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

<span class="kw">let </span><font color="Maroon" title="c1">N</font> be   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="metric_1.html#L1" title="METRIC_1:struct.1">MetrStruct</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">S2</font> being   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c1">N</font>  st <font color="Maroon" title="c1">N</font> is  <a href="metric_1.html#V9" title="METRIC_1:attr.9">triangle</a>  &amp; <font color="Maroon" title="c1">N</font> is  <a href="metric_1.html#V8" title="METRIC_1:attr.8">symmetric</a>  &amp; <font color="Olive" title="b1">S2</font> is  <a href="tbsp_1.html#V2" title="TBSP_1:attr.2">convergent</a>  holds <br/><font color="Olive" title="b1">S2</font> is  <a href="tbsp_1.html#V3" title="TBSP_1:attr.3">Cauchy</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">S2</font> be   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c1">N</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="c1">N</font> is  <a href="metric_1.html#V9" title="METRIC_1:attr.9">triangle</a>  &amp; <font color="Maroon" title="c1">N</font> is  <a href="metric_1.html#V8" title="METRIC_1:attr.8">symmetric</a>  &amp; <font color="Maroon" title="c2">S2</font> is  <a href="tbsp_1.html#V2" title="TBSP_1:attr.2">convergent</a>  implies <font color="Maroon" title="c2">S2</font> is  <a href="tbsp_1.html#V3" title="TBSP_1:attr.3">Cauchy</a>  )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:11"/><span class="lab"><font color="Green" title="E6">A1</font></span>: 
<font color="Maroon" title="c1">N</font> is  <a href="metric_1.html#V9" title="METRIC_1:attr.9">triangle</a> 
 <span class="kw">and </span><br/><a NAME="E2:11"/><span class="lab"><font color="Green" title="E7">A2</font></span>: 
<font color="Maroon" title="c1">N</font> is  <a href="metric_1.html#V8" title="METRIC_1:attr.8">symmetric</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"> (  not <font color="Maroon" title="c2">S2</font> is  <a href="tbsp_1.html#V2" title="TBSP_1:attr.2">convergent</a>  or <font color="Maroon" title="c2">S2</font> is  <a href="tbsp_1.html#V3" title="TBSP_1:attr.3">Cauchy</a>  )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c3">N</font> = <font color="Maroon" title="c1">N</font> as   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="metric_1.html#V8" title="METRIC_1:attr.8">symmetric</a>   <a href="metric_1.html#L1" title="METRIC_1:struct.1">MetrStruct</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:11"><span class="lab"><font color="Green" title="E7">A2</font></span></a></span>;<br/>
<span class="kw">assume </span><a NAME="E4:11"/><span class="lab"><font color="Green" title="E8">A3</font></span>: 
<font color="Maroon" title="c2">S2</font> is  <a href="tbsp_1.html#V2" title="TBSP_1:attr.2">convergent</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"> <font color="Maroon" title="c2">S2</font> is  <a href="tbsp_1.html#V3" title="TBSP_1:attr.3">Cauchy</a> </span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c4">S2</font> = <font color="Maroon" title="c2">S2</font> as   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c3">N</font> ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c5">g</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">N</font><span class="kw"> such that </span><br/><a NAME="E7:11"/><span class="lab"><font color="Green" title="E9">A4</font></span>: 
 for <font color="Olive" title="b1">r</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">r</font> holds <br/> ex <font color="Olive" title="b2">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/> for <font color="Olive" title="b3">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">m</font> holds <br/> <a href="metric_1.html#K3" title="METRIC_1:func.3">dist</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">S2</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b3">m</font></span>)</span>,<font color="Maroon" title="c5">g</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:11"><span class="lab"><font color="Green" title="E8">A3</font></span></a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c6">r</font> be   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tbsp_1.html#D4" target="_self" onmouseover="rs('tbsp_1/D4')" onmouseout="rh()">TBSP_1:def 4</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"> ( <font color="Maroon" title="c6">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  implies  ex <font color="Olive" title="b1">p</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/> for <font color="Olive" title="b2">n</font>, <font color="Olive" title="b3">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">n</font> &amp; <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">m</font> holds <br/> <a href="metric_1.html#K1" title="METRIC_1:func.1">dist</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">S2</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b2">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">S2</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b3">m</font></span>)</span>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">r</font> )</span><br/>

<span class="kw">assume </span><a NAME="E8:11"/>
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">r</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">p</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/> for <font color="Olive" title="b2">n</font>, <font color="Olive" title="b3">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">n</font> &amp; <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">m</font> holds <br/> <a href="metric_1.html#K1" title="METRIC_1:func.1">dist</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">S2</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b2">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">S2</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b3">m</font></span>)</span>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">r</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E10:11"/><span class="lab"><font color="Green" title="E10">A5</font></span>: 
 for <font color="Olive" title="b1">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon" title="c7">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> holds <br/> <a href="metric_1.html#K3" title="METRIC_1:func.3">dist</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">S2</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">m</font></span>)</span>,<font color="Maroon" title="c5">g</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">r</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:11"><span class="lab"><font color="Green" title="E9">A4</font></span></a>, <a class="ref" href="xreal_1.html#T215" onmouseover="rs('xreal_1/T215')" onmouseout="rh()">XREAL_1:215</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c7">n</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">n</font>, <font color="Olive" title="b2">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon" title="c7">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> &amp; <font color="Maroon" title="c7">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">m</font> holds <br/> <a href="metric_1.html#K1" title="METRIC_1:func.1">dist</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">S2</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">S2</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b2">m</font></span>)</span>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">r</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c8">m</font>, <font color="Maroon" title="c9">m9</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</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"> ( <font color="Maroon" title="c7">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">m</font> &amp; <font color="Maroon" title="c7">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">m9</font> implies  <a href="metric_1.html#K1" title="METRIC_1:func.1">dist</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">S2</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c8">m</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">S2</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c9">m9</font></span>)</span>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">r</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E11:11"/><span class="lab"><font color="Green" title="E11">A6</font></span>: 
<font color="Maroon" title="c8">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">n</font>
 <span class="kw">and </span><br/><a NAME="E12:11"/><span class="lab"><font color="Green" title="E12">A7</font></span>: 
<font color="Maroon" title="c9">m9</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">n</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="metric_1.html#K1" title="METRIC_1:func.1">dist</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">S2</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c8">m</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">S2</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c9">m9</font></span>)</span>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">r</font></span><br/>

<a NAME="E13:11"/><span class="lab"><font color="Green" title="E13">A8</font></span>: 
 <a href="metric_1.html#K3" title="METRIC_1:func.3">dist</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">S2</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c9">m9</font></span>)</span>,<font color="Maroon" title="c5">g</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">r</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:11"><span class="lab"><font color="Green" title="E10">A5</font></span></a>, <a class="txt" href="#E12:11"><span class="lab"><font color="Green" title="E12">A7</font></span></a></span>;<br/>
<a NAME="E14:11"/>
 <a href="metric_1.html#K3" title="METRIC_1:func.3">dist</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">S2</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c8">m</font></span>)</span>,<font color="Maroon" title="c5">g</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">r</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:11"><span class="lab"><font color="Green" title="E10">A5</font></span></a>, <a class="txt" href="#E11:11"><span class="lab"><font color="Green" title="E11">A6</font></span></a></span>;<br/>
<a NAME="E15:11"/><span class="kw">then </span><span class="lab"><font color="Green" title="E14">A9</font></span>: 
<span class="p1">(<span class="default"><a href="metric_1.html#K3" title="METRIC_1:func.3">dist</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c4">S2</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c8">m</font></span>)</span>,<font color="Maroon" title="c5">g</font>)</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K3" title="METRIC_1:func.3">dist</a> (<font color="Maroon" title="c5">g</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">S2</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c9">m9</font></span>)</span>)</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">r</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">r</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:11"><span class="lab"><font color="Green" title="E13">A8</font></span></a>, <a class="ref" href="xreal_1.html#T8" onmouseover="rs('xreal_1/T8')" onmouseout="rh()">XREAL_1:8</a></span>;<br/>
<a NAME="E16:11"/>
 <a href="metric_1.html#K3" title="METRIC_1:func.3">dist</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">S2</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c8">m</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">S2</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c9">m9</font></span>)</span>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K3" title="METRIC_1:func.3">dist</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c4">S2</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c8">m</font></span>)</span>,<font color="Maroon" title="c5">g</font>)</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K3" title="METRIC_1:func.3">dist</a> (<font color="Maroon" title="c5">g</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">S2</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c9">m9</font></span>)</span>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:11"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="ref" href="metric_1.html#T4" onmouseover="rs('metric_1/T4')" onmouseout="rh()">METRIC_1:4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E17:11"/>
 <a href="metric_1.html#K1" title="METRIC_1:func.1">dist</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">S2</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c8">m</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">S2</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c9">m9</font></span>)</span>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:11"><span class="lab"><font color="Green" title="E14">A9</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</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>
