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<span class="kw">let </span><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#V6" title="METRIC_1:attr.6">Reflexive</a>   <a href="metric_1.html#V8" title="METRIC_1:attr.8">symmetric</a>   <a href="metric_1.html#V9" title="METRIC_1:attr.9">triangle</a>   <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="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">S</font>, <font color="Olive" title="b2">CL</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">M</font>  st <font color="Olive" title="b1">S</font> is  <a href="tbsp_1.html#V6" title="TBSP_1:attr.6">bounded</a>  holds <br/> for <font color="Olive" title="b3">S9</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span>  st <font color="Olive" title="b1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">S9</font> &amp; <font color="Olive" title="b2">CL</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Olive" title="b3">S9</font> holds <br/>( <font color="Olive" title="b2">CL</font> is  <a href="tbsp_1.html#V6" title="TBSP_1:attr.6">bounded</a>  &amp;  <a href="tbsp_1.html#K3" title="TBSP_1:func.3">diameter</a> <font color="Olive" title="b1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tbsp_1.html#K3" title="TBSP_1:func.3">diameter</a> <font color="Olive" title="b2">CL</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">S</font>, <font color="Maroon" title="c3">CL</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">M</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">S</font> is  <a href="tbsp_1.html#V6" title="TBSP_1:attr.6">bounded</a>  implies  for <font color="Olive" title="b1">S9</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span>  st <font color="Maroon" title="c2">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">S9</font> &amp; <font color="Maroon" title="c3">CL</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Olive" title="b1">S9</font> holds <br/>( <font color="Maroon" title="c3">CL</font> is  <a href="tbsp_1.html#V6" title="TBSP_1:attr.6">bounded</a>  &amp;  <a href="tbsp_1.html#K3" title="TBSP_1:func.3">diameter</a> <font color="Maroon" title="c2">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tbsp_1.html#K3" title="TBSP_1:func.3">diameter</a> <font color="Maroon" title="c3">CL</font> ) )</span><br/>



<span class="kw">assume </span><a NAME="E1:21"/><span class="lab"><font color="Green" title="E17">A1</font></span>: 
<font color="Maroon" title="c2">S</font> is  <a href="tbsp_1.html#V6" title="TBSP_1:attr.6">bounded</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">S9</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span>  st <font color="Maroon" title="c2">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">S9</font> &amp; <font color="Maroon" title="c3">CL</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Olive" title="b1">S9</font> holds <br/>( <font color="Maroon" title="c3">CL</font> is  <a href="tbsp_1.html#V6" title="TBSP_1:attr.6">bounded</a>  &amp;  <a href="tbsp_1.html#K3" title="TBSP_1:func.3">diameter</a> <font color="Maroon" title="c2">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tbsp_1.html#K3" title="TBSP_1:func.3">diameter</a> <font color="Maroon" title="c3">CL</font> )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c4">d</font> =  <a href="tbsp_1.html#K3" title="TBSP_1:func.3">diameter</a> <font color="Maroon" title="c2">S</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">T</font> =  <a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font>;<br/>
<span class="kw">let </span><font color="Maroon" title="c6">S9</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">S9</font> &amp; <font color="Maroon" title="c3">CL</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c6">S9</font> implies ( <font color="Maroon" title="c3">CL</font> is  <a href="tbsp_1.html#V6" title="TBSP_1:attr.6">bounded</a>  &amp;  <a href="tbsp_1.html#K3" title="TBSP_1:func.3">diameter</a> <font color="Maroon" title="c2">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tbsp_1.html#K3" title="TBSP_1:func.3">diameter</a> <font color="Maroon" title="c3">CL</font> ) )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:21"/><span class="lab"><font color="Green" title="E18">A2</font></span>: 
<font color="Maroon" title="c2">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">S9</font>
 <span class="kw">and </span><br/><a NAME="E3:21"/><span class="lab"><font color="Green" title="E19">A3</font></span>: 
<font color="Maroon" title="c3">CL</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c6">S9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">CL</font> is  <a href="tbsp_1.html#V6" title="TBSP_1:attr.6">bounded</a>  &amp;  <a href="tbsp_1.html#K3" title="TBSP_1:func.3">diameter</a> <font color="Maroon" title="c2">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tbsp_1.html#K3" title="TBSP_1:func.3">diameter</a> <font color="Maroon" title="c3">CL</font> )</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:21_1"/>
( <font color="Maroon" title="c2">S</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c2">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1"><span class="kw">suppose </span></a><a NAME="E1:21_1_1"/><span class="lab"><font color="Green" title="E20">A4</font></span>: 
<font color="Maroon" title="c2">S</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">CL</font> is  <a href="tbsp_1.html#V6" title="TBSP_1:attr.6">bounded</a>  &amp;  <a href="tbsp_1.html#K3" title="TBSP_1:func.3">diameter</a> <font color="Maroon" title="c2">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tbsp_1.html#K3" title="TBSP_1:func.3">diameter</a> <font color="Maroon" title="c3">CL</font> )</span><br/><div class="add"><div><span class="lab"><font color="Green" title="E21">A5</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">x</font>, <font color="Maroon" title="c8">y</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">CL</font> &amp; <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">CL</font> implies  not  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</font>) <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="tbsp_1.html#K3" title="TBSP_1:func.3">diameter</a> <font color="Maroon" title="c2">S</font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:21_1_1_1"/><span class="lab"><font color="Green" title="E22">A6</font></span>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">CL</font>
 <span class="kw">and </span><br/><a NAME="E2:21_1_1_1"/><span class="lab"><font color="Green" title="E23">A7</font></span>: 
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">CL</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</font>) <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="tbsp_1.html#K3" title="TBSP_1:func.3">diameter</a> <font color="Maroon" title="c2">S</font></span><br/><span class="kw">reconsider </span><font color="Maroon" title="c9">X</font> = <font color="Maroon" title="c7">x</font>, <font color="Maroon" title="c10">Y</font> = <font color="Maroon" title="c8">y</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span> ;<br/><span class="kw">set </span><font color="Maroon" title="c11">dxy</font> =  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</font>);<br/><span class="kw">set </span><font color="Maroon" title="c12">dd</font> = <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</font>)</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="tbsp_1.html#K3" title="TBSP_1:func.3">diameter</a> <font color="Maroon" title="c2">S</font></span>)</span>;<br/><span class="kw">set </span><font color="Maroon" title="c13">dd2</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</font>)</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="tbsp_1.html#K3" title="TBSP_1:func.3">diameter</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2;<br/><span class="kw">set </span><font color="Maroon" title="c14">Bx</font> =  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<font color="Maroon" title="c7">x</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</font>)</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="tbsp_1.html#K3" title="TBSP_1:func.3">diameter</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>);<br/><span class="kw">set </span><font color="Maroon" title="c15">By</font> =  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<font color="Maroon" title="c8">y</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</font>)</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="tbsp_1.html#K3" title="TBSP_1:func.3">diameter</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>);<br/><span class="kw">reconsider </span><font color="Maroon" title="c16">BX</font> =  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<font color="Maroon" title="c7">x</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</font>)</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="tbsp_1.html#K3" title="TBSP_1:func.3">diameter</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>), <font color="Maroon" title="c17">BY</font> =  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<font color="Maroon" title="c8">y</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</font>)</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="tbsp_1.html#K3" title="TBSP_1:func.3">diameter</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>) as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span> ;<br/><span class="kw">assume </span><a NAME="E5:21_1_1_1"/>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</font>) <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="tbsp_1.html#K3" title="TBSP_1:func.3">diameter</a> <font color="Maroon" title="c2">S</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E6:21_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</font>)</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="tbsp_1.html#K3" title="TBSP_1:func.3">diameter</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><a href="tbsp_1.html#K3" title="TBSP_1:func.3">diameter</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="tbsp_1.html#K3" title="TBSP_1:func.3">diameter</a> <font color="Maroon" title="c2">S</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T14" title="XREAL_1:th.14">XREAL_1:14</a></span>;<br/><a NAME="E7:21_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A8</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</font>)</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="tbsp_1.html#K3" title="TBSP_1:func.3">diameter</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T74" title="XREAL_1:th.74">XREAL_1:74</a></span>;<br/><a NAME="E8:21_1_1_1"/>
 <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<font color="Maroon" title="c8">y</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</font>)</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="tbsp_1.html#K3" title="TBSP_1:func.3">diameter</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pcomps_1.html#K2" title="PCOMPS_1:func.2">Family_open_set</a> <font color="Maroon" title="c1">M</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pcomps_1.html#T29" title="PCOMPS_1:th.29">PCOMPS_1:29</a></span>;<br/><a NAME="E9:21_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A9</font></span>: 
<font color="Maroon" title="c17">BY</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D2" title="PRE_TOPC:def.2">PRE_TOPC:def 2</a></span>;<br/><a NAME="E10:21_1_1_1"/>
 <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<font color="Maroon" title="c7">x</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</font>)</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="tbsp_1.html#K3" title="TBSP_1:func.3">diameter</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pcomps_1.html#K2" title="PCOMPS_1:func.2">Family_open_set</a> <font color="Maroon" title="c1">M</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pcomps_1.html#T29" title="PCOMPS_1:th.29">PCOMPS_1:29</a></span>;<br/><a NAME="E11:21_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A10</font></span>: 
<font color="Maroon" title="c16">BX</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D2" title="PRE_TOPC:def.2">PRE_TOPC:def 2</a></span>;<br/><a NAME="E12:21_1_1_1"/>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<font color="Maroon" title="c8">y</font>,<font color="Maroon" title="c8">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#T1" title="METRIC_1:th.1">METRIC_1:1</a></span>;<br/><a NAME="E13:21_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">BY</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:21_1_1_1"><span class="lab"><font color="Green" title="E24">A8</font></span></a>, <a class="ref" href="metric_1.html#T11" title="METRIC_1:th.11">METRIC_1:11</a></span>;<br/><a NAME="E14:21_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c17">BY</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c6">S9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21"><span class="lab"><font color="Green" title="E19">A3</font></span></a>, <a class="txt" href="#E2:21_1_1_1"><span class="lab"><font color="Green" title="E23">A7</font></span></a>, <a class="txt" href="#E9:21_1_1_1"><span class="lab"><font color="Green" title="E25">A9</font></span></a>, <a class="ref" href="tops_1.html#T12" title="TOPS_1:th.12">TOPS_1:12</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c18">y1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E16:21_1_1_1"/><span class="lab"><font color="Green" title="E27">A11</font></span>: 
<font color="Maroon" title="c18">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">BY</font>
 <span class="kw">and </span><br/><a NAME="E17:21_1_1_1"/><span class="lab"><font color="Green" title="E28">A12</font></span>: 
<font color="Maroon" title="c18">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">S9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/><a NAME="E18:21_1_1_1"/>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c7">x</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#T1" title="METRIC_1:th.1">METRIC_1:1</a></span>;<br/><a NAME="E19:21_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">BX</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:21_1_1_1"><span class="lab"><font color="Green" title="E24">A8</font></span></a>, <a class="ref" href="metric_1.html#T11" title="METRIC_1:th.11">METRIC_1:11</a></span>;<br/><a NAME="E20:21_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c16">BX</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c6">S9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21"><span class="lab"><font color="Green" title="E19">A3</font></span></a>, <a class="txt" href="#E1:21_1_1_1"><span class="lab"><font color="Green" title="E22">A6</font></span></a>, <a class="txt" href="#E11:21_1_1_1"><span class="lab"><font color="Green" title="E26">A10</font></span></a>, <a class="ref" href="tops_1.html#T12" title="TOPS_1:th.12">TOPS_1:12</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c19">x1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E22:21_1_1_1"/><span class="lab"><font color="Green" title="E29">A13</font></span>: 
<font color="Maroon" title="c19">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">BX</font>
 <span class="kw">and </span><br/><a NAME="E23:21_1_1_1"/><span class="lab"><font color="Green" title="E30">A14</font></span>: 
<font color="Maroon" title="c19">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">S9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c20">x1</font> = <font color="Maroon" title="c19">x1</font>, <font color="Maroon" title="c21">y1</font> = <font color="Maroon" title="c18">y1</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:21_1_1_1"><span class="lab"><font color="Green" title="E29">A13</font></span></a>, <a class="txt" href="#E16:21_1_1_1"><span class="lab"><font color="Green" title="E27">A11</font></span></a></span>;<br/><a NAME="E25:21_1_1_1"/><span class="lab"><font color="Green" title="E31">A15</font></span>: 
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c20">x1</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</font>)</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="tbsp_1.html#K3" title="TBSP_1:func.3">diameter</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:21_1_1_1"><span class="lab"><font color="Green" title="E29">A13</font></span></a>, <a class="ref" href="metric_1.html#T11" title="METRIC_1:th.11">METRIC_1:11</a></span>;<br/><a NAME="E26:21_1_1_1"/>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<font color="Maroon" title="c20">x1</font>,<font color="Maroon" title="c21">y1</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="tbsp_1.html#K3" title="TBSP_1:func.3">diameter</a> <font color="Maroon" title="c2">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21"><span class="lab"><font color="Green" title="E17">A1</font></span></a>, <a class="txt" href="#E2:21"><span class="lab"><font color="Green" title="E18">A2</font></span></a>, <a class="txt" href="#E23:21_1_1_1"><span class="lab"><font color="Green" title="E30">A14</font></span></a>, <a class="txt" href="#E17:21_1_1_1"><span class="lab"><font color="Green" title="E28">A12</font></span></a>, <a class="ref" href="tbsp_1.html#D8" title="TBSP_1:def.8">TBSP_1:def 8</a></span>;<br/><a NAME="E27:21_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A16</font></span>: 
<span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c20">x1</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#K4" title="METRIC_1:func.4">dist</a> (<font color="Maroon" title="c20">x1</font>,<font color="Maroon" title="c21">y1</font>)</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</font>)</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="tbsp_1.html#K3" title="TBSP_1:func.3">diameter</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="tbsp_1.html#K3" title="TBSP_1:func.3">diameter</a> <font color="Maroon" title="c2">S</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:21_1_1_1"><span class="lab"><font color="Green" title="E31">A15</font></span></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><a NAME="E28:21_1_1_1"/><span class="lab"><font color="Green" title="E33">A17</font></span>: 
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<font color="Maroon" title="c8">y</font>,<font color="Maroon" title="c21">y1</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</font>)</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="tbsp_1.html#K3" title="TBSP_1:func.3">diameter</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:21_1_1_1"><span class="lab"><font color="Green" title="E27">A11</font></span></a>, <a class="ref" href="metric_1.html#T11" title="METRIC_1:th.11">METRIC_1:11</a></span>;<br/><a NAME="E29:21_1_1_1"/>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c21">y1</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c20">x1</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#K4" title="METRIC_1:func.4">dist</a> (<font color="Maroon" title="c20">x1</font>,<font color="Maroon" title="c21">y1</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#T4" title="METRIC_1:th.4">METRIC_1:4</a></span>;<br/><a NAME="E30:21_1_1_1"/><span class="kw">then </span>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c21">y1</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</font>)</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="tbsp_1.html#K3" title="TBSP_1:func.3">diameter</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="tbsp_1.html#K3" title="TBSP_1:func.3">diameter</a> <font color="Maroon" title="c2">S</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:21_1_1_1"><span class="lab"><font color="Green" title="E32">A16</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E31:21_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c21">y1</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#K4" title="METRIC_1:func.4">dist</a> (<font color="Maroon" title="c21">y1</font>,<font color="Maroon" title="c8">y</font>)</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</font>)</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><a href="tbsp_1.html#K3" title="TBSP_1:func.3">diameter</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="tbsp_1.html#K3" title="TBSP_1:func.3">diameter</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</font>)</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="tbsp_1.html#K3" title="TBSP_1:func.3">diameter</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:21_1_1_1"><span class="lab"><font color="Green" title="E33">A17</font></span></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><span class="kw">hence </span><a NAME="E32:21_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#T4" title="METRIC_1:th.4">METRIC_1:4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><span class="lab"><font color="Green" title="E22">A18</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_2"><span class="kw">now </span></a><div class="add"><a NAME="E1:21_1_1_2"/><span class="lab"><font color="Green" title="E23">A19</font></span>: 
<span class="p1">(<span class="default"><a href="tbsp_1.html#K3" title="TBSP_1:func.3">diameter</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="tbsp_1.html#K3" title="TBSP_1:func.3">diameter</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><span class="kw">let </span><font color="Maroon" title="c7">x</font>, <font color="Maroon" title="c8">y</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">CL</font> &amp; <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">CL</font> implies  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="tbsp_1.html#K3" title="TBSP_1:func.3">diameter</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1 )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:21_1_1_2"/><span class="lab"><font color="Green" title="E24">A20</font></span>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">CL</font>
 <span class="kw">and </span><br/><a NAME="E3:21_1_1_2"/><span class="lab"><font color="Green" title="E25">A21</font></span>: 
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">CL</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="tbsp_1.html#K3" title="TBSP_1:func.3">diameter</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span><br/><a NAME="E4:21_1_1_2"/>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="tbsp_1.html#K3" title="TBSP_1:func.3">diameter</a> <font color="Maroon" title="c2">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:21_1_1"><span class="lab"><font color="Green" title="E21">A5</font></span></a>, <a class="txt" href="#E2:21_1_1_2"><span class="lab"><font color="Green" title="E24">A20</font></span></a>, <a class="txt" href="#E3:21_1_1_2"><span class="lab"><font color="Green" title="E25">A21</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E5:21_1_1_2"/>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="tbsp_1.html#K3" title="TBSP_1:func.3">diameter</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_1_1_2"><span class="lab"><font color="Green" title="E23">A19</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><span class="lab"><font color="Green" title="E23">A22</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_3"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">s</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">CL</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">CL</font> holds <br/> <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">s</font> ) implies  <a href="tbsp_1.html#K3" title="TBSP_1:func.3">diameter</a> <font color="Maroon" title="c2">S</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">s</font> )</span><br/><span class="kw">assume </span><a NAME="E1:21_1_1_3"/><span class="lab"><font color="Green" title="E24">A23</font></span>: 
 for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">CL</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">CL</font> holds <br/> <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">s</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="tbsp_1.html#K3" title="TBSP_1:func.3">diameter</a> <font color="Maroon" title="c2">S</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">s</font></span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_3_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c8">x</font>, <font color="Maroon" title="c9">y</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> &amp; <font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> implies  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">y</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">s</font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:21_1_1_3_1"/><span class="lab"><font color="Green" title="E25">A24</font></span>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font>
 <span class="kw">and </span><br/><a NAME="E2:21_1_1_3_1"/><span class="lab"><font color="Green" title="E26">A25</font></span>: 
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">y</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">s</font></span><br/><a NAME="E3:21_1_1_3_1"/>
<font color="Maroon" title="c2">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">CL</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:21"><span class="lab"><font color="Green" title="E18">A2</font></span></a>, <a class="txt" href="#E3:21"><span class="lab"><font color="Green" title="E19">A3</font></span></a>, <a class="ref" href="pre_topc.html#T18" title="PRE_TOPC:th.18">PRE_TOPC:18</a></span>;<br/><span class="kw">hence </span><a NAME="E4:21_1_1_3_1"/>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">y</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_1_1_3"><span class="lab"><font color="Green" title="E24">A23</font></span></a>, <a class="txt" href="#E1:21_1_1_3_1"><span class="lab"><font color="Green" title="E25">A24</font></span></a>, <a class="txt" href="#E2:21_1_1_3_1"><span class="lab"><font color="Green" title="E26">A25</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:21_1_1_3"/>
 <a href="tbsp_1.html#K3" title="TBSP_1:func.3">diameter</a> <font color="Maroon" title="c2">S</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21"><span class="lab"><font color="Green" title="E17">A1</font></span></a>, <a class="txt" href="#E1:21_1_1"><span class="lab"><font color="Green" title="E20">A4</font></span></a>, <a class="ref" href="tbsp_1.html#D8" title="TBSP_1:def.8">TBSP_1:def 8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E5:21_1_1"/><span class="lab"><font color="Green" title="E24">A26</font></span>: 
<font color="Maroon" title="c3">CL</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:21"><span class="lab"><font color="Green" title="E18">A2</font></span></a>, <a class="txt" href="#E3:21"><span class="lab"><font color="Green" title="E19">A3</font></span></a>, <a class="txt" href="#E1:21_1_1"><span class="lab"><font color="Green" title="E20">A4</font></span></a>, <a class="ref" href="pcomps_1.html#T2" title="PCOMPS_1:th.2">PCOMPS_1:2</a></span>;<br/><a NAME="E6:21_1_1"/>
<span class="p1">(<span class="default"><a href="tbsp_1.html#K3" title="TBSP_1:func.3">diameter</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21"><span class="lab"><font color="Green" title="E17">A1</font></span></a>, <a class="ref" href="tbsp_1.html#T21" title="TBSP_1:th.21">TBSP_1:21</a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><a NAME="E7:21_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">CL</font> is  <a href="tbsp_1.html#V6" title="TBSP_1:attr.6">bounded</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21_1_1"><span class="lab"><font color="Green" title="E22">A18</font></span></a>, <a class="ref" href="tbsp_1.html#D7" title="TBSP_1:def.7">TBSP_1:def 7</a></span>;<br/><span class="kw">hence </span><a NAME="E8:21_1_1"/>
( <font color="Maroon" title="c3">CL</font> is  <a href="tbsp_1.html#V6" title="TBSP_1:attr.6">bounded</a>  &amp;  <a href="tbsp_1.html#K3" title="TBSP_1:func.3">diameter</a> <font color="Maroon" title="c2">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tbsp_1.html#K3" title="TBSP_1:func.3">diameter</a> <font color="Maroon" title="c3">CL</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:21_1_1"><span class="lab"><font color="Green" title="E24">A26</font></span></a>, <a class="txt" href="#E2:21_1_1"><span class="lab"><font color="Green" title="E21">A5</font></span></a>, <a class="txt" href="#E4:21_1_1"><span class="lab"><font color="Green" title="E23">A22</font></span></a>, <a class="ref" href="tbsp_1.html#D8" title="TBSP_1:def.8">TBSP_1:def 8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_2"><span class="kw">suppose </span></a><a NAME="E1:21_1_2"/>
<font color="Maroon" title="c2">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">CL</font> is  <a href="tbsp_1.html#V6" title="TBSP_1:attr.6">bounded</a>  &amp;  <a href="tbsp_1.html#K3" title="TBSP_1:func.3">diameter</a> <font color="Maroon" title="c2">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tbsp_1.html#K3" title="TBSP_1:func.3">diameter</a> <font color="Maroon" title="c3">CL</font> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:21_1_2"/>
( <font color="Maroon" title="c3">CL</font> is  <a href="tbsp_1.html#V6" title="TBSP_1:attr.6">bounded</a>  &amp;  <a href="tbsp_1.html#K3" title="TBSP_1:func.3">diameter</a> <font color="Maroon" title="c2">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tbsp_1.html#K3" title="TBSP_1:func.3">diameter</a> <font color="Maroon" title="c3">CL</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:21"><span class="lab"><font color="Green" title="E18">A2</font></span></a>, <a class="txt" href="#E3:21"><span class="lab"><font color="Green" title="E19">A3</font></span></a>, <a class="ref" href="pcomps_1.html#T2" title="PCOMPS_1:th.2">PCOMPS_1:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div>
