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<b>let </b><font color="Maroon">x</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">y</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><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">e1</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">n</font></span>)</span>, <font color="Maroon">e2</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">n</font></span>)</span>, <font color="Maroon">e3</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">n</font></span>)</span>, <font color="Maroon">e4</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">n</font></span>)</span>, <font color="Maroon">e5</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">n</font></span>)</span>, <font color="Maroon">e6</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">n</font></span>)</span>;<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">p3</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">p4</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><b>that </b><br/><a NAME="E1:12"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">e1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> &amp; <font color="Maroon">e2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> &amp; <font color="Maroon">e3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p3</font> &amp; <font color="Maroon">e4</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p4</font> )
 <b>and </b><br/><a NAME="E2:12"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">e5</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> <a href="euclid.html#K17">+</a> <font color="Maroon">p3</font>
 <b>and </b><br/><a NAME="E3:12"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">e6</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> <a href="euclid.html#K17">+</a> <font color="Maroon">p4</font>
 <b>and </b><br/><a NAME="E4:12"/><i><font color="Green">E48</font></i>: 
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">e1</font>,<font color="Maroon">e2</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">x</font>
 <b>and </b><br/><a NAME="E5:12"/><i><font color="Green">E49</font></i>: 
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">e3</font>,<font color="Maroon">e4</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">y</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">f1</font> = <font color="Maroon">e1</font>, <font color="Maroon">f2</font> = <font color="Maroon">e2</font>, <font color="Maroon">f3</font> = <font color="Maroon">e3</font>, <font color="Maroon">f4</font> = <font color="Maroon">e4</font>, <font color="Maroon">f5</font> = <font color="Maroon">e5</font>, <font color="Maroon">f6</font> = <font color="Maroon">e6</font> as    <a href="finseq_2.html#M2">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="E7:12"/><i><font color="Green">E50</font></i>: 
 <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="ref" href="euclid.html#D1">EUCLID:def 1</a></i>;<br/>
<i><font color="Green">E51</font></i>:  <a href="metric_1.html#K4">dist</a> <font color="Maroon">e5</font>,<font color="Maroon">e6</font> = 
<span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f5</font> <a href="euclid.html#K8">-</a> <font color="Maroon">f6</font></span>)</span></span><a href="euclid.html#K12">.|</a></span>
<b>by </b><i><a class="ref" href="sppol_1.html#T20">SPPOL_1:20</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f1</font> <a href="euclid.html#K7">+</a> <font color="Maroon">f3</font></span>)</span> <a href="euclid.html#K8">-</a> <font color="Maroon">f6</font></span>)</span></span><a href="euclid.html#K12">.|</a></span>
<b>by </b><i>, , <a class="ref" href="euclid.html#D10">EUCLID:def 10</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f1</font> <a href="euclid.html#K7">+</a> <font color="Maroon">f3</font></span>)</span> <a href="euclid.html#K8">-</a> <span class="p3">(<span class="default"><font color="Maroon">f2</font> <a href="euclid.html#K7">+</a> <font color="Maroon">f4</font></span>)</span></span>)</span></span><a href="euclid.html#K12">.|</a></span>
<b>by </b><i>, , <a class="ref" href="euclid.html#D10">EUCLID:def 10</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">f2</font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p3">(<span class="default"><font color="Maroon">f3</font> <a href="euclid.html#K8">-</a> <font color="Maroon">f4</font></span>)</span></span>)</span></span><a href="euclid.html#K12">.|</a></span>
<b>by </b><i>, </i>
;<br/>
<a NAME="E9:12"/><i><font color="Green">E52</font></i>: 
(  <a href="metric_1.html#K4">dist</a> <font color="Maroon">e1</font>,<font color="Maroon">e2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">f2</font></span>)</span></span><a href="euclid.html#K12">.|</a></span> &amp;  <a href="metric_1.html#K4">dist</a> <font color="Maroon">e3</font>,<font color="Maroon">e4</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f3</font> <a href="euclid.html#K8">-</a> <font color="Maroon">f4</font></span>)</span></span><a href="euclid.html#K12">.|</a></span> )
 
<b>by </b><i><a class="ref" href="sppol_1.html#T20">SPPOL_1:20</a></i>;<br/>
<a NAME="E10:12"/><i><font color="Green">E81</font></i>: 
<span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">f2</font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p3">(<span class="default"><font color="Maroon">f3</font> <a href="euclid.html#K8">-</a> <font color="Maroon">f4</font></span>)</span></span>)</span></span><a href="euclid.html#K12">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">f2</font></span>)</span></span><a href="euclid.html#K12">.|</a></span> <a href="real_1.html#K3">+</a> <span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f3</font> <a href="euclid.html#K8">-</a> <font color="Maroon">f4</font></span>)</span></span><a href="euclid.html#K12">.|</a></span>
 
<b>by </b><i><a class="ref" href="euclid.html#T15">EUCLID:15</a></i>;<br/>
<a NAME="E11:12"/>
<span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">e1</font>,<font color="Maroon">e2</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">e3</font>,<font color="Maroon">e4</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">x</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">y</font>
 
<b>by </b><i>, , <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<b>hence </b><a NAME="E12:12"/>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">e5</font>,<font color="Maroon">e6</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">x</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">y</font>
 <b>by </b><i>, , , <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>


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