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<b>let </b><font color="Maroon" title="c1">M</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="metric_1.html#NM1" title="METRIC_1:NM.1">MetrSpace</a>;<br/><b>let </b><font color="Maroon" title="c2">z</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font>;<br/><b>let </b><font color="Maroon" title="c3">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/>





<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:67_1"/>
( <font color="Maroon" title="c3">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 or <font color="Maroon" title="c3">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 0 or <font color="Maroon" title="c3">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_1"><b>suppose </b></a><a NAME="E1:67_1_1"/><i><font color="Green" title="E32">A1</font></i>: 
<font color="Maroon" title="c3">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon" title="c4">r1</font> = <font color="Maroon" title="c3">r</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/><b>take </b>
<font color="Maroon" title="c4">r1</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1
; <i><font color="Red">:: according to </font></i><a class="ref" href="metric_6.html#D10" title="METRIC_6:def.10">METRIC_6:def 10</a><br/><b>take </b>
<font color="Maroon" title="c2">z</font>
;<br/><a NAME="E3:67_1_1"/>
<font color="Maroon" title="c3">r</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="topreal6.html#E1:67_1_1"><i><font color="Green" title="E32">A1</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><b>hence </b><a NAME="E4:67_1_1"/>
0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">r1</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1
 ;<br/><b>let </b><font color="Maroon" title="c5">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/><b>assume </b><a NAME="E5:67_1_1"/><i><font color="Green" title="E33">A2</font></i>: 
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="metric_1.html#K10" title="METRIC_1:func.10">cl_Ball</a> <font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">r</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c6">e</font> = <font color="Maroon" title="c5">a</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font> ;<br/><a NAME="E7:67_1_1"/><i><font color="Green" title="E34">A3</font></i>: 
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c2">z</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">r</font>
 <b>by </b><i><a class="txt" href="topreal6.html#E5:67_1_1"><i><font color="Green" title="E33">A2</font></i></a>, <a class="ref" href="metric_1.html#T13" title="METRIC_1:th.13">METRIC_1:13</a></i>;<br/><a NAME="E8:67_1_1"/>
<font color="Maroon" title="c3">r</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">r</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E9:67_1_1"/><b>then </b>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c2">z</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">r</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="topreal6.html#E7:67_1_1"><i><font color="Green" title="E34">A3</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E10:67_1_1"/>
<font color="Maroon" title="c5">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="c2">z</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">r1</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="metric_1.html#T12" title="METRIC_1:th.12">METRIC_1:12</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_2"><b>suppose </b></a><a NAME="E1:67_1_2"/><i><font color="Green" title="E32">A4</font></i>: 
<font color="Maroon" title="c3">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 0
 ;<br/><div class="add"><a NAME="E2:67_1_2"/>
 <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c1">M</font> is <a href="tbsp_1.html#V6" title="TBSP_1:attr.6">bounded</a>
 <b>by </b><i><a class="ref" href="tbsp_1.html#T14" title="TBSP_1:th.14">TBSP_1:14</a></i>;<br/><b>hence </b><a NAME="E3:67_1_2"/>
 <a href="metric_1.html#K10" title="METRIC_1:func.10">cl_Ball</a> <font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">r</font> is <a href="tbsp_1.html#V6" title="TBSP_1:attr.6">bounded</a>
 <b>by </b><i><a class="txt" href="topreal6.html#E1:67_1_2"><i><font color="Green" title="E32">A4</font></i></a>, <a class="ref" href="topreal6.html#T63" target="_self" title="TOPREAL6:th.63">Th63</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_3"><b>suppose </b></a><a NAME="E1:67_1_3"/>
<font color="Maroon" title="c3">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:67_1_3"/><b>then </b>
 <a href="metric_1.html#K10" title="METRIC_1:func.10">cl_Ball</a> <font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="topreal6.html#T64" target="_self" title="TOPREAL6:th.64">Th64</a></i>;<br/><b>hence </b><a NAME="E3:67_1_3"/>
 <a href="metric_1.html#K10" title="METRIC_1:func.10">cl_Ball</a> <font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">r</font> is <a href="tbsp_1.html#V6" title="TBSP_1:attr.6">bounded</a>
 <b>by </b><i><a class="ref" href="tbsp_1.html#T22" title="TBSP_1:th.22">TBSP_1:22</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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