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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#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> being   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">M</font> holds   <a href="simplex2.html#K1" title="SIMPLEX2:func.1">diameter</a> (<font color="Maroon" title="c1">M</font>,<span class="p1">(<span class="default"><a href="simplex0.html#K5" title="SIMPLEX0:func.5">Complex_of</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b1">S</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>) <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="b1">S</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">S</font> be   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <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">  <a href="simplex2.html#K1" title="SIMPLEX2:func.1">diameter</a> (<font color="Maroon" title="c1">M</font>,<span class="p1">(<span class="default"><a href="simplex0.html#K5" title="SIMPLEX0:func.5">Complex_of</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">S</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>) <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="c2">S</font></span><br/>



<span class="kw">set </span><font color="Maroon" title="c3">C</font> =  <a href="simplex0.html#K5" title="SIMPLEX0:func.5">Complex_of</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">S</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c4">C</font> =  <a href="simplex0.html#K5" title="SIMPLEX0:func.5">Complex_of</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">S</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> as   non  <a href="pencil_1.html#V3" title="PENCIL_1:attr.3">void</a>  <font color="Maroon" title="c1">M</font> <a href="simplex2.html#V1" title="SIMPLEX2:attr.1">bounded</a>  <a href="simplex1.html#NM2" title="SIMPLEX1:NM.2">SimplicialComplex</a> of <font color="Maroon" title="c1">M</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="simplex2.html#T6" target="_self" title="SIMPLEX2:th.6">Th6</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">s</font> = <font color="Maroon" title="c2">S</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c4">C</font> ;<br/>
<a NAME="E3:19"/><span class="lab"><font color="Green" title="E9">A1</font></span>: 
 the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c4">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Maroon" title="c2">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="simplex0.html#T4" title="SIMPLEX0:th.4">SIMPLEX0:4</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c6">W</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="c6">W</font> is   <a href="simplex0.html#NM5" title="SIMPLEX0:NM.5">Simplex</a> of <font color="Maroon" title="c4">C</font> implies  <a href="tbsp_1.html#K3" title="TBSP_1:func.3">diameter</a> <font color="Maroon" title="c6">W</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><br/><span class="kw">assume </span><a NAME="E1:19_1"/>
<font color="Maroon" title="c6">W</font> is   <a href="simplex0.html#NM5" title="SIMPLEX0:NM.5">Simplex</a> of <font color="Maroon" title="c4">C</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="c6">W</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><br/><a NAME="E2:19_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Maroon" title="c2">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:19"><span class="lab"><font color="Green" title="E9">A1</font></span></a>, <a class="ref" href="pre_topc.html#D2" title="PRE_TOPC:def.2">PRE_TOPC:def 2</a></span>;<br/><span class="kw">hence </span><a NAME="E3:19_1"/>
 <a href="tbsp_1.html#K3" title="TBSP_1:func.3">diameter</a> <font color="Maroon" title="c6">W</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="ref" href="tbsp_1.html#T24" title="TBSP_1:th.24">TBSP_1:24</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="E5:19"/><span class="kw">then </span><span class="lab"><font color="Green" title="E10">A2</font></span>: 
 <a href="simplex2.html#K2" title="SIMPLEX2:func.2">diameter</a> (<font color="Maroon" title="c1">M</font>,<font color="Maroon" title="c4">C</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="ref" href="simplex2.html#D4" target="_self" title="SIMPLEX2:def.4">Def4</a></span>;<br/>
<a NAME="E6:19"/>
<font color="Maroon" title="c2">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">S</font>
 
;<br/>
<a NAME="E7:19"/><span class="kw">then </span>
<font color="Maroon" title="c5">s</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="txt" href="#E3:19"><span class="lab"><font color="Green" title="E9">A1</font></span></a>, <a class="ref" href="pre_topc.html#D2" title="PRE_TOPC:def.2">PRE_TOPC:def 2</a></span>;<br/>
<a NAME="E8:19"/><span class="kw">then </span>
 <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>  <a href="simplex2.html#K2" title="SIMPLEX2:func.2">diameter</a> (<font color="Maroon" title="c1">M</font>,<font color="Maroon" title="c4">C</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="simplex2.html#D4" target="_self" title="SIMPLEX2:def.4">Def4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:19"/>
 <a href="simplex2.html#K1" title="SIMPLEX2:func.1">diameter</a> (<font color="Maroon" title="c1">M</font>,<span class="p1">(<span class="default"><a href="simplex0.html#K5" title="SIMPLEX0:func.5">Complex_of</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">S</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>) <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="c2">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:19"><span class="lab"><font color="Green" title="E10">A2</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</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/>


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