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

<span class="kw">let </span><font color="Maroon" title="c1">PM</font> be    <a href="metric_1.html#L1" title="METRIC_1:struct.1">MetrStruct</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">y</font>, <font color="Olive" title="b2">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">PM</font><br/> for <font color="Olive" title="b3">r</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Maroon" title="c1">PM</font> is  <a href="metric_1.html#V9" title="METRIC_1:attr.9">triangle</a>  &amp; <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Olive" title="b2">x</font>,<font color="Olive" title="b3">r</font> holds <br/> ex <font color="Olive" title="b4">p</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b4">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Olive" title="b1">y</font>,<font color="Olive" title="b4">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Olive" title="b2">x</font>,<font color="Olive" title="b3">r</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">y</font>, <font color="Maroon" title="c3">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">PM</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">r</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Maroon" title="c1">PM</font> is  <a href="metric_1.html#V9" title="METRIC_1:attr.9">triangle</a>  &amp; <font color="Maroon" title="c2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c3">x</font>,<font color="Olive" title="b1">r</font> holds <br/> ex <font color="Olive" title="b2">p</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b2">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c2">y</font>,<font color="Olive" title="b2">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c3">x</font>,<font color="Olive" title="b1">r</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">r</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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">PM</font> is  <a href="metric_1.html#V9" title="METRIC_1:attr.9">triangle</a>  &amp; <font color="Maroon" title="c2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">r</font> implies  ex <font color="Olive" title="b1">p</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c2">y</font>,<font color="Olive" title="b1">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">r</font> ) )</span><br/>







<span class="kw">reconsider </span><font color="Maroon" title="c5">r9</font> = <font color="Maroon" title="c4">r</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:31"/><span class="lab"><font color="Green" title="E22">A1</font></span>: 
<font color="Maroon" title="c1">PM</font> is  <a href="metric_1.html#V9" title="METRIC_1:attr.9">triangle</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">r</font> or  ex <font color="Olive" title="b1">p</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c2">y</font>,<font color="Olive" title="b1">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">r</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E3:31"/><span class="lab"><font color="Green" title="E23">A2</font></span>: 
<font color="Maroon" title="c2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">r</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">p</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c2">y</font>,<font color="Olive" title="b1">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">r</font> )</span><br/>

<a NAME="E4:31"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A3</font></span>: 
 <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#T12" title="METRIC_1:th.12">METRIC_1:12</a></span>;<br/>
<a NAME="E5:31"/><span class="lab"><font color="Green" title="E25">A4</font></span>: 
not <font color="Maroon" title="c1">PM</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:31"><span class="lab"><font color="Green" title="E23">A2</font></span></a></span>;<br/>
<span class="kw">thus </span><a NAME="E6:31"/>
 ex <font color="Olive" title="b1">p</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c2">y</font>,<font color="Olive" title="b1">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">r</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c6">p</font> = <font color="Maroon" title="c5">r9</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">y</font></span>)</span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c5">r9</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">y</font></span>)</span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">r9</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">y</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c2">y</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">r9</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">y</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">r</font> )</span><br/>

<span class="kw">thus </span><a NAME="E1:31_1"/>
<font color="Maroon" title="c5">r9</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">y</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:31"><span class="lab"><font color="Green" title="E24">A3</font></span></a>, <a class="ref" href="xreal_1.html#T52" title="XREAL_1:th.52">XREAL_1:52</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c2">y</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">r9</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">y</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">r</font></span><br/>

<a NAME="E2:31_1"/>
for <font color="Olive" title="b1">z</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">PM</font>  st <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c2">y</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">r9</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">y</font></span>)</span></span>)</span> holds <br/><font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">r</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">z</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">PM</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c2">y</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">r9</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">y</font></span>)</span></span>)</span> implies <font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">r</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:31_1_1"/>
<font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c2">y</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">r9</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">y</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">r</font></span><br/>

<a NAME="E2:31_1_1"/><span class="kw">then </span>
 <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c7">z</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">r9</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">y</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#T12" title="METRIC_1:th.12">METRIC_1:12</a></span>;<br/>
<a NAME="E3:31_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A5</font></span>: 
<span class="p1">(<span class="default"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">y</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c7">z</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T22" title="XREAL_1:th.22">XREAL_1:22</a></span>;<br/>
<a NAME="E4:31_1_1"/>
<span class="p1">(<span class="default"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">y</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c7">z</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c7">z</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:31"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="ref" href="metric_1.html#T4" title="METRIC_1:th.4">METRIC_1:4</a></span>;<br/>
<a NAME="E5:31_1_1"/><span class="kw">then </span>
 <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c7">z</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:31_1_1"><span class="lab"><font color="Green" title="E26">A5</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:31_1_1"/>
<font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:31"><span class="lab"><font color="Green" title="E25">A4</font></span></a>, <a class="ref" href="metric_1.html#T12" title="METRIC_1:th.12">METRIC_1:12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:31_1"/>
 <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c2">y</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">r9</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">y</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">r</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="subset_1.html#T7" title="SUBSET_1:th.7">SUBSET_1:7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>


</div>
