.8">is_line</a> 
 <span class="kw">and </span><br/><a NAME="E3:32"/><span class="lab"><font color="Green" title="E27">A15</font></span>: 
<font color="Maroon" title="c3">A9</font> <a href="gtarski3.html#R8" title="GTARSKI3:pred.8">is_line</a> 
 <span class="kw">and </span><br/><a NAME="E4:32"/><span class="lab"><font color="Green" title="E28">A16</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E5:32"/><span class="lab"><font color="Green" title="E29">A17</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E6:32"/><span class="lab"><font color="Green" title="E30">A18</font></span>: 
 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> )
 ; <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#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></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c5">u9</font>, <font color="Maroon" title="c6">v9</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="E8:32"/><span class="lab"><font color="Green" title="E31">A19</font></span>: 
<font color="Maroon" title="c5">u9</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="E9:32"/><span class="lab"><font color="Green" title="E32">A20</font></span>: 
<font color="Maroon" title="c6">v9</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="E10:32"/><span class="lab"><font color="Green" title="E33">A21</font></span>: 
<font color="Maroon" title="c5">u9</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="E11:32"/><span class="lab"><font color="Green" title="E34">A22</font></span>: 
<font color="Maroon" title="c6">v9</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="E12:32"/><span class="lab"><font color="Green" title="E35">A23</font></span>: 
 <a href="gtarski4.html#R1" title="GTARSKI4:pred.1">right_angle</a> <font color="Maroon" title="c5">u9</font>,<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c6">v9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:32"><span class="lab"><font color="Green" title="E30">A18</font></span></a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="32_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">  for <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 <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> holds <br/> <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></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">u</font>, <font color="Maroon" title="c8">v</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"> ( <font color="Maroon" title="c7">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c8">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A9</font> implies  <a href="gtarski4.html#R1" title="GTARSKI4:pred.1">right_angle</a> <font color="Maroon" title="c7">u</font>,<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c8">v</font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:32_2"/><span class="lab"><font color="Green" title="E36">A24</font></span>: 
<font color="Maroon" title="c7">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="E2:32_2"/><span class="lab"><font color="Green" title="E37">A25</font></span>: 
<font color="Maroon" title="c8">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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">  <a href="gtarski4.html#R1" title="GTARSKI4:pred.1">right_angle</a> <font color="Maroon" title="c7">u</font>,<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c8">v</font></span><br/><a NAME="E3:32_2"/>
 <a href="gtarski1.html#R6" title="GTARSKI1:pred.6">Collinear</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">u9</font>,<font color="Maroon" title="c7">u</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:32"><span class="lab"><font color="Green" title="E26">A14</font></span></a>, <a class="txt" href="#E4:32"><span class="lab"><font color="Green" title="E28">A16</font></span></a>, <a class="txt" href="#E8:32"><span class="lab"><font color="Green" title="E31">A19</font></span></a>, <a class="txt" href="#E1:32_2"><span class="lab"><font color="Green" title="E36">A24</font></span></a>, <a class="txt" href="#E10:32"><span class="lab"><font color="Green" title="E33">A21</font></span></a>, <a class="ref" href="gtarski3.html#T90" onmouseover="rs('gtarski3/T90')" onmouseout="rh()">GTARSKI3:90</a></span>;<br/><a NAME="E4:32_2"/><span class="kw">then </span>
 <a href="gtarski4.html#R1" title="GTARSKI4:pred.1">right_angle</a> <font color="Maroon" title="c7">u</font>,<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c6">v9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:32"><span class="lab"><font color="Green" title="E35">A23</font></span></a>, <a class="txt" href="#E10:32"><span class="lab"><font color="Green" title="E33">A21</font></span></a>, <a class="ref" href="gtarski4.html#T15" target="_self" onmouseover="rs('gtarski4/T15')" onmouseout="rh()">Satz8p3</a></span>;<br/><a NAME="E5:32_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A26</font></span>: 
 <a href="gtarski4.html#R1" title="GTARSKI4:pred.1">right_angle</a> <font color="Maroon" title="c6">v9</font>,<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c7">u</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="E6:32_2"/>
 <a href="gtarski1.html#R6" title="GTARSKI1:pred.6">Collinear</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c6">v9</font>,<font color="Maroon" title="c8">v</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:32"><span class="lab"><font color="Green" title="E27">A15</font></span></a>, <a class="txt" href="#E5:32"><span class="lab"><font color="Green" title="E29">A17</font></span></a>, <a class="txt" href="#E9:32"><span class="lab"><font color="Green" title="E32">A20</font></span></a>, <a class="txt" href="#E2:32_2"><span class="lab"><font color="Green" title="E37">A25</font></span></a>, <a class="txt" href="#E10:32"><span class="lab"><font color="Green" title="E33">A21</font></span></a>, <a class="ref" href="gtarski3.html#T90" onmouseover="rs('gtarski3/T90')" onmouseout="rh()">GTARSKI3:90</a></span>;<br/><a NAME="E7:32_2"/><span class="kw">then </span>
 <a href="gtarski4.html#R1" title="GTARSKI4:pred.1">right_angle</a> <font color="Maroon" title="c8">v</font>,<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c7">u</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:32"><span class="lab"><font color="Green" title="E34">A22</font></span></a>, <a class="txt" href="#E5:32_2"><span class="lab"><font color="Green" title="E38">A26</font></span></a>, <a class="ref" href="gtarski4.html#T15" target="_self" onmouseover="rs('gtarski4/T15')" onmouseout="rh()">Satz8p3</a></span>;<br/><span class="kw">hence </span><a NAME="E8:32_2"/>
 <a href="gtarski4.html#R1" title="GTARSKI4:pred.1">right_angle</a> <font color="Maroon" title="c7">u</font>,<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c8">v</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>; <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="E14:32"/>
 <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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:32"><span class="lab"><font color="Green" title="E26">A14</font></span></a>, <a class="txt" href="#E3:32"><span class="lab"><font color="Green" title="E27">A15</font></span></a>, <a class="txt" href="#E4:32"><span class="lab"><font color="Green" title="E28">A16</font></span></a>, <a class="txt" href="#E5:32"><span class="lab"><font color="Green" title="E29">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>
