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<b>let </b><font color="Maroon" title="c1">x</font>, <font color="Maroon" title="c2">y</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/><b>let </b><font color="Maroon" title="c3">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c4">e1</font>, <font color="Maroon" title="c5">e2</font>, <font color="Maroon" title="c6">e3</font>, <font color="Maroon" title="c7">e4</font>, <font color="Maroon" title="c8">e5</font>, <font color="Maroon" title="c9">e6</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c3">n</font></span>)</span>;<br/><b>let </b><font color="Maroon" title="c10">p1</font>, <font color="Maroon" title="c11">p2</font>, <font color="Maroon" title="c12">p3</font>, <font color="Maroon" title="c13">p4</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c3">n</font></span>)</span>;<br/>









<b>assume </b><b>that </b><br/><a NAME="E1:12"/><i><font color="Green" title="E10">A1</font></i>: 
( <font color="Maroon" title="c4">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">p1</font> &amp; <font color="Maroon" title="c5">e2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p2</font> &amp; <font color="Maroon" title="c6">e3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">p3</font> &amp; <font color="Maroon" title="c7">e4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">p4</font> )
 <b>and </b><br/><a NAME="E2:12"/><i><font color="Green" title="E11">A2</font></i>: 
<font color="Maroon" title="c8">e5</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <font color="Maroon" title="c12">p3</font>
 <b>and </b><br/><a NAME="E3:12"/><i><font color="Green" title="E12">A3</font></i>: 
<font color="Maroon" title="c9">e6</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <font color="Maroon" title="c13">p4</font>
 <b>and </b><br/><a NAME="E4:12"/><i><font color="Green" title="E13">A4</font></i>: 
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c4">e1</font>,<font color="Maroon" title="c5">e2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">x</font>
 <b>and </b><br/><a NAME="E5:12"/><i><font color="Green" title="E14">A5</font></i>: 
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c6">e3</font>,<font color="Maroon" title="c7">e4</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c2">y</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c14">f1</font> = <font color="Maroon" title="c4">e1</font>, <font color="Maroon" title="c15">f2</font> = <font color="Maroon" title="c5">e2</font>, <font color="Maroon" title="c16">f3</font> = <font color="Maroon" title="c6">e3</font>, <font color="Maroon" title="c17">f4</font> = <font color="Maroon" title="c7">e4</font>, <font color="Maroon" title="c18">f5</font> = <font color="Maroon" title="c8">e5</font>, <font color="Maroon" title="c19">f6</font> = <font color="Maroon" title="c9">e6</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c3">n</font> <b>by </b><i><a class="txt" href="topalg_1.html#E1:12"><i><font color="Green" title="E10">A1</font></i></a>, <a class="txt" href="topalg_1.html#E2:12"><i><font color="Green" title="E11">A2</font></i></a>, <a class="txt" href="topalg_1.html#E3:12"><i><font color="Green" title="E12">A3</font></i></a>, <a class="ref" href="euclid.html#T25" title="EUCLID:th.25">EUCLID:25</a></i>;<br/>
<a NAME="E7:12"/><i><font color="Green" title="E15">A6</font></i>: 
 <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<b>by </b><i><a class="ref" href="euclid.html#D1" title="EUCLID:def.1">EUCLID:def 1</a></i>;<br/>
<i><font color="Green" title="E16">A7</font></i>:  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c8">e5</font>,<font color="Maroon" title="c9">e6</font> = 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">f5</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c19">f6</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
<b>by </b><i><a class="ref" href="sppol_1.html#T20" title="SPPOL_1:th.20">SPPOL_1:20</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c14">f1</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c16">f3</font></span>)</span> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c19">f6</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
<b>by </b><i><a class="txt" href="topalg_1.html#E1:12"><i><font color="Green" title="E10">A1</font></i></a>, <a class="txt" href="topalg_1.html#E2:12"><i><font color="Green" title="E11">A2</font></i></a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c14">f1</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c16">f3</font></span>)</span> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c15">f2</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c17">f4</font></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
<b>by </b><i><a class="txt" href="topalg_1.html#E1:12"><i><font color="Green" title="E10">A1</font></i></a>, <a class="txt" href="topalg_1.html#E3:12"><i><font color="Green" title="E12">A3</font></i></a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c14">f1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c15">f2</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c16">f3</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c17">f4</font></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
<b>by </b><i><a class="txt" href="topalg_1.html#E7:12"><i><font color="Green" title="E15">A6</font></i></a>, <a class="ref" href="topalg_1.html#T9" target="_self" title="TOPALG_1:th.9">Th9</a></i>
;<br/>
<a NAME="E9:12"/><i><font color="Green" title="E17">A8</font></i>: 
(  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c4">e1</font>,<font color="Maroon" title="c5">e2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">f1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c15">f2</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> &amp;  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c6">e3</font>,<font color="Maroon" title="c7">e4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">f3</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c17">f4</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> )
 
<b>by </b><i><a class="ref" href="sppol_1.html#T20" title="SPPOL_1:th.20">SPPOL_1:20</a></i>;<br/>
<a NAME="E10:12"/><i><font color="Green" title="E18">A9</font></i>: 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c14">f1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c15">f2</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c16">f3</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c17">f4</font></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">f1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c15">f2</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">f3</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c17">f4</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
 
<b>by </b><i><a class="ref" href="euclid.html#T15" title="EUCLID:th.15">EUCLID:15</a></i>;<br/>
<a NAME="E11:12"/>
<span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c4">e1</font>,<font color="Maroon" title="c5">e2</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c6">e3</font>,<font color="Maroon" title="c7">e4</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">x</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">y</font>
 
<b>by </b><i><a class="txt" href="topalg_1.html#E4:12"><i><font color="Green" title="E13">A4</font></i></a>, <a class="txt" href="topalg_1.html#E5:12"><i><font color="Green" title="E14">A5</font></i></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/>
<b>hence </b><a NAME="E12:12"/>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c8">e5</font>,<font color="Maroon" title="c9">e6</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">x</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">y</font>
 <b>by </b><i><a class="txt" href="topalg_1.html#E8:12"><i><font color="Green" title="E16">A7</font></i></a>, <a class="txt" href="topalg_1.html#E9:12"><i><font color="Green" title="E17">A8</font></i></a>, <a class="txt" href="topalg_1.html#E10:12"><i><font color="Green" title="E18">A9</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>


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