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<span class="kw">let </span><font color="Maroon" title="c1">R</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#L1" title="METRIC_1:struct.1">MetrStruct</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">S</font>, <font color="Olive" title="b2">V</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">R</font>  st <font color="Olive" title="b1">S</font> is  <a href="tbsp_1.html#V6" title="TBSP_1:attr.6">bounded</a>  &amp; <font color="Olive" title="b2">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">S</font> holds <br/> <a href="tbsp_1.html#K2" title="TBSP_1:func.2">diameter</a> <font color="Olive" title="b2">V</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="tbsp_1.html#K2" title="TBSP_1:func.2">diameter</a> <font color="Olive" title="b1">S</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">S</font>, <font color="Maroon" title="c3">V</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">R</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">S</font> is  <a href="tbsp_1.html#V6" title="TBSP_1:attr.6">bounded</a>  &amp; <font color="Maroon" title="c3">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">S</font> implies  <a href="tbsp_1.html#K2" title="TBSP_1:func.2">diameter</a> <font color="Maroon" title="c3">V</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="tbsp_1.html#K2" title="TBSP_1:func.2">diameter</a> <font color="Maroon" title="c2">S</font> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:42"/><span class="lab"><font color="Green" title="E19">A1</font></span>: 
<font color="Maroon" title="c2">S</font> is  <a href="tbsp_1.html#V6" title="TBSP_1:attr.6">bounded</a> 
 <span class="kw">and </span><br/><a NAME="E2:42"/><span class="lab"><font color="Green" title="E20">A2</font></span>: 
<font color="Maroon" title="c3">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">S</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="tbsp_1.html#K2" title="TBSP_1:func.2">diameter</a> <font color="Maroon" title="c3">V</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="tbsp_1.html#K2" title="TBSP_1:func.2">diameter</a> <font color="Maroon" title="c2">S</font></span><br/>

<a NAME="E3:42"/><span class="lab"><font color="Green" title="E21">A3</font></span>: 
<font color="Maroon" title="c3">V</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="#E1:42"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="txt" href="#E2:42"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="ref" href="tbsp_1.html#T14" target="_self" onmouseover="rs('tbsp_1/T14')" onmouseout="rh()">Th14</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:42_1"/>
( <font color="Maroon" title="c3">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c3">V</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>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_1"><span class="kw">suppose </span></a><a NAME="E1:42_1_1"/>
<font color="Maroon" title="c3">V</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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="tbsp_1.html#K2" title="TBSP_1:func.2">diameter</a> <font color="Maroon" title="c3">V</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="tbsp_1.html#K2" title="TBSP_1:func.2">diameter</a> <font color="Maroon" title="c2">S</font></span><br/><div class="add"><a NAME="E2:42_1_1"/><span class="kw">then </span>
 <a href="tbsp_1.html#K2" title="TBSP_1:func.2">diameter</a> <font color="Maroon" title="c3">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tbsp_1.html#D8" target="_self" onmouseover="rs('tbsp_1/D8')" onmouseout="rh()">Def8</a></span>;<br/><span class="kw">hence </span><a NAME="E3:42_1_1"/>
 <a href="tbsp_1.html#K2" title="TBSP_1:func.2">diameter</a> <font color="Maroon" title="c3">V</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="tbsp_1.html#K2" title="TBSP_1:func.2">diameter</a> <font color="Maroon" title="c2">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="ref" href="tbsp_1.html#T21" target="_self" onmouseover="rs('tbsp_1/T21')" onmouseout="rh()">Th21</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_2"><span class="kw">suppose </span></a><a NAME="E1:42_1_2"/><span class="lab"><font color="Green" title="E22">A4</font></span>: 
<font color="Maroon" title="c3">V</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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="tbsp_1.html#K2" title="TBSP_1:func.2">diameter</a> <font color="Maroon" title="c3">V</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="tbsp_1.html#K2" title="TBSP_1:func.2">diameter</a> <font color="Maroon" title="c2">S</font></span><br/><div class="add"><a NAME="E2:42_1_2"/>
 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">R</font>  st <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">V</font> &amp; <font color="Olive" title="b2">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">V</font> holds <br/> <a href="metric_1.html#K1" title="METRIC_1:func.1">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>  <a href="tbsp_1.html#K2" title="TBSP_1:func.2">diameter</a> <font color="Maroon" title="c2">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="txt" href="#E2:42"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="ref" href="tbsp_1.html#D8" target="_self" onmouseover="rs('tbsp_1/D8')" onmouseout="rh()">Def8</a></span>;<br/><span class="kw">hence </span><a NAME="E3:42_1_2"/>
 <a href="tbsp_1.html#K2" title="TBSP_1:func.2">diameter</a> <font color="Maroon" title="c3">V</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="tbsp_1.html#K2" title="TBSP_1:func.2">diameter</a> <font color="Maroon" title="c2">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:42"><span class="lab"><font color="Green" title="E21">A3</font></span></a>, <a class="txt" href="#E1:42_1_2"><span class="lab"><font color="Green" title="E22">A4</font></span></a>, <a class="ref" href="tbsp_1.html#D8" target="_self" onmouseover="rs('tbsp_1/D8')" onmouseout="rh()">Def8</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

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