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<span class="kw">let </span><font color="Maroon" title="c1">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</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">A</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> holds <br/> ( <font color="Olive" title="b1">A</font> is  <a href="jordan2c.html#V1" title="JORDAN2C:attr.1">Bounded</a>  iff  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <font color="Olive" title="b1">A</font> is  <a href="jordan2c.html#V1" title="JORDAN2C:attr.1">Bounded</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</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">A</font> is  <a href="jordan2c.html#V1" title="JORDAN2C:attr.1">Bounded</a>  iff  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <font color="Maroon" title="c2">A</font> is  <a href="jordan2c.html#V1" title="JORDAN2C:attr.1">Bounded</a>  )</span><br/>



<a NAME="E1:27"/>
 <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>, the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c1">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#D8" title="EUCLID:def.8">EUCLID:def 8</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c3">cA</font> =  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <font color="Maroon" title="c2">A</font>, <font color="Maroon" title="c4">a</font> = <font color="Maroon" title="c2">A</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c1">n</font></span>)</span> ;<br/>
<div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <font color="Maroon" title="c2">A</font> is  <a href="jordan2c.html#V1" title="JORDAN2C:attr.1">Bounded</a>  implies <font color="Maroon" title="c2">A</font> is  <a href="jordan2c.html#V1" title="JORDAN2C:attr.1">Bounded</a>  )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:27_1"/>
<font color="Maroon" title="c2">A</font> is  <a href="jordan2c.html#V1" title="JORDAN2C:attr.1">Bounded</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <font color="Maroon" title="c2">A</font> is  <a href="jordan2c.html#V1" title="JORDAN2C:attr.1">Bounded</a> </span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">a</font> = <font color="Maroon" title="c4">a</font> as   <a href="tbsp_1.html#V6" title="TBSP_1:attr.6">bounded</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c1">n</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="jordan2c.html#D1" title="JORDAN2C:def.1">JORDAN2C:def 1</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c6">D</font> =  <a href="tbsp_1.html#K3" title="TBSP_1:func.3">diameter</a> <font color="Maroon" title="c5">a</font>;<br/><div><span class="lab"><font color="Green" title="E16">A1</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="27_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 <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c1">n</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="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">cA</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">cA</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="c5">a</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1 )</span><br/><span class="kw">assume </span><a NAME="E1:27_1_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">cA</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">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">cA</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="c5">a</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1 )</span><br/><a NAME="E2:27_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E17">A2</font></span>: 
 <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="metric_1.html#K10" title="METRIC_1:func.10">cl_Ball</a> (<font color="Maroon" title="c7">x</font>,<span class="p1">(<span class="default"><a href="tbsp_1.html#K3" title="TBSP_1:func.3">diameter</a> <font color="Maroon" title="c5">a</font></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="simplex2.html#T13" target="_self" title="SIMPLEX2:th.13">Th13</a></span>;<br/><span class="kw">assume </span><a NAME="E3:27_1_1"/>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">cA</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="c5">a</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span><br/><a NAME="E4:27_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="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="c5">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:27_1_1"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="ref" href="metric_1.html#T12" title="METRIC_1:th.12">METRIC_1:12</a></span>;<br/><span class="kw">hence </span><a NAME="E5:27_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#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="c5">a</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#T39" title="XREAL_1:th.39">XREAL_1:39</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="E4:27_1"/>
 <a href="tbsp_1.html#K3" title="TBSP_1:func.3">diameter</a> <font color="Maroon" title="c5">a</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</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="tbsp_1.html#T21" title="TBSP_1:th.21">TBSP_1:21</a></span>;<br/><a NAME="E5:27_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">cA</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="#E3:27_1"><span class="lab"><font color="Green" title="E16">A1</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="E6:27_1"/>
 <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <font color="Maroon" title="c2">A</font> is  <a href="jordan2c.html#V1" title="JORDAN2C:attr.1">Bounded</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jordan2c.html#D1" title="JORDAN2C:def.1">JORDAN2C:def 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/></div>
<span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E4:27"/>
 <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <font color="Maroon" title="c2">A</font> is  <a href="jordan2c.html#V1" title="JORDAN2C:attr.1">Bounded</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="c2">A</font> is  <a href="jordan2c.html#V1" title="JORDAN2C:attr.1">Bounded</a> </span><br/>

<a NAME="E5:27"/><span class="kw">then </span>
<font color="Maroon" title="c3">cA</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="ref" href="jordan2c.html#D1" title="JORDAN2C:def.1">JORDAN2C:def 1</a></span>;<br/>
<a NAME="E6:27"/><span class="kw">then </span>
<font color="Maroon" title="c4">a</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="ref" href="convex1.html#T41" title="CONVEX1:th.41">CONVEX1:41</a>, <a class="ref" href="tbsp_1.html#T14" title="TBSP_1:th.14">TBSP_1:14</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:27"/>
<font color="Maroon" title="c2">A</font> is  <a href="jordan2c.html#V1" title="JORDAN2C:attr.1">Bounded</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jordan2c.html#D1" title="JORDAN2C:def.1">JORDAN2C:def 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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