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

<b>let </b><font color="Maroon" title="c1">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> ;<br/><b>let </b><font color="Maroon" title="c2">M</font> be   <a href="metric_1.html#NM1" title="METRIC_1:NM.1">MetrSpace</a>;<br/><b>let </b><font color="Maroon" title="c3">A</font> be    <a href="topmetr.html#M1" title="TOPMETR:mode.1">SubSpace</a> of <font color="Maroon" title="c2">M</font>;<br/><b>let </b><font color="Maroon" title="c4">x</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">M</font>;<br/><b>let </b><font color="Maroon" title="c5">x'</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c3">A</font>;<br/>









<b>assume </b><a NAME="E1:16"/><i><font color="Green" title="E10">A1</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x'</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:16_1"/><i><font color="Green" title="E11">A2</font></i>: 
<font color="Maroon" title="c6">a</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="c5">x'</font>,<font color="Maroon" title="c1">r</font>
 ;<br/><a NAME="E2:16_1"/><b>then </b><i><font color="Green" title="E12">A3</font></i>: 
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">A</font>
 ;<br/><b>reconsider </b><font color="Maroon" title="c7">y'</font> = <font color="Maroon" title="c6">a</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c3">A</font> <b>by </b><i><a class="txt" href="topmetr.html#E1:16_1"><i><font color="Green" title="E11">A2</font></i></a></i>;<br/><a NAME="E4:16_1"/><i><font color="Green" title="E13">A4</font></i>: 
not <font color="Maroon" title="c3">A</font> is <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:16_1_1"/>
not the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">A</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 <b>by </b><i><a class="txt" href="topmetr.html#E1:16_1"><i><font color="Green" title="E11">A2</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="struct_0.html#D1" title="STRUCT_0:def.1">STRUCT_0:def 1</a><br/>


</div><b>end;</b></div><a NAME="E5:16_1"/><i><font color="Green" title="E14">A5</font></i>: 
not <font color="Maroon" title="c2">M</font> is <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:16_1_2"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">M</font>
 
<b>by </b><i><a class="ref" href="topmetr.html#D1" target="_self" title="TOPMETR:def.1">Def1</a></i>;<br/>
<b>hence </b><a NAME="E2:16_1_2"/>
not the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">M</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 <b>by </b><i><a class="txt" href="topmetr.html#E2:16_1"><i><font color="Green" title="E12">A3</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="struct_0.html#D1" title="STRUCT_0:def.1">STRUCT_0:def 1</a><br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c8">y</font> = <font color="Maroon" title="c7">y'</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">M</font> <b>by </b><i><a class="txt" href="topmetr.html#E4:16_1"><i><font color="Green" title="E13">A4</font></i></a>, <a class="ref" href="topmetr.html#T12" target="_self" title="TOPMETR:th.12">Th12</a></i>;<br/><a NAME="E7:16_1"/>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x'</font>,<font color="Maroon" title="c7">y'</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">r</font>
 <b>by </b><i><a class="txt" href="topmetr.html#E1:16_1"><i><font color="Green" title="E11">A2</font></i></a>, <a class="ref" href="metric_1.html#T12" title="METRIC_1:th.12">METRIC_1:12</a></i>;<br/><a NAME="E8:16_1"/><b>then </b>
the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <font color="Maroon" title="c3">A</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c5">x'</font>,<font color="Maroon" title="c7">y'</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">r</font>
 <b>by </b><i><a class="ref" href="metric_1.html#D1" title="METRIC_1:def.1">METRIC_1:def 1</a></i>;<br/><a NAME="E9:16_1"/><b>then </b>
the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <font color="Maroon" title="c2">M</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c8">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">r</font>
 <b>by </b><i><a class="txt" href="topmetr.html#E1:16"><i><font color="Green" title="E10">A1</font></i></a>, <a class="ref" href="topmetr.html#D1" target="_self" title="TOPMETR:def.1">Def1</a></i>;<br/><a NAME="E10:16_1"/><b>then </b>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c8">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">r</font>
 <b>by </b><i><a class="ref" href="metric_1.html#D1" title="METRIC_1:def.1">METRIC_1:def 1</a></i>;<br/><a NAME="E11:16_1"/><b>then </b>
<font color="Maroon" title="c6">a</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="c4">x</font>,<font color="Maroon" title="c1">r</font>
 <b>by </b><i><a class="txt" href="topmetr.html#E5:16_1"><i><font color="Green" title="E14">A5</font></i></a>, <a class="ref" href="metric_1.html#T12" title="METRIC_1:th.12">METRIC_1:12</a></i>;<br/><b>hence </b><a NAME="E12:16_1"/>
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c1">r</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">A</font>
 <b>by </b><i><a class="txt" href="topmetr.html#E1:16_1"><i><font color="Green" title="E11">A2</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div>
<a NAME="E3:16"/><b>then </b><i><font color="Green" title="E11">A6</font></i>: 
 <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c5">x'</font>,<font color="Maroon" title="c1">r</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c1">r</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">A</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:16_2"/><i><font color="Green" title="E12">A7</font></i>: 
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c1">r</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">A</font>
 ;<br/><a NAME="E2:16_2"/><b>then </b><i><font color="Green" title="E13">A8</font></i>: 
( <font color="Maroon" title="c6">a</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="c4">x</font>,<font color="Maroon" title="c1">r</font> &amp; <font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">A</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E3:16_2"/><i><font color="Green" title="E14">A9</font></i>: 
not <font color="Maroon" title="c3">A</font> is <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:16_2_1"/>
not the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">A</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 <b>by </b><i><a class="txt" href="topmetr.html#E1:16_2"><i><font color="Green" title="E12">A7</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="struct_0.html#D1" title="STRUCT_0:def.1">STRUCT_0:def 1</a><br/>


</div><b>end;</b></div><a NAME="E4:16_2"/><i><font color="Green" title="E15">A10</font></i>: 
not <font color="Maroon" title="c2">M</font> is <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:16_2_2"/>
not the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">M</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 <b>by </b><i><a class="txt" href="topmetr.html#E2:16_2"><i><font color="Green" title="E13">A8</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="struct_0.html#D1" title="STRUCT_0:def.1">STRUCT_0:def 1</a><br/>


</div><b>end;</b></div><b>reconsider </b><font color="Maroon" title="c7">y'</font> = <font color="Maroon" title="c6">a</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c3">A</font> <b>by </b><i><a class="txt" href="topmetr.html#E1:16_2"><i><font color="Green" title="E12">A7</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c8">y</font> = <font color="Maroon" title="c7">y'</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">M</font> <b>by </b><i><a class="txt" href="topmetr.html#E3:16_2"><i><font color="Green" title="E14">A9</font></i></a>, <a class="txt" href="topmetr.html#E4:16_2"><i><font color="Green" title="E15">A10</font></i></a>, <a class="ref" href="topmetr.html#T12" target="_self" title="TOPMETR:th.12">Th12</a></i>;<br/><a NAME="E7:16_2"/>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c8">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">r</font>
 <b>by </b><i><a class="txt" href="topmetr.html#E2:16_2"><i><font color="Green" title="E13">A8</font></i></a>, <a class="ref" href="metric_1.html#T12" title="METRIC_1:th.12">METRIC_1:12</a></i>;<br/><a NAME="E8:16_2"/><b>then </b>
the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <font color="Maroon" title="c2">M</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c8">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">r</font>
 <b>by </b><i><a class="ref" href="metric_1.html#D1" title="METRIC_1:def.1">METRIC_1:def 1</a></i>;<br/><a NAME="E9:16_2"/><b>then </b>
the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <font color="Maroon" title="c3">A</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c5">x'</font>,<font color="Maroon" title="c7">y'</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">r</font>
 <b>by </b><i><a class="txt" href="topmetr.html#E1:16"><i><font color="Green" title="E10">A1</font></i></a>, <a class="ref" href="topmetr.html#D1" target="_self" title="TOPMETR:def.1">Def1</a></i>;<br/><a NAME="E10:16_2"/><b>then </b>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x'</font>,<font color="Maroon" title="c7">y'</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">r</font>
 <b>by </b><i><a class="ref" href="metric_1.html#D1" title="METRIC_1:def.1">METRIC_1:def 1</a></i>;<br/><b>hence </b><a NAME="E11:16_2"/>
<font color="Maroon" title="c6">a</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="c5">x'</font>,<font color="Maroon" title="c1">r</font>
 <b>by </b><i><a class="txt" href="topmetr.html#E3:16_2"><i><font color="Green" title="E14">A9</font></i></a>, <a class="ref" href="metric_1.html#T12" title="METRIC_1:th.12">METRIC_1:12</a></i>;<br/></div><b>end;</b></div>
<a NAME="E5:16"/><b>then </b>
<span class="p1">(<span class="default"><a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c1">r</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">A</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="c5">x'</font>,<font color="Maroon" title="c1">r</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<b>hence </b><a NAME="E6:16"/>
 <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c5">x'</font>,<font color="Maroon" title="c1">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c1">r</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">A</font>
 <b>by </b><i><a class="txt" href="topmetr.html#E3:16"><i><font color="Green" title="E11">A6</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>


</div>
