<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">R</font> be   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> ;<br/><b>let </b><font color="Maroon">p1</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span>, <font color="Maroon">p2</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span>, <font color="Maroon">q1</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span>;<br/>





<b>assume </b><a NAME="E1:76"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">R</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid_4.html#K9" target="_self">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">q1</font> <a href="euclid.html#K20">-</a> <font color="Olive">p</font></span>)</span></span><a href="euclid_4.html#K9" target="_self">.|</a></span> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> : <font color="Olive">p</font> <a href="hidden.html#R2">in</a>  <a href="euclid_4.html#K7" target="_self">Line</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font> </span>}</span> 
 ;<br/>

<b>consider </b><font color="Maroon">x1</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font>, <font color="Maroon">x2</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font><b> such that </b><br/><a NAME="E3:76"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> &amp; <font color="Maroon">p2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font> &amp;  <a href="euclid_4.html#K2" target="_self">Line</a> <font color="Maroon">x1</font>,<font color="Maroon">x2</font> <a href="hidden.html#R1">=</a>  <a href="euclid_4.html#K7" target="_self">Line</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font> )
 <b>by </b><i><a class="txt" href="euclid_4.html#E1:76"><i><font color="Green">E41</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">y1</font> = <font color="Maroon">q1</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font> <b>by </b><i><a class="ref" href="euclid.html#T25">EUCLID:25</a></i>;<br/>
<b>consider </b><font color="Maroon">y2</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font><b> such that </b><br/><a NAME="E6:76"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">y2</font> <a href="hidden.html#R2">in</a>  <a href="euclid_4.html#K2" target="_self">Line</a> <font color="Maroon">x1</font>,<font color="Maroon">x2</font>
 <b>and </b><br/><a NAME="E7:76"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">x1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x2</font>,<font color="Maroon">y1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">y2</font> <a href="euclid_4.html#R1" target="_self">are_orthogonal</a> 
 <b>and </b><br/><a NAME="E8:76"/><i><font color="Green">E61</font></i>: 
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="euclid_4.html#K2" target="_self">Line</a> <font color="Maroon">x1</font>,<font color="Maroon">x2</font> holds <br/><span class="p1"><a href="euclid_4.html#K4" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">y1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">y2</font></span>)</span></span><a href="euclid_4.html#K4" target="_self">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="euclid_4.html#K4" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">y1</font> <a href="euclid.html#K8">-</a> <font color="Olive">x</font></span>)</span></span><a href="euclid_4.html#K4" target="_self">.|</a></span>
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">q2</font> = <font color="Maroon">y2</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> <b>by </b><i><a class="ref" href="euclid.html#T25">EUCLID:25</a></i>;<br/>
<a NAME="E10:76"/>
( <font color="Maroon">p1</font> <a href="euclid.html#K20">-</a> <font color="Maroon">p2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x2</font> &amp; <font color="Maroon">y1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">y2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q1</font> <a href="euclid.html#K20">-</a> <font color="Maroon">q2</font> )
 
<b>by </b><i><a class="txt" href="euclid_4.html#E10"><i><font color="Green">Lemma51</font></i></a>, <a class="ref" href="euclid.html#D13">EUCLID:def 13</a></i>;<br/>
<a NAME="E11:76"/><b>then </b><i><font color="Green">E62</font></i>: 
(  <a href="euclid_4.html#K8" target="_self">TPn2Rn</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K20">-</a> <font color="Maroon">p2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x2</font> &amp;  <a href="euclid_4.html#K8" target="_self">TPn2Rn</a> <span class="p1">(<span class="default"><font color="Maroon">q1</font> <a href="euclid.html#K20">-</a> <font color="Maroon">q2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">y1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">y2</font> )
 
;<br/>
<a NAME="E12:76"/>
<span class="p1"><a href="euclid_4.html#K10" target="_self">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K20">-</a> <font color="Maroon">p2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">q1</font> <a href="euclid.html#K20">-</a> <font color="Maroon">q2</font></span>)</span></span><a href="euclid_4.html#K10" target="_self">)|</a></span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="euclid_4.html#T9" target="_self">Th9</a>, <a class="txt" href="euclid_4.html#E13"><i><font color="Green">Lemma55</font></i></a>, </i>;<br/>
<a NAME="E13:76"/><b>then </b><i><font color="Green">E63</font></i>: 
<font color="Maroon">p1</font> <a href="euclid.html#K20">-</a> <font color="Maroon">p2</font>,<font color="Maroon">q1</font> <a href="euclid.html#K20">-</a> <font color="Maroon">q2</font> <a href="euclid_4.html#R2" target="_self">are_orthogonal</a> 
 
<b>by </b><i><a class="txt" href="euclid_4.html#E3:76"><i><font color="Green">E49</font></i></a></i>;<br/>
<a NAME="E14:76"/><i><font color="Green">E97</font></i>: 
for <font color="Olive">p</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span>  st <font color="Olive">p</font> <a href="hidden.html#R2">in</a>  <a href="euclid_4.html#K7" target="_self">Line</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font> holds <br/><span class="p1"><a href="euclid_4.html#K9" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">q1</font> <a href="euclid.html#K20">-</a> <font color="Maroon">q2</font></span>)</span></span><a href="euclid_4.html#K9" target="_self">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="euclid_4.html#K9" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">q1</font> <a href="euclid.html#K20">-</a> <font color="Olive">p</font></span>)</span></span><a href="euclid_4.html#K9" target="_self">.|</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:76_1"/><i><font color="Green">E98</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="euclid_4.html#K7" target="_self">Line</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">p</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font> <b>by </b><i><a class="ref" href="euclid.html#T25">EUCLID:25</a></i>;<br/>
<a NAME="E3:76_1"/>
<font color="Maroon">q1</font> <a href="euclid.html#K20">-</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="euclid.html#D13">EUCLID:def 13</a></i>;<br/>
<a NAME="E4:76_1"/><b>then </b><i><font color="Green">E99</font></i>: 
 <a href="euclid_4.html#K8" target="_self">TPn2Rn</a> <span class="p1">(<span class="default"><font color="Maroon">q1</font> <a href="euclid.html#K20">-</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">y1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x</font>
 
;<br/>
<a NAME="E5:76_1"/>
<span class="p1"><a href="euclid_4.html#K4" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">y1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">y2</font></span>)</span></span><a href="euclid_4.html#K4" target="_self">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="euclid_4.html#K4" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">y1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x</font></span>)</span></span><a href="euclid_4.html#K4" target="_self">.|</a></span>
 
<b>by </b><i><a class="txt" href="euclid_4.html#E10"><i><font color="Green">Lemma51</font></i></a>, <a class="ref" href="euclid_4.html#T10" target="_self">Th10</a>, <a class="txt" href="euclid_4.html#E6:76"><i><font color="Green">E58</font></i></a></i>;<br/>
<a NAME="E6:76_1"/><b>then </b>
<span class="p1"><a href="euclid_4.html#K9" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">q1</font> <a href="euclid.html#K20">-</a> <font color="Maroon">q2</font></span>)</span></span><a href="euclid_4.html#K9" target="_self">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="euclid_4.html#K4" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><a href="euclid_4.html#K8" target="_self">TPn2Rn</a> <span class="p3">(<span class="default"><font color="Maroon">q1</font> <a href="euclid.html#K20">-</a> <font color="Maroon">p</font></span>)</span></span>)</span></span><a href="euclid_4.html#K4" target="_self">.|</a></span>
 
<b>by </b><i><a class="txt" href="euclid_4.html#E13"><i><font color="Green">Lemma55</font></i></a>, <a class="ref" href="euclid_4.html#T12" target="_self">Th12</a></i>;<br/>
<b>hence </b><a NAME="E7:76_1"/>
<span class="p1"><a href="euclid_4.html#K9" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">q1</font> <a href="euclid.html#K20">-</a> <font color="Maroon">q2</font></span>)</span></span><a href="euclid_4.html#K9" target="_self">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="euclid_4.html#K9" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">q1</font> <a href="euclid.html#K20">-</a> <font color="Maroon">p</font></span>)</span></span><a href="euclid_4.html#K9" target="_self">.|</a></span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E15:76"/>
<font color="Maroon">p1</font> <a href="hidden.html#R2">in</a>  <a href="euclid_4.html#K7" target="_self">Line</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>
 
<b>by </b><i><a class="ref" href="euclid_4.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E16:76"/><b>then </b><i><font color="Green">E115</font></i>: 
<span class="p1"><a href="euclid_4.html#K9" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">q1</font> <a href="euclid.html#K20">-</a> <font color="Maroon">p1</font></span>)</span></span><a href="euclid_4.html#K9" target="_self">.|</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 
<b>by </b><i/>;<br/>
<a NAME="E17:76"/><i><font color="Green">E116</font></i>: 
for <font color="Olive">s</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">s</font> holds <br/> ex <font color="Olive">r</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>( <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1"><a href="euclid_4.html#K9" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">q1</font> <a href="euclid.html#K20">-</a> <font color="Maroon">q2</font></span>)</span></span><a href="euclid_4.html#K9" target="_self">.|</a></span> <a href="xcmplx_0.html#K2">+</a> <font color="Olive">s</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">s</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><a NAME="E1:76_2"/><i><font color="Green">E117</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">s</font>
 ;<br/>

<b>take </b><font color="Maroon">r</font> = <span class="p1"><a href="euclid_4.html#K9" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">q1</font> <a href="euclid.html#K20">-</a> <font color="Maroon">q2</font></span>)</span></span><a href="euclid_4.html#K9" target="_self">.|</a></span>;<br/>

<b>thus </b><a NAME="E2:76_2"/>
( <font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> &amp; <font color="Maroon">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1"><a href="euclid_4.html#K9" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">q1</font> <a href="euclid.html#K20">-</a> <font color="Maroon">q2</font></span>)</span></span><a href="euclid_4.html#K9" target="_self">.|</a></span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">s</font> )
 <b>by </b><i>, <a class="txt" href="euclid_4.html#E10"><i><font color="Green">Lemma51</font></i></a>, , <a class="txt" href="euclid_4.html#E7:76"><i><font color="Green">E60</font></i></a>, <a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E18:76"/><i><font color="Green">E118</font></i>: 
for <font color="Olive">r</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> holds <br/><span class="p1"><a href="euclid_4.html#K9" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">q1</font> <a href="euclid.html#K20">-</a> <font color="Maroon">q2</font></span>)</span></span><a href="euclid_4.html#K9" target="_self">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">r</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><a NAME="E1:76_3"/>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 ;<br/>

<b>then consider </b><font color="Maroon">p0</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span><b> such that </b><br/><a NAME="E3:76_3"/><i><font color="Green">E120</font></i>: 
( <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid_4.html#K9" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">q1</font> <a href="euclid.html#K20">-</a> <font color="Maroon">p0</font></span>)</span></span><a href="euclid_4.html#K9" target="_self">.|</a></span> &amp; <font color="Maroon">p0</font> <a href="hidden.html#R2">in</a>  <a href="euclid_4.html#K7" target="_self">Line</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font> )
 <b>by </b><i/>;<br/>
<b>thus </b><a NAME="E4:76_3"/>
<span class="p1"><a href="euclid_4.html#K9" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">q1</font> <a href="euclid.html#K20">-</a> <font color="Maroon">q2</font></span>)</span></span><a href="euclid_4.html#K9" target="_self">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="euclid_4.html#T11" target="_self">Th11</a>, <a class="txt" href="euclid_4.html#E8:76"><i><font color="Green">E61</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E19:76"/><b>then </b>
<font color="Maroon">R</font> is <a href="seq_4.html#V2">bounded_below</a>
 
<b>by </b><i><a class="ref" href="seq_4.html#D2">SEQ_4:def 2</a></i>;<br/>
<a NAME="E20:76"/><b>then </b>
<span class="p1"><a href="euclid_4.html#K9" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">q1</font> <a href="euclid.html#K20">-</a> <font color="Maroon">q2</font></span>)</span></span><a href="euclid_4.html#K9" target="_self">.|</a></span> <a href="hidden.html#R1">=</a>  <a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">R</font>
 
<b>by </b><i><a class="txt" href="euclid_4.html#E6:76"><i><font color="Green">E58</font></i></a>, <a class="ref" href="euclid_4.html#T12" target="_self">Th12</a>, <a class="txt" href="euclid_4.html#E7:76"><i><font color="Green">E60</font></i></a>, <a class="ref" href="seq_4.html#D5">SEQ_4:def 5</a></i>;<br/>
<b>hence </b><a NAME="E21:76"/>
 ex <font color="Olive">q2</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> st <br/>( <font color="Olive">q2</font> <a href="hidden.html#R2">in</a>  <a href="euclid_4.html#K7" target="_self">Line</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font> &amp; <span class="p1"><a href="euclid_4.html#K9" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">q1</font> <a href="euclid.html#K20">-</a> <font color="Olive">q2</font></span>)</span></span><a href="euclid_4.html#K9" target="_self">.|</a></span> <a href="hidden.html#R1">=</a>  <a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">R</font> &amp; <font color="Maroon">p1</font> <a href="euclid.html#K20">-</a> <font color="Maroon">p2</font>,<font color="Maroon">q1</font> <a href="euclid.html#K20">-</a> <font color="Olive">q2</font> <a href="euclid_4.html#R2" target="_self">are_orthogonal</a>  )
 <b>by </b><i><a class="txt" href="euclid_4.html#E10"><i><font color="Green">Lemma51</font></i></a>, , </i>;<br/>


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