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<span class="kw">let </span><font color="Maroon" title="c1">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</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">M</font> being   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> <br/>  for <font color="Olive" title="b2">K</font> being   <a href="matroid0.html#V1" title="MATROID0:attr.1">subset-closed</a>  <font color="Olive">b<sub>1</sub></font> <a href="simplex2.html#V1" title="SIMPLEX2:attr.1">bounded</a>   <a href="simplex0.html#M1" title="SIMPLEX0:mode.1">SimplicialComplexStr</a> of <font color="Maroon" title="c1">X</font><br/>  for <font color="Olive" title="b3">i</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> holds   <a href="simplex2.html#K1" title="SIMPLEX2:func.1">diameter</a> (<font color="Olive" title="b1">M</font>,<span class="p1">(<span class="default"><a href="simplex0.html#K9" title="SIMPLEX0:func.9">Skeleton_of</a> (<font color="Olive" title="b2">K</font>,<font color="Olive" title="b3">i</font>)</span>)</span>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="simplex2.html#K1" title="SIMPLEX2:func.1">diameter</a> (<font color="Olive" title="b1">M</font>,<font color="Olive" title="b2">K</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c2">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">K</font> being   <a href="matroid0.html#V1" title="MATROID0:attr.1">subset-closed</a>  <font color="Maroon" title="c2">M</font> <a href="simplex2.html#V1" title="SIMPLEX2:attr.1">bounded</a>   <a href="simplex0.html#M1" title="SIMPLEX0:mode.1">SimplicialComplexStr</a> of <font color="Maroon" title="c1">X</font><br/>  for <font color="Olive" title="b2">i</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> holds   <a href="simplex2.html#K1" title="SIMPLEX2:func.1">diameter</a> (<font color="Maroon" title="c2">M</font>,<span class="p1">(<span class="default"><a href="simplex0.html#K9" title="SIMPLEX0:func.9">Skeleton_of</a> (<font color="Olive" title="b1">K</font>,<font color="Olive" title="b2">i</font>)</span>)</span>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="simplex2.html#K1" title="SIMPLEX2:func.1">diameter</a> (<font color="Maroon" title="c2">M</font>,<font color="Olive" title="b1">K</font>)</span><br/>



<span class="kw">set </span><font color="Maroon" title="c3">r</font> =  the   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>
<span class="kw">let </span><font color="Maroon" title="c4">K</font> be   <a href="matroid0.html#V1" title="MATROID0:attr.1">subset-closed</a>  <font color="Maroon" title="c2">M</font> <a href="simplex2.html#V1" title="SIMPLEX2:attr.1">bounded</a>   <a href="simplex0.html#M1" title="SIMPLEX0:mode.1">SimplicialComplexStr</a> of <font color="Maroon" title="c1">X</font>; <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">i</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> holds   <a href="simplex2.html#K1" title="SIMPLEX2:func.1">diameter</a> (<font color="Maroon" title="c2">M</font>,<span class="p1">(<span class="default"><a href="simplex0.html#K9" title="SIMPLEX0:func.9">Skeleton_of</a> (<font color="Maroon" title="c4">K</font>,<font color="Olive" title="b1">i</font>)</span>)</span>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="simplex2.html#K1" title="SIMPLEX2:func.1">diameter</a> (<font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c4">K</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c5">i</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>; <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="c2">M</font>,<span class="p1">(<span class="default"><a href="simplex0.html#K9" title="SIMPLEX0:func.9">Skeleton_of</a> (<font color="Maroon" title="c4">K</font>,<font color="Maroon" title="c5">i</font>)</span>)</span>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="simplex2.html#K1" title="SIMPLEX2:func.1">diameter</a> (<font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c4">K</font>)</span><br/>



<span class="kw">set </span><font color="Maroon" title="c6">SK</font> =  <a href="simplex0.html#K9" title="SIMPLEX0:func.9">Skeleton_of</a> (<font color="Maroon" title="c4">K</font>,<font color="Maroon" title="c5">i</font>);<br/>
<a NAME="E1:17"/><span class="lab"><font color="Green" title="E8">A1</font></span>: 
 the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="simplex0.html#K9" title="SIMPLEX0:func.9">Skeleton_of</a> (<font color="Maroon" title="c4">K</font>,<font color="Maroon" title="c5">i</font>)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c4">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="simplex0.html#D13" title="SIMPLEX0:def.13">SIMPLEX0:def 13</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_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="E1:17_1"/>
(  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="simplex0.html#K9" title="SIMPLEX0:func.9">Skeleton_of</a> (<font color="Maroon" title="c4">K</font>,<font color="Maroon" title="c5">i</font>)</span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">M</font></span>)</span> or  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="simplex0.html#K9" title="SIMPLEX0:func.9">Skeleton_of</a> (<font color="Maroon" title="c4">K</font>,<font color="Maroon" title="c5">i</font>)</span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">M</font></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1"><span class="kw">suppose </span></a><a NAME="E1:17_1_1"/><span class="lab"><font color="Green" title="E9">A2</font></span>: 
 the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="simplex0.html#K9" title="SIMPLEX0:func.9">Skeleton_of</a> (<font color="Maroon" title="c4">K</font>,<font color="Maroon" title="c5">i</font>)</span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">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">  <a href="simplex2.html#K1" title="SIMPLEX2:func.1">diameter</a> (<font color="Maroon" title="c2">M</font>,<span class="p1">(<span class="default"><a href="simplex0.html#K9" title="SIMPLEX0:func.9">Skeleton_of</a> (<font color="Maroon" title="c4">K</font>,<font color="Maroon" title="c5">i</font>)</span>)</span>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="simplex2.html#K1" title="SIMPLEX2:func.1">diameter</a> (<font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c4">K</font>)</span><br/><div class="add"><a NAME="E2:17_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E10">A3</font></span>: 
 the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c4">K</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">M</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:17"><span class="lab"><font color="Green" title="E8">A1</font></span></a>, <a class="ref" href="xboole_1.html#T63" title="XBOOLE_1:th.63">XBOOLE_1:63</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">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">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="simplex0.html#K9" title="SIMPLEX0:func.9">Skeleton_of</a> (<font color="Maroon" title="c4">K</font>,<font color="Maroon" title="c5">i</font>)</span>)</span> implies  <a href="tbsp_1.html#K3" title="TBSP_1:func.3">diameter</a> <font color="Maroon" title="c7">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="simplex2.html#K1" title="SIMPLEX2:func.1">diameter</a> (<font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c4">K</font>) )</span><br/><a NAME="E1:17_1_1_1"/>
 <a href="matroid0.html#K1" title="MATROID0:func.1">the_family_of</a> <font color="Maroon" title="c4">K</font> is  <a href="classes1.html#V1" title="CLASSES1:attr.1">subset-closed</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matroid0.html#D3" title="MATROID0:def.3">MATROID0:def 3</a></span>;<br/><a NAME="E2:17_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E11">A4</font></span>: 
 the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c4">K</font> is  <a href="classes1.html#V1" title="CLASSES1:attr.1">subset-closed</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matroid0.html#D1" title="MATROID0:def.1">MATROID0:def 1</a></span>;<br/><span class="kw">assume </span><a NAME="E3:17_1_1_1"/>
<font color="Maroon" title="c7">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="simplex0.html#K9" title="SIMPLEX0:func.9">Skeleton_of</a> (<font color="Maroon" title="c4">K</font>,<font color="Maroon" title="c5">i</font>)</span>)</span>
 ; <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="c7">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="simplex2.html#K1" title="SIMPLEX2:func.1">diameter</a> (<font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c4">K</font>)</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">w</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E5:17_1_1_1"/><span class="lab"><font color="Green" title="E12">A5</font></span>: 
<font color="Maroon" title="c7">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">w</font>
 <span class="kw">and </span><br/><a NAME="E6:17_1_1_1"/><span class="lab"><font color="Green" title="E13">A6</font></span>: 
<font color="Maroon" title="c8">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="simplex0.html#NK4" title="SIMPLEX0:NK.4">the_subsets_with_limited_card</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c5">i</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span>, the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c4">K</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="simplex0.html#T2" title="SIMPLEX0:th.2">SIMPLEX0:2</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c9">w</font> = <font color="Maroon" title="c8">w</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c4">K</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:17_1_1_1"><span class="lab"><font color="Green" title="E13">A6</font></span></a></span>;<br/><a NAME="E8:17_1_1_1"/>
<font color="Maroon" title="c9">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c4">K</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:17_1_1_1"><span class="lab"><font color="Green" title="E13">A6</font></span></a>, <a class="ref" href="simplex0.html#D2" title="SIMPLEX0:def.2">SIMPLEX0:def 2</a></span>;<br/><a NAME="E9:17_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c4">K</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:17_1_1_1"><span class="lab"><font color="Green" title="E11">A4</font></span></a>, <a class="txt" href="#E5:17_1_1_1"><span class="lab"><font color="Green" title="E12">A5</font></span></a>, <a class="ref" href="classes1.html#D1" title="CLASSES1:def.1">CLASSES1:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E10:17_1_1_1"/>
 <a href="tbsp_1.html#K3" title="TBSP_1:func.3">diameter</a> <font color="Maroon" title="c7">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="simplex2.html#K1" title="SIMPLEX2:func.1">diameter</a> (<font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c4">K</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:17_1_1"><span class="lab"><font color="Green" title="E10">A3</font></span></a>, <a class="ref" href="simplex2.html#D3" target="_self" title="SIMPLEX2:def.3">Def3</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="E4:17_1_1"/>
 <a href="simplex2.html#K1" title="SIMPLEX2:func.1">diameter</a> (<font color="Maroon" title="c2">M</font>,<span class="p1">(<span class="default"><a href="simplex0.html#K9" title="SIMPLEX0:func.9">Skeleton_of</a> (<font color="Maroon" title="c4">K</font>,<font color="Maroon" title="c5">i</font>)</span>)</span>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="simplex2.html#K1" title="SIMPLEX2:func.1">diameter</a> (<font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c4">K</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:17_1_1"><span class="lab"><font color="Green" title="E9">A2</font></span></a>, <a class="ref" href="simplex2.html#D3" target="_self" title="SIMPLEX2:def.3">Def3</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="17_1_2"><span class="kw">suppose </span></a><a NAME="E1:17_1_2"/>
 the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="simplex0.html#K9" title="SIMPLEX0:func.9">Skeleton_of</a> (<font color="Maroon" title="c4">K</font>,<font color="Maroon" title="c5">i</font>)</span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">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">  <a href="simplex2.html#K1" title="SIMPLEX2:func.1">diameter</a> (<font color="Maroon" title="c2">M</font>,<span class="p1">(<span class="default"><a href="simplex0.html#K9" title="SIMPLEX0:func.9">Skeleton_of</a> (<font color="Maroon" title="c4">K</font>,<font color="Maroon" title="c5">i</font>)</span>)</span>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="simplex2.html#K1" title="SIMPLEX2:func.1">diameter</a> (<font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c4">K</font>)</span><br/><div class="add"><a NAME="E2:17_1_2"/><span class="kw">then </span>
 <a href="simplex2.html#K1" title="SIMPLEX2:func.1">diameter</a> (<font color="Maroon" title="c2">M</font>,<span class="p1">(<span class="default"><a href="simplex0.html#K9" title="SIMPLEX0:func.9">Skeleton_of</a> (<font color="Maroon" title="c4">K</font>,<font color="Maroon" title="c5">i</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="simplex2.html#D3" target="_self" title="SIMPLEX2:def.3">Def3</a></span>;<br/><span class="kw">hence </span><a NAME="E3:17_1_2"/>
 <a href="simplex2.html#K1" title="SIMPLEX2:func.1">diameter</a> (<font color="Maroon" title="c2">M</font>,<span class="p1">(<span class="default"><a href="simplex0.html#K9" title="SIMPLEX0:func.9">Skeleton_of</a> (<font color="Maroon" title="c4">K</font>,<font color="Maroon" title="c5">i</font>)</span>)</span>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="simplex2.html#K1" title="SIMPLEX2:func.1">diameter</a> (<font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c4">K</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="simplex2.html#T7" target="_self" title="SIMPLEX2:th.7">Th7</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>

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