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<span class="kw">let </span><font color="Maroon" title="c1">T</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">P</font>, <font color="Olive" title="b2">Q</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b1">P</font> is  <a href="tbsp_1.html#V6" title="TBSP_1:attr.6">bounded</a>  &amp; <font color="Olive" title="b2">Q</font> is  <a href="tbsp_1.html#V6" title="TBSP_1:attr.6">bounded</a>  &amp; <font color="Olive" title="b1">P</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Olive" title="b2">Q</font> holds <br/> <a href="tbsp_1.html#K3" title="TBSP_1:func.3">diameter</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">P</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b2">Q</font></span>)</span> <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="Olive" title="b1">P</font></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="Olive" title="b2">Q</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">P</font>, <font color="Maroon" title="c3">Q</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</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">P</font> is  <a href="tbsp_1.html#V6" title="TBSP_1:attr.6">bounded</a>  &amp; <font color="Maroon" title="c3">Q</font> is  <a href="tbsp_1.html#V6" title="TBSP_1:attr.6">bounded</a>  &amp; <font color="Maroon" title="c2">P</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">Q</font> implies  <a href="tbsp_1.html#K3" title="TBSP_1:func.3">diameter</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">Q</font></span>)</span> <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">P</font></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="c3">Q</font></span>)</span> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:44"/><span class="lab"><font color="Green" title="E25">A1</font></span>: 
<font color="Maroon" title="c2">P</font> is  <a href="tbsp_1.html#V6" title="TBSP_1:attr.6">bounded</a> 
 <span class="kw">and </span><br/><a NAME="E2:44"/><span class="lab"><font color="Green" title="E26">A2</font></span>: 
<font color="Maroon" title="c3">Q</font> is  <a href="tbsp_1.html#V6" title="TBSP_1:attr.6">bounded</a> 
 <span class="kw">and </span><br/><a NAME="E3:44"/><span class="lab"><font color="Green" title="E27">A3</font></span>: 
<font color="Maroon" title="c2">P</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">Q</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="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a> <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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">Q</font></span>)</span> <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">P</font></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="c3">Q</font></span>)</span></span><br/>

<span class="kw">set </span><font color="Maroon" title="c4">g</font> =  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">P</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">Q</font>;<br/>
<a NAME="E4:44"/><span class="lab"><font color="Green" title="E28">A4</font></span>: 
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">P</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">Q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Q</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:44"><span class="lab"><font color="Green" title="E27">A3</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">s</font> = <span class="p1">(<span class="default"><a href="tbsp_1.html#K3" title="TBSP_1:func.3">diameter</a> <font color="Maroon" title="c2">P</font></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="c3">Q</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">b</font> =  <a href="tbsp_1.html#K3" title="TBSP_1:func.3">diameter</a> <font color="Maroon" title="c3">Q</font>;<br/>
<a NAME="E5:44"/><span class="lab"><font color="Green" title="E29">A5</font></span>: 
 <a href="tbsp_1.html#K3" title="TBSP_1:func.3">diameter</a> <font color="Maroon" title="c3">Q</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">P</font></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="c3">Q</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:44"><span class="lab"><font color="Green" title="E25">A1</font></span></a>, <a class="ref" href="tbsp_1.html#T21" target="_self" title="TBSP_1:th.21">Th29</a>, <a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">a</font> =  <a href="tbsp_1.html#K3" title="TBSP_1:func.3">diameter</a> <font color="Maroon" title="c2">P</font>;<br/>
<a NAME="E6:44"/><span class="lab"><font color="Green" title="E30">A6</font></span>: 
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">P</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">Q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:44"><span class="lab"><font color="Green" title="E27">A3</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">g</font> =  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">P</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">Q</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">T</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:44"><span class="lab"><font color="Green" title="E27">A3</font></span></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/>
<a NAME="E8:44"/><span class="lab"><font color="Green" title="E31">A7</font></span>: 
 <a href="tbsp_1.html#K3" title="TBSP_1:func.3">diameter</a> <font color="Maroon" title="c2">P</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">P</font></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="c3">Q</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:44"><span class="lab"><font color="Green" title="E26">A2</font></span></a>, <a class="ref" href="tbsp_1.html#T21" target="_self" title="TBSP_1:th.21">Th29</a>, <a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></span>;<br/>
<a NAME="E9:44"/><span class="lab"><font color="Green" title="E32">A8</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">T</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">Q</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">Q</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> <span class="p1">(<span class="default"><a href="tbsp_1.html#K3" title="TBSP_1:func.3">diameter</a> <font color="Maroon" title="c2">P</font></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="c3">Q</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">x</font>, <font color="Maroon" title="c10">y</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</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="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">Q</font> &amp; <font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">Q</font> implies  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c10">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">P</font></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="c3">Q</font></span>)</span> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:44_1"/><span class="lab"><font color="Green" title="E33">A9</font></span>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">Q</font>
 <span class="kw">and </span><br/><a NAME="E2:44_1"/><span class="lab"><font color="Green" title="E34">A10</font></span>: 
<font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">Q</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="c9">x</font>,<font color="Maroon" title="c10">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">P</font></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="c3">Q</font></span>)</span></span><br/>

<a NAME="E3:44_1"/><span class="lab"><font color="Green" title="E35">A11</font></span>: 
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c10">y</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="c9">x</font>,<font color="Maroon" title="c8">g</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="c8">g</font>,<font color="Maroon" title="c10">y</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/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_1_1_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="E1:44_1_1_1"/>
( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P</font> or <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Q</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:44_1"><span class="lab"><font color="Green" title="E33">A9</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:44_1_1_1_1"/><span class="lab"><font color="Green" title="E36">A12</font></span>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P</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="c9">x</font>,<font color="Maroon" title="c10">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">P</font></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="c3">Q</font></span>)</span></span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_1_1_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_1_1_1_1_1_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="E1:44_1_1_1_1_1_1"/>
( <font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P</font> or <font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Q</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:44_1"><span class="lab"><font color="Green" title="E34">A10</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_1_1_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:44_1_1_1_1_1_1_1"/>
<font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P</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="c9">x</font>,<font color="Maroon" title="c10">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">P</font></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="c3">Q</font></span>)</span></span><br/><div class="add"><a NAME="E2:44_1_1_1_1_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="c9">x</font>,<font color="Maroon" title="c10">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">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:44"><span class="lab"><font color="Green" title="E25">A1</font></span></a>, <a class="txt" href="#E1:44_1_1_1_1"><span class="lab"><font color="Green" title="E36">A12</font></span></a>, <a class="ref" href="tbsp_1.html#D8" target="_self" title="TBSP_1:def.8">Def10</a></span>;<br/><span class="kw">hence </span><a NAME="E3:44_1_1_1_1_1_1_1"/>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c10">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">P</font></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="c3">Q</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:44"><span class="lab"><font color="Green" title="E31">A7</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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_1_1_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:44_1_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E37">A13</font></span>: 
<font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Q</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="c9">x</font>,<font color="Maroon" title="c10">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">P</font></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="c3">Q</font></span>)</span></span><br/><div class="add"><a NAME="E2:44_1_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E38">A14</font></span>: 
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c8">g</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">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:44"><span class="lab"><font color="Green" title="E25">A1</font></span></a>, <a class="txt" href="#E6:44"><span class="lab"><font color="Green" title="E30">A6</font></span></a>, <a class="txt" href="#E1:44_1_1_1_1"><span class="lab"><font color="Green" title="E36">A12</font></span></a>, <a class="ref" href="tbsp_1.html#D8" target="_self" title="TBSP_1:def.8">Def10</a></span>;<br/><a NAME="E3:44_1_1_1_1_1_1_2"/>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<font color="Maroon" title="c8">g</font>,<font color="Maroon" title="c10">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="c3">Q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:44"><span class="lab"><font color="Green" title="E26">A2</font></span></a>, <a class="txt" href="#E4:44"><span class="lab"><font color="Green" title="E28">A4</font></span></a>, <a class="txt" href="#E1:44_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E37">A13</font></span></a>, <a class="ref" href="tbsp_1.html#D8" target="_self" title="TBSP_1:def.8">Def10</a></span>;<br/><a NAME="E4:44_1_1_1_1_1_1_2"/><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="c9">x</font>,<font color="Maroon" title="c8">g</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="c8">g</font>,<font color="Maroon" title="c10">y</font>)</span>)</span> <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">P</font></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="c3">Q</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:44_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E38">A14</font></span></a>, <a class="ref" href="xreal_1.html#T7" title="XREAL_1:th.7">XREAL_1:7</a></span>;<br/><span class="kw">hence </span><a NAME="E5:44_1_1_1_1_1_1_2"/>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c10">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">P</font></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="c3">Q</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:44_1"><span class="lab"><font color="Green" title="E35">A11</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="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:44_1_1_1_1"/>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c10">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">P</font></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="c3">Q</font></span>)</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="44_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:44_1_1_1_2"/><span class="lab"><font color="Green" title="E36">A15</font></span>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Q</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="c9">x</font>,<font color="Maroon" title="c10">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">P</font></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="c3">Q</font></span>)</span></span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_1_1_2_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_1_1_1_2_1_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="E1:44_1_1_1_2_1_1"/>
( <font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P</font> or <font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Q</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:44_1"><span class="lab"><font color="Green" title="E34">A10</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_1_1_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:44_1_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E37">A16</font></span>: 
<font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P</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="c9">x</font>,<font color="Maroon" title="c10">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">P</font></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="c3">Q</font></span>)</span></span><br/><div class="add"><a NAME="E2:44_1_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E38">A17</font></span>: 
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c8">g</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="c3">Q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:44"><span class="lab"><font color="Green" title="E26">A2</font></span></a>, <a class="txt" href="#E4:44"><span class="lab"><font color="Green" title="E28">A4</font></span></a>, <a class="txt" href="#E1:44_1_1_1_2"><span class="lab"><font color="Green" title="E36">A15</font></span></a>, <a class="ref" href="tbsp_1.html#D8" target="_self" title="TBSP_1:def.8">Def10</a></span>;<br/><a NAME="E3:44_1_1_1_2_1_1_1"/>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<font color="Maroon" title="c8">g</font>,<font color="Maroon" title="c10">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">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:44"><span class="lab"><font color="Green" title="E25">A1</font></span></a>, <a class="txt" href="#E6:44"><span class="lab"><font color="Green" title="E30">A6</font></span></a>, <a class="txt" href="#E1:44_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E37">A16</font></span></a>, <a class="ref" href="tbsp_1.html#D8" target="_self" title="TBSP_1:def.8">Def10</a></span>;<br/><a NAME="E4:44_1_1_1_2_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="c9">x</font>,<font color="Maroon" title="c8">g</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="c8">g</font>,<font color="Maroon" title="c10">y</font>)</span>)</span> <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="c3">Q</font></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">P</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:44_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E38">A17</font></span></a>, <a class="ref" href="xreal_1.html#T7" title="XREAL_1:th.7">XREAL_1:7</a></span>;<br/><span class="kw">hence </span><a NAME="E5:44_1_1_1_2_1_1_1"/>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c10">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">P</font></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="c3">Q</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:44_1"><span class="lab"><font color="Green" title="E35">A11</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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_1_1_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:44_1_1_1_2_1_1_2"/>
<font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Q</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="c9">x</font>,<font color="Maroon" title="c10">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">P</font></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="c3">Q</font></span>)</span></span><br/><div class="add"><a NAME="E2:44_1_1_1_2_1_1_2"/><span class="kw">then </span>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c10">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="c3">Q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:44"><span class="lab"><font color="Green" title="E26">A2</font></span></a>, <a class="txt" href="#E1:44_1_1_1_2"><span class="lab"><font color="Green" title="E36">A15</font></span></a>, <a class="ref" href="tbsp_1.html#D8" target="_self" title="TBSP_1:def.8">Def10</a></span>;<br/><span class="kw">hence </span><a NAME="E3:44_1_1_1_2_1_1_2"/>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c10">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">P</font></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="c3">Q</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:44"><span class="lab"><font color="Green" title="E29">A5</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="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:44_1_1_1_2"/>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c10">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">P</font></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="c3">Q</font></span>)</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><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E5:44_1"/>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c10">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">P</font></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="c3">Q</font></span>)</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="E10:44"/>
( <font color="Maroon" title="c2">P</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>  &amp; <font color="Maroon" title="c2">P</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">Q</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="#E1:44"><span class="lab"><font color="Green" title="E25">A1</font></span></a>, <a class="txt" href="#E2:44"><span class="lab"><font color="Green" title="E26">A2</font></span></a>, <a class="txt" href="#E3:44"><span class="lab"><font color="Green" title="E27">A3</font></span></a>, <a class="ref" href="tbsp_1.html#T13" target="_self" title="TBSP_1:th.13">Th20</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:44"/>
 <a href="tbsp_1.html#K3" title="TBSP_1:func.3">diameter</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">Q</font></span>)</span> <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">P</font></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="c3">Q</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:44"><span class="lab"><font color="Green" title="E32">A8</font></span></a>, <a class="ref" href="tbsp_1.html#D8" target="_self" title="TBSP_1:def.8">Def10</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>
