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<span class="kw">let </span><font color="Maroon" title="c1">M</font> be    <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="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">a</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font><br/> for <font color="Olive" title="b2">X</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( (  <a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">X</font> is  <a href="metric_1.html#V2" title="METRIC_1:attr.2">Reflexive</a>  implies <font color="Maroon" title="c1">M</font> is  <a href="metric_1.html#V6" title="METRIC_1:attr.6">Reflexive</a>  ) &amp; (  <a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">X</font> is  <a href="metric_1.html#V4" title="METRIC_1:attr.4">symmetric</a>  implies <font color="Maroon" title="c1">M</font> is  <a href="metric_1.html#V8" title="METRIC_1:attr.8">symmetric</a>  ) &amp; (  <a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">X</font> is  <a href="metric_1.html#V5" title="METRIC_1:attr.5">triangle</a>  &amp;  <a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">X</font> is  <a href="metric_1.html#V2" title="METRIC_1:attr.2">Reflexive</a>  implies <font color="Maroon" title="c1">M</font> is  <a href="metric_1.html#V9" title="METRIC_1:attr.9">triangle</a>  ) &amp; (  <a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">X</font> is  <a href="metric_1.html#V3" title="METRIC_1:attr.3">discerning</a>  &amp;  <a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">X</font> is  <a href="metric_1.html#V2" title="METRIC_1:attr.2">Reflexive</a>  implies <font color="Maroon" title="c1">M</font> is  <a href="metric_1.html#V7" title="METRIC_1:attr.7">discerning</a>  ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">A</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">X</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( (  <a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">A</font>,<font color="Olive" title="b1">X</font> is  <a href="metric_1.html#V2" title="METRIC_1:attr.2">Reflexive</a>  implies <font color="Maroon" title="c1">M</font> is  <a href="metric_1.html#V6" title="METRIC_1:attr.6">Reflexive</a>  ) &amp; (  <a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">A</font>,<font color="Olive" title="b1">X</font> is  <a href="metric_1.html#V4" title="METRIC_1:attr.4">symmetric</a>  implies <font color="Maroon" title="c1">M</font> is  <a href="metric_1.html#V8" title="METRIC_1:attr.8">symmetric</a>  ) &amp; (  <a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">A</font>,<font color="Olive" title="b1">X</font> is  <a href="metric_1.html#V5" title="METRIC_1:attr.5">triangle</a>  &amp;  <a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">A</font>,<font color="Olive" title="b1">X</font> is  <a href="metric_1.html#V2" title="METRIC_1:attr.2">Reflexive</a>  implies <font color="Maroon" title="c1">M</font> is  <a href="metric_1.html#V9" title="METRIC_1:attr.9">triangle</a>  ) &amp; (  <a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">A</font>,<font color="Olive" title="b1">X</font> is  <a href="metric_1.html#V3" title="METRIC_1:attr.3">discerning</a>  &amp;  <a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">A</font>,<font color="Olive" title="b1">X</font> is  <a href="metric_1.html#V2" title="METRIC_1:attr.2">Reflexive</a>  implies <font color="Maroon" title="c1">M</font> is  <a href="metric_1.html#V7" title="METRIC_1:attr.7">discerning</a>  ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">X</font> be  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( (  <a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">X</font> is  <a href="metric_1.html#V2" title="METRIC_1:attr.2">Reflexive</a>  implies <font color="Maroon" title="c1">M</font> is  <a href="metric_1.html#V6" title="METRIC_1:attr.6">Reflexive</a>  ) &amp; (  <a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">X</font> is  <a href="metric_1.html#V4" title="METRIC_1:attr.4">symmetric</a>  implies <font color="Maroon" title="c1">M</font> is  <a href="metric_1.html#V8" title="METRIC_1:attr.8">symmetric</a>  ) &amp; (  <a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">X</font> is  <a href="metric_1.html#V5" title="METRIC_1:attr.5">triangle</a>  &amp;  <a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">X</font> is  <a href="metric_1.html#V2" title="METRIC_1:attr.2">Reflexive</a>  implies <font color="Maroon" title="c1">M</font> is  <a href="metric_1.html#V9" title="METRIC_1:attr.9">triangle</a>  ) &amp; (  <a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">X</font> is  <a href="metric_1.html#V3" title="METRIC_1:attr.3">discerning</a>  &amp;  <a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">X</font> is  <a href="metric_1.html#V2" title="METRIC_1:attr.2">Reflexive</a>  implies <font color="Maroon" title="c1">M</font> is  <a href="metric_1.html#V7" title="METRIC_1:attr.7">discerning</a>  ) )</span><br/>





<span class="kw">consider </span><font color="Maroon" title="c4">x0</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E2:61"/><span class="lab"><font color="Green" title="E51">A1</font></span>: 
<font color="Maroon" title="c4">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">w</font> =  <a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">X</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">XX</font> = <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c3">X</font>,<span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c2">A</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>;<br/>
<span class="kw">thus </span><a NAME="E3:61"/><span class="lab"><font color="Green" title="E52">A2</font></span>: 
(  <a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">X</font> is  <a href="metric_1.html#V2" title="METRIC_1:attr.2">Reflexive</a>  implies <font color="Maroon" title="c1">M</font> is  <a href="metric_1.html#V6" title="METRIC_1:attr.6">Reflexive</a>  )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( (  <a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">X</font> is  <a href="metric_1.html#V4" title="METRIC_1:attr.4">symmetric</a>  implies <font color="Maroon" title="c1">M</font> is  <a href="metric_1.html#V8" title="METRIC_1:attr.8">symmetric</a>  ) &amp; (  <a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">X</font> is  <a href="metric_1.html#V5" title="METRIC_1:attr.5">triangle</a>  &amp;  <a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">X</font> is  <a href="metric_1.html#V2" title="METRIC_1:attr.2">Reflexive</a>  implies <font color="Maroon" title="c1">M</font> is  <a href="metric_1.html#V9" title="METRIC_1:attr.9">triangle</a>  ) &amp; (  <a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">X</font> is  <a href="metric_1.html#V3" title="METRIC_1:attr.3">discerning</a>  &amp;  <a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">X</font> is  <a href="metric_1.html#V2" title="METRIC_1:attr.2">Reflexive</a>  implies <font color="Maroon" title="c1">M</font> is  <a href="metric_1.html#V7" title="METRIC_1:attr.7">discerning</a>  ) )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:61_1"/><span class="lab"><font color="Green" title="E53">A3</font></span>: 
 <a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">X</font> is  <a href="metric_1.html#V2" title="METRIC_1:attr.2">Reflexive</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">M</font> is  <a href="metric_1.html#V6" title="METRIC_1:attr.6">Reflexive</a> </span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_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#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">M</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_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="E1:61_1_1_1_1"/>
( <font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> or <font color="Maroon" title="c7">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">A</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:61_1_1_1_1_1"/>
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/><div class="add"><a NAME="E2:61_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A4</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c7">a</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c3">X</font>,<span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c2">A</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="compl_sp.html#T38" target="_self" title="COMPL_SP:th.38">Th38</a></span>;<br/><span class="kw">hence </span> <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c7">a</font> = 
<span class="p1">(<span class="default"><a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">X</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c7">a</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c7">a</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="compl_sp.html#D10" target="_self" title="COMPL_SP:def.10">Def10</a></span>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61_1"><span class="lab"><font color="Green" title="E53">A3</font></span></a>, <a class="txt" href="#E2:61_1_1_1_1_1"><span class="lab"><font color="Green" title="E54">A4</font></span></a>, <a class="ref" href="metric_1.html#D3" title="METRIC_1:def.3">METRIC_1:def 3</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="61_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:61_1_1_1_1_2"/>
<font color="Maroon" title="c7">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/><div class="add"><a NAME="E2:61_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A5</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x0</font>,<font color="Maroon" title="c7">a</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c3">X</font>,<span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c2">A</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:61"><span class="lab"><font color="Green" title="E51">A1</font></span></a>, <a class="ref" href="compl_sp.html#T38" target="_self" title="COMPL_SP:th.38">Th38</a></span>;<br/><span class="kw">hence </span> <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c7">a</font> = 
<span class="p1">(<span class="default"><a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">X</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x0</font>,<font color="Maroon" title="c7">a</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x0</font>,<font color="Maroon" title="c7">a</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="compl_sp.html#D10" target="_self" title="COMPL_SP:def.10">Def10</a></span>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61_1"><span class="lab"><font color="Green" title="E53">A3</font></span></a>, <a class="txt" href="#E2:61_1_1_1_1_2"><span class="lab"><font color="Green" title="E54">A5</font></span></a>, <a class="ref" href="metric_1.html#D3" title="METRIC_1:def.3">METRIC_1:def 3</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="E2:61_1_1"/>
 <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:61_1"/>
<font color="Maroon" title="c1">M</font> is  <a href="metric_1.html#V6" title="METRIC_1:attr.6">Reflexive</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#T1" title="METRIC_1:th.1">METRIC_1:1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E4:61"/>
(  <a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">X</font> is  <a href="metric_1.html#V4" title="METRIC_1:attr.4">symmetric</a>  implies <font color="Maroon" title="c1">M</font> is  <a href="metric_1.html#V8" title="METRIC_1:attr.8">symmetric</a>  )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( (  <a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">X</font> is  <a href="metric_1.html#V5" title="METRIC_1:attr.5">triangle</a>  &amp;  <a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">X</font> is  <a href="metric_1.html#V2" title="METRIC_1:attr.2">Reflexive</a>  implies <font color="Maroon" title="c1">M</font> is  <a href="metric_1.html#V9" title="METRIC_1:attr.9">triangle</a>  ) &amp; (  <a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">X</font> is  <a href="metric_1.html#V3" title="METRIC_1:attr.3">discerning</a>  &amp;  <a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">X</font> is  <a href="metric_1.html#V2" title="METRIC_1:attr.2">Reflexive</a>  implies <font color="Maroon" title="c1">M</font> is  <a href="metric_1.html#V7" title="METRIC_1:attr.7">discerning</a>  ) )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:61_2"/><span class="lab"><font color="Green" title="E53">A6</font></span>: 
 <a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">X</font> is  <a href="metric_1.html#V4" title="METRIC_1:attr.4">symmetric</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">M</font> is  <a href="metric_1.html#V8" title="METRIC_1:attr.8">symmetric</a> </span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">a</font>, <font color="Maroon" title="c8">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">M</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c7">a</font></span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_2_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_2_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="E1:61_2_1_1_1"/>
( ( <font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c8">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> ) or ( <font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c8">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">A</font> ) or ( <font color="Maroon" title="c7">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c8">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> ) or ( <font color="Maroon" title="c7">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c8">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">A</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_2_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:61_2_1_1_1_1"/>
( <font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c8">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c7">a</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:61_2_1_1_1_1"/>
 <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c7">a</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="61_2_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:61_2_1_1_1_2"/><span class="lab"><font color="Green" title="E54">A7</font></span>: 
( <font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c8">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">A</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c7">a</font></span><br/><div class="add"><a NAME="E2:61_2_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A8</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x0</font>,<font color="Maroon" title="c8">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c3">X</font>,<span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c2">A</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:61"><span class="lab"><font color="Green" title="E51">A1</font></span></a>, <a class="ref" href="compl_sp.html#T38" target="_self" title="COMPL_SP:th.38">Th38</a></span>;<br/><a NAME="E3:61_2_1_1_1_2"/><span class="lab"><font color="Green" title="E56">A9</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c2">A</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c3">X</font>,<span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c2">A</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="compl_sp.html#T38" target="_self" title="COMPL_SP:th.38">Th38</a></span>;<br/><a NAME="E4:61_2_1_1_1_2"/><span class="lab"><font color="Green" title="E57">A10</font></span>: 
<font color="Maroon" title="c3">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">x0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:61"><span class="lab"><font color="Green" title="E51">A1</font></span></a></span>;<br/><span class="kw">then </span><span class="p1">(<span class="default"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c2">A</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#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c8">b</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">X</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c2">A</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x0</font>,<font color="Maroon" title="c8">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:61_2_1_1_1_2"><span class="lab"><font color="Green" title="E56">A9</font></span></a>, <a class="txt" href="#E2:61_2_1_1_1_2"><span class="lab"><font color="Green" title="E55">A8</font></span></a>, <a class="ref" href="compl_sp.html#D10" target="_self" title="COMPL_SP:def.10">Def10</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">X</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x0</font>,<font color="Maroon" title="c8">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c2">A</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61_2"><span class="lab"><font color="Green" title="E53">A6</font></span></a>, <a class="txt" href="#E3:61_2_1_1_1_2"><span class="lab"><font color="Green" title="E56">A9</font></span></a>, <a class="txt" href="#E2:61_2_1_1_1_2"><span class="lab"><font color="Green" title="E55">A8</font></span></a>, <a class="ref" href="metric_1.html#D5" title="METRIC_1:def.5">METRIC_1:def 5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c2">A</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#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c2">A</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:61_2_1_1_1_2"><span class="lab"><font color="Green" title="E56">A9</font></span></a>, <a class="txt" href="#E2:61_2_1_1_1_2"><span class="lab"><font color="Green" title="E55">A8</font></span></a>, <a class="txt" href="#E4:61_2_1_1_1_2"><span class="lab"><font color="Green" title="E57">A10</font></span></a>, <a class="ref" href="compl_sp.html#D10" target="_self" title="COMPL_SP:def.10">Def10</a></span>
;<br/><span class="kw">hence </span><a NAME="E6:61_2_1_1_1_2"/>
 <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c7">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61_2_1_1_1_2"><span class="lab"><font color="Green" title="E54">A7</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="61_2_1_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:61_2_1_1_1_3"/><span class="lab"><font color="Green" title="E54">A11</font></span>: 
( <font color="Maroon" title="c7">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c8">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c7">a</font></span><br/><div class="add"><a NAME="E2:61_2_1_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A12</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x0</font>,<font color="Maroon" title="c7">a</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c3">X</font>,<span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c2">A</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:61"><span class="lab"><font color="Green" title="E51">A1</font></span></a>, <a class="ref" href="compl_sp.html#T38" target="_self" title="COMPL_SP:th.38">Th38</a></span>;<br/><a NAME="E3:61_2_1_1_1_3"/><span class="lab"><font color="Green" title="E56">A13</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c2">A</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c3">X</font>,<span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c2">A</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="compl_sp.html#T38" target="_self" title="COMPL_SP:th.38">Th38</a></span>;<br/><a NAME="E4:61_2_1_1_1_3"/><span class="lab"><font color="Green" title="E57">A14</font></span>: 
<font color="Maroon" title="c3">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">x0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:61"><span class="lab"><font color="Green" title="E51">A1</font></span></a></span>;<br/><span class="kw">then </span><span class="p1">(<span class="default"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c2">A</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#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c7">a</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">X</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c2">A</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x0</font>,<font color="Maroon" title="c7">a</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:61_2_1_1_1_3"><span class="lab"><font color="Green" title="E56">A13</font></span></a>, <a class="txt" href="#E2:61_2_1_1_1_3"><span class="lab"><font color="Green" title="E55">A12</font></span></a>, <a class="ref" href="compl_sp.html#D10" target="_self" title="COMPL_SP:def.10">Def10</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">X</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x0</font>,<font color="Maroon" title="c7">a</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c2">A</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61_2"><span class="lab"><font color="Green" title="E53">A6</font></span></a>, <a class="txt" href="#E3:61_2_1_1_1_3"><span class="lab"><font color="Green" title="E56">A13</font></span></a>, <a class="txt" href="#E2:61_2_1_1_1_3"><span class="lab"><font color="Green" title="E55">A12</font></span></a>, <a class="ref" href="metric_1.html#D5" title="METRIC_1:def.5">METRIC_1:def 5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c2">A</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#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c2">A</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:61_2_1_1_1_3"><span class="lab"><font color="Green" title="E56">A13</font></span></a>, <a class="txt" href="#E2:61_2_1_1_1_3"><span class="lab"><font color="Green" title="E55">A12</font></span></a>, <a class="txt" href="#E4:61_2_1_1_1_3"><span class="lab"><font color="Green" title="E57">A14</font></span></a>, <a class="ref" href="compl_sp.html#D10" target="_self" title="COMPL_SP:def.10">Def10</a></span>
;<br/><span class="kw">hence </span><a NAME="E6:61_2_1_1_1_3"/>
 <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c7">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61_2_1_1_1_3"><span class="lab"><font color="Green" title="E54">A11</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="61_2_1_1_1_4"><span class="kw">suppose </span></a><a NAME="E1:61_2_1_1_1_4"/><span class="lab"><font color="Green" title="E54">A15</font></span>: 
( <font color="Maroon" title="c7">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c8">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">A</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c7">a</font></span><br/><div class="add"><a NAME="E2:61_2_1_1_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A16</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x0</font>,<font color="Maroon" title="c8">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c3">X</font>,<span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c2">A</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:61"><span class="lab"><font color="Green" title="E51">A1</font></span></a>, <a class="ref" href="compl_sp.html#T38" target="_self" title="COMPL_SP:th.38">Th38</a></span>;<br/><a NAME="E3:61_2_1_1_1_4"/><span class="lab"><font color="Green" title="E56">A17</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x0</font>,<font color="Maroon" title="c7">a</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c3">X</font>,<span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c2">A</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:61"><span class="lab"><font color="Green" title="E51">A1</font></span></a>, <a class="txt" href="#E1:61_2_1_1_1_4"><span class="lab"><font color="Green" title="E54">A15</font></span></a>, <a class="ref" href="compl_sp.html#T38" target="_self" title="COMPL_SP:th.38">Th38</a></span>;<br/><span class="kw">hence </span> <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font> = 
<span class="p1">(<span class="default"><a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">X</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x0</font>,<font color="Maroon" title="c7">a</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x0</font>,<font color="Maroon" title="c8">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:61_2_1_1_1_4"><span class="lab"><font color="Green" title="E55">A16</font></span></a>, <a class="ref" href="compl_sp.html#D10" target="_self" title="COMPL_SP:def.10">Def10</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">X</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x0</font>,<font color="Maroon" title="c8">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x0</font>,<font color="Maroon" title="c7">a</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61_2"><span class="lab"><font color="Green" title="E53">A6</font></span></a>, <a class="txt" href="#E3:61_2_1_1_1_4"><span class="lab"><font color="Green" title="E56">A17</font></span></a>, <a class="txt" href="#E2:61_2_1_1_1_4"><span class="lab"><font color="Green" title="E55">A16</font></span></a>, <a class="ref" href="metric_1.html#D5" title="METRIC_1:def.5">METRIC_1:def 5</a></span>
<br/>.= 
 <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c7">a</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:61_2_1_1_1_4"><span class="lab"><font color="Green" title="E56">A17</font></span></a>, <a class="txt" href="#E2:61_2_1_1_1_4"><span class="lab"><font color="Green" title="E55">A16</font></span></a>, <a class="ref" href="compl_sp.html#D10" target="_self" title="COMPL_SP:def.10">Def10</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="E2:61_2_1"/>
 <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c7">a</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:61_2"/>
<font color="Maroon" title="c1">M</font> is  <a href="metric_1.html#V8" title="METRIC_1:attr.8">symmetric</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#T3" title="METRIC_1:th.3">METRIC_1:3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E5:61"/>
(  <a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">X</font> is  <a href="metric_1.html#V5" title="METRIC_1:attr.5">triangle</a>  &amp;  <a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">X</font> is  <a href="metric_1.html#V2" title="METRIC_1:attr.2">Reflexive</a>  implies <font color="Maroon" title="c1">M</font> is  <a href="metric_1.html#V9" title="METRIC_1:attr.9">triangle</a>  )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">X</font> is  <a href="metric_1.html#V3" title="METRIC_1:attr.3">discerning</a>  &amp;  <a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">X</font> is  <a href="metric_1.html#V2" title="METRIC_1:attr.2">Reflexive</a>  implies <font color="Maroon" title="c1">M</font> is  <a href="metric_1.html#V7" title="METRIC_1:attr.7">discerning</a>  )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:61_3"/><span class="lab"><font color="Green" title="E53">A18</font></span>: 
(  <a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">X</font> is  <a href="metric_1.html#V5" title="METRIC_1:attr.5">triangle</a>  &amp;  <a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">X</font> is  <a href="metric_1.html#V2" title="METRIC_1:attr.2">Reflexive</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">M</font> is  <a href="metric_1.html#V9" title="METRIC_1:attr.9">triangle</a> </span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_3_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">a</font>, <font color="Maroon" title="c8">b</font>, <font color="Maroon" title="c9">c</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c9">c</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#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</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#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font></span>)</span></span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_3_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_3_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="E1:61_3_1_1_1"/>
( ( <font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c8">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c9">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> ) or ( <font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c8">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c9">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">A</font> ) or ( <font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c8">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c9">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> ) or ( <font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c8">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c9">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">A</font> ) or ( <font color="Maroon" title="c7">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c8">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c9">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> ) or ( <font color="Maroon" title="c7">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c8">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c9">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">A</font> ) or ( <font color="Maroon" title="c7">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c8">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c9">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> ) or ( <font color="Maroon" title="c7">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c8">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c9">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">A</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_3_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:61_3_1_1_1_1"/>
( <font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c8">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c9">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c9">c</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#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</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#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font></span>)</span></span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">Xa</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c7">a</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>, <font color="Maroon" title="c11">Xb</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c8">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>, <font color="Maroon" title="c12">Xc</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c9">c</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c3">X</font>,<span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c2">A</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <span class="kw">by</span> <span class="lab"><a class="ref" href="compl_sp.html#T38" target="_self" title="COMPL_SP:th.38">Th38</a></span>;<br/><a NAME="E3:61_3_1_1_1_1"/><span class="lab"><font color="Green" title="E54">A19</font></span>: 
 <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c9">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">X</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c10">Xa</font>,<font color="Maroon" title="c12">Xc</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="compl_sp.html#D10" target="_self" title="COMPL_SP:def.10">Def10</a></span>;<br/><a NAME="E4:61_3_1_1_1_1"/><span class="lab"><font color="Green" title="E55">A20</font></span>: 
 <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">X</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c10">Xa</font>,<font color="Maroon" title="c11">Xb</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="compl_sp.html#D10" target="_self" title="COMPL_SP:def.10">Def10</a></span>;<br/><a NAME="E5:61_3_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">X</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c10">Xa</font>,<font color="Maroon" title="c12">Xc</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">X</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c10">Xa</font>,<font color="Maroon" title="c11">Xb</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">X</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c11">Xb</font>,<font color="Maroon" title="c12">Xc</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61_3"><span class="lab"><font color="Green" title="E53">A18</font></span></a>, <a class="ref" href="metric_1.html#D6" title="METRIC_1:def.6">METRIC_1:def 6</a></span>;<br/><span class="kw">hence </span><a NAME="E6:61_3_1_1_1_1"/>
 <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c9">c</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#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</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#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:61_3_1_1_1_1"><span class="lab"><font color="Green" title="E54">A19</font></span></a>, <a class="txt" href="#E4:61_3_1_1_1_1"><span class="lab"><font color="Green" title="E55">A20</font></span></a>, <a class="ref" href="compl_sp.html#D10" target="_self" title="COMPL_SP:def.10">Def10</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="61_3_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:61_3_1_1_1_2"/><span class="lab"><font color="Green" title="E54">A21</font></span>: 
( <font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c8">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c9">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">A</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c9">c</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#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</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#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font></span>)</span></span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">Xa</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c7">a</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>, <font color="Maroon" title="c11">Xb</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c8">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>, <font color="Maroon" title="c12">Xc</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x0</font>,<font color="Maroon" title="c9">c</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c3">X</font>,<span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c2">A</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:61"><span class="lab"><font color="Green" title="E51">A1</font></span></a>, <a class="ref" href="compl_sp.html#T38" target="_self" title="COMPL_SP:th.38">Th38</a></span>;<br/><a NAME="E3:61_3_1_1_1_2"/>
 <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:61"><span class="lab"><font color="Green" title="E52">A2</font></span></a>, <a class="txt" href="#E1:61_3"><span class="lab"><font color="Green" title="E53">A18</font></span></a>, <a class="ref" href="metric_1.html#T1" title="METRIC_1:th.1">METRIC_1:1</a></span>;<br/><span class="kw">hence </span><a NAME="E4:61_3_1_1_1_2"/>
 <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c9">c</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#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</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#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61_3_1_1_1_2"><span class="lab"><font color="Green" title="E54">A21</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="61_3_1_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:61_3_1_1_1_3"/><span class="lab"><font color="Green" title="E54">A22</font></span>: 
( <font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c8">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c9">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c9">c</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#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</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#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font></span>)</span></span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">Xa</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c7">a</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>, <font color="Maroon" title="c11">Xb</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x0</font>,<font color="Maroon" title="c8">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>, <font color="Maroon" title="c12">Xc</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c9">c</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c3">X</font>,<span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c2">A</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:61"><span class="lab"><font color="Green" title="E51">A1</font></span></a>, <a class="ref" href="compl_sp.html#T38" target="_self" title="COMPL_SP:th.38">Th38</a></span>;<br/><a NAME="E3:61_3_1_1_1_3"/><span class="lab"><font color="Green" title="E55">A23</font></span>: 
<font color="Maroon" title="c4">x0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:61"><span class="lab"><font color="Green" title="E51">A1</font></span></a></span>;<br/><a NAME="E4:61_3_1_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A24</font></span>: 
<span class="p1">(<span class="default"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</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#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c7">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">X</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c11">Xb</font>,<font color="Maroon" title="c12">Xc</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61_3_1_1_1_3"><span class="lab"><font color="Green" title="E54">A22</font></span></a>, <a class="ref" href="compl_sp.html#D10" target="_self" title="COMPL_SP:def.10">Def10</a></span>;<br/><a NAME="E5:61_3_1_1_1_3"/><span class="lab"><font color="Green" title="E57">A25</font></span>: 
 <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:61"><span class="lab"><font color="Green" title="E52">A2</font></span></a>, <a class="txt" href="#E1:61_3"><span class="lab"><font color="Green" title="E53">A18</font></span></a>, <a class="ref" href="metric_1.html#T1" title="METRIC_1:th.1">METRIC_1:1</a></span>;<br/><a NAME="E6:61_3_1_1_1_3"/><span class="lab"><font color="Green" title="E58">A26</font></span>: 
<span class="p1">(<span class="default"><a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">X</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c10">Xa</font>,<font color="Maroon" title="c12">Xc</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">X</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c10">Xa</font>,<font color="Maroon" title="c11">Xb</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">X</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c11">Xb</font>,<font color="Maroon" title="c12">Xc</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61_3"><span class="lab"><font color="Green" title="E53">A18</font></span></a>, <a class="ref" href="metric_1.html#D6" title="METRIC_1:def.6">METRIC_1:def 6</a></span>;<br/><a NAME="E7:61_3_1_1_1_3"/>
<span class="p1">(<span class="default"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c7">a</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#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">X</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c10">Xa</font>,<font color="Maroon" title="c11">Xb</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61_3_1_1_1_3"><span class="lab"><font color="Green" title="E54">A22</font></span></a>, <a class="txt" href="#E3:61_3_1_1_1_3"><span class="lab"><font color="Green" title="E55">A23</font></span></a>, <a class="ref" href="compl_sp.html#D10" target="_self" title="COMPL_SP:def.10">Def10</a></span>;<br/><span class="kw">hence </span><a NAME="E8:61_3_1_1_1_3"/>
 <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c9">c</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#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</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#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:61_3_1_1_1_3"><span class="lab"><font color="Green" title="E58">A26</font></span></a>, <a class="txt" href="#E4:61_3_1_1_1_3"><span class="lab"><font color="Green" title="E56">A24</font></span></a>, <a class="txt" href="#E5:61_3_1_1_1_3"><span class="lab"><font color="Green" title="E57">A25</font></span></a>, <a class="ref" href="compl_sp.html#D10" target="_self" title="COMPL_SP:def.10">Def10</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="61_3_1_1_1_4"><span class="kw">suppose </span></a><a NAME="E1:61_3_1_1_1_4"/><span class="lab"><font color="Green" title="E54">A27</font></span>: 
( <font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c8">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c9">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">A</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c9">c</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#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</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#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font></span>)</span></span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">Xa</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c7">a</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>, <font color="Maroon" title="c11">Xb</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x0</font>,<font color="Maroon" title="c8">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>, <font color="Maroon" title="c12">Xc</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x0</font>,<font color="Maroon" title="c9">c</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c3">X</font>,<span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c2">A</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:61"><span class="lab"><font color="Green" title="E51">A1</font></span></a>, <a class="ref" href="compl_sp.html#T38" target="_self" title="COMPL_SP:th.38">Th38</a></span>;<br/><a NAME="E3:61_3_1_1_1_4"/><span class="lab"><font color="Green" title="E55">A28</font></span>: 
<font color="Maroon" title="c4">x0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:61"><span class="lab"><font color="Green" title="E51">A1</font></span></a></span>;<br/><a NAME="E4:61_3_1_1_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A29</font></span>: 
<span class="p1">(<span class="default"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c7">a</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#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">X</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c10">Xa</font>,<font color="Maroon" title="c11">Xb</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61_3_1_1_1_4"><span class="lab"><font color="Green" title="E54">A27</font></span></a>, <a class="ref" href="compl_sp.html#D10" target="_self" title="COMPL_SP:def.10">Def10</a></span>;<br/><a NAME="E5:61_3_1_1_1_4"/><span class="lab"><font color="Green" title="E57">A30</font></span>: 
 <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:61"><span class="lab"><font color="Green" title="E52">A2</font></span></a>, <a class="txt" href="#E1:61_3"><span class="lab"><font color="Green" title="E53">A18</font></span></a>, <a class="ref" href="metric_1.html#T1" title="METRIC_1:th.1">METRIC_1:1</a></span>;<br/><a NAME="E6:61_3_1_1_1_4"/><span class="lab"><font color="Green" title="E58">A31</font></span>: 
<span class="p1">(<span class="default"><a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">X</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c10">Xa</font>,<font color="Maroon" title="c12">Xc</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">X</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c10">Xa</font>,<font color="Maroon" title="c11">Xb</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">X</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c11">Xb</font>,<font color="Maroon" title="c12">Xc</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61_3"><span class="lab"><font color="Green" title="E53">A18</font></span></a>, <a class="ref" href="metric_1.html#D6" title="METRIC_1:def.6">METRIC_1:def 6</a></span>;<br/><a NAME="E7:61_3_1_1_1_4"/>
<span class="p1">(<span class="default"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c7">a</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#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c9">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">X</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c10">Xa</font>,<font color="Maroon" title="c12">Xc</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61_3_1_1_1_4"><span class="lab"><font color="Green" title="E54">A27</font></span></a>, <a class="txt" href="#E3:61_3_1_1_1_4"><span class="lab"><font color="Green" title="E55">A28</font></span></a>, <a class="ref" href="compl_sp.html#D10" target="_self" title="COMPL_SP:def.10">Def10</a></span>;<br/><span class="kw">hence </span><a NAME="E8:61_3_1_1_1_4"/>
 <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c9">c</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#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</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#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:61_3_1_1_1_4"><span class="lab"><font color="Green" title="E58">A31</font></span></a>, <a class="txt" href="#E4:61_3_1_1_1_4"><span class="lab"><font color="Green" title="E56">A29</font></span></a>, <a class="txt" href="#E5:61_3_1_1_1_4"><span class="lab"><font color="Green" title="E57">A30</font></span></a>, <a class="ref" href="compl_sp.html#D10" target="_self" title="COMPL_SP:def.10">Def10</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="61_3_1_1_1_5"><span class="kw">suppose </span></a><a NAME="E1:61_3_1_1_1_5"/><span class="lab"><font color="Green" title="E54">A32</font></span>: 
( <font color="Maroon" title="c7">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c8">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c9">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c9">c</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#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</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#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font></span>)</span></span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">Xa</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x0</font>,<font color="Maroon" title="c7">a</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>, <font color="Maroon" title="c11">Xb</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c8">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>, <font color="Maroon" title="c12">Xc</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c9">c</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c3">X</font>,<span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c2">A</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:61"><span class="lab"><font color="Green" title="E51">A1</font></span></a>, <a class="ref" href="compl_sp.html#T38" target="_self" title="COMPL_SP:th.38">Th38</a></span>;<br/><a NAME="E3:61_3_1_1_1_5"/>
 <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c9">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:61"><span class="lab"><font color="Green" title="E52">A2</font></span></a>, <a class="txt" href="#E1:61_3"><span class="lab"><font color="Green" title="E53">A18</font></span></a>, <a class="ref" href="metric_1.html#T1" title="METRIC_1:th.1">METRIC_1:1</a></span>;<br/><span class="kw">hence </span><a NAME="E4:61_3_1_1_1_5"/>
 <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c9">c</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#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</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#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61_3_1_1_1_5"><span class="lab"><font color="Green" title="E54">A32</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="61_3_1_1_1_6"><span class="kw">suppose </span></a><a NAME="E1:61_3_1_1_1_6"/><span class="lab"><font color="Green" title="E54">A33</font></span>: 
( <font color="Maroon" title="c7">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c8">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c9">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">A</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c9">c</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#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</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#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font></span>)</span></span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">Xa</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x0</font>,<font color="Maroon" title="c7">a</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>, <font color="Maroon" title="c11">Xb</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c8">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>, <font color="Maroon" title="c12">Xc</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x0</font>,<font color="Maroon" title="c9">c</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c3">X</font>,<span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c2">A</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:61"><span class="lab"><font color="Green" title="E51">A1</font></span></a>, <a class="ref" href="compl_sp.html#T38" target="_self" title="COMPL_SP:th.38">Th38</a></span>;<br/><a NAME="E3:61_3_1_1_1_6"/><span class="lab"><font color="Green" title="E55">A34</font></span>: 
<font color="Maroon" title="c4">x0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:61"><span class="lab"><font color="Green" title="E51">A1</font></span></a></span>;<br/><a NAME="E4:61_3_1_1_1_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A35</font></span>: 
<span class="p1">(<span class="default"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c8">b</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#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">X</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c11">Xb</font>,<font color="Maroon" title="c12">Xc</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61_3_1_1_1_6"><span class="lab"><font color="Green" title="E54">A33</font></span></a>, <a class="ref" href="compl_sp.html#D10" target="_self" title="COMPL_SP:def.10">Def10</a></span>;<br/><a NAME="E5:61_3_1_1_1_6"/><span class="lab"><font color="Green" title="E57">A36</font></span>: 
 <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c8">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:61"><span class="lab"><font color="Green" title="E52">A2</font></span></a>, <a class="txt" href="#E1:61_3"><span class="lab"><font color="Green" title="E53">A18</font></span></a>, <a class="ref" href="metric_1.html#T1" title="METRIC_1:th.1">METRIC_1:1</a></span>;<br/><a NAME="E6:61_3_1_1_1_6"/><span class="lab"><font color="Green" title="E58">A37</font></span>: 
<span class="p1">(<span class="default"><a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">X</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c10">Xa</font>,<font color="Maroon" title="c12">Xc</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">X</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c10">Xa</font>,<font color="Maroon" title="c11">Xb</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">X</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c11">Xb</font>,<font color="Maroon" title="c12">Xc</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61_3"><span class="lab"><font color="Green" title="E53">A18</font></span></a>, <a class="ref" href="metric_1.html#D6" title="METRIC_1:def.6">METRIC_1:def 6</a></span>;<br/><a NAME="E7:61_3_1_1_1_6"/>
<span class="p1">(<span class="default"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</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#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c8">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">X</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c10">Xa</font>,<font color="Maroon" title="c11">Xb</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61_3_1_1_1_6"><span class="lab"><font color="Green" title="E54">A33</font></span></a>, <a class="txt" href="#E3:61_3_1_1_1_6"><span class="lab"><font color="Green" title="E55">A34</font></span></a>, <a class="ref" href="compl_sp.html#D10" target="_self" title="COMPL_SP:def.10">Def10</a></span>;<br/><span class="kw">hence </span><a NAME="E8:61_3_1_1_1_6"/>
 <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c9">c</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#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</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#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:61_3_1_1_1_6"><span class="lab"><font color="Green" title="E58">A37</font></span></a>, <a class="txt" href="#E4:61_3_1_1_1_6"><span class="lab"><font color="Green" title="E56">A35</font></span></a>, <a class="txt" href="#E5:61_3_1_1_1_6"><span class="lab"><font color="Green" title="E57">A36</font></span></a>, <a class="ref" href="compl_sp.html#D10" target="_self" title="COMPL_SP:def.10">Def10</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="61_3_1_1_1_7"><span class="kw">suppose </span></a><a NAME="E1:61_3_1_1_1_7"/><span class="lab"><font color="Green" title="E54">A38</font></span>: 
( <font color="Maroon" title="c7">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c8">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c9">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c9">c</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#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</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#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font></span>)</span></span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">Xa</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x0</font>,<font color="Maroon" title="c7">a</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>, <font color="Maroon" title="c11">Xb</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x0</font>,<font color="Maroon" title="c8">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>, <font color="Maroon" title="c12">Xc</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c9">c</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c3">X</font>,<span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c2">A</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:61"><span class="lab"><font color="Green" title="E51">A1</font></span></a>, <a class="ref" href="compl_sp.html#T38" target="_self" title="COMPL_SP:th.38">Th38</a></span>;<br/><a NAME="E3:61_3_1_1_1_7"/><span class="lab"><font color="Green" title="E55">A39</font></span>: 
<font color="Maroon" title="c4">x0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:61"><span class="lab"><font color="Green" title="E51">A1</font></span></a></span>;<br/><a NAME="E4:61_3_1_1_1_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A40</font></span>: 
<span class="p1">(<span class="default"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</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#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c9">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">X</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c11">Xb</font>,<font color="Maroon" title="c12">Xc</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61_3_1_1_1_7"><span class="lab"><font color="Green" title="E54">A38</font></span></a>, <a class="ref" href="compl_sp.html#D10" target="_self" title="COMPL_SP:def.10">Def10</a></span>;<br/><a NAME="E5:61_3_1_1_1_7"/><span class="lab"><font color="Green" title="E57">A41</font></span>: 
 <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c9">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:61"><span class="lab"><font color="Green" title="E52">A2</font></span></a>, <a class="txt" href="#E1:61_3"><span class="lab"><font color="Green" title="E53">A18</font></span></a>, <a class="ref" href="metric_1.html#T1" title="METRIC_1:th.1">METRIC_1:1</a></span>;<br/><a NAME="E6:61_3_1_1_1_7"/><span class="lab"><font color="Green" title="E58">A42</font></span>: 
<span class="p1">(<span class="default"><a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">X</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c10">Xa</font>,<font color="Maroon" title="c12">Xc</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">X</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c10">Xa</font>,<font color="Maroon" title="c11">Xb</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">X</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c11">Xb</font>,<font color="Maroon" title="c12">Xc</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61_3"><span class="lab"><font color="Green" title="E53">A18</font></span></a>, <a class="ref" href="metric_1.html#D6" title="METRIC_1:def.6">METRIC_1:def 6</a></span>;<br/><a NAME="E7:61_3_1_1_1_7"/>
<span class="p1">(<span class="default"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c9">c</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#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c9">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">X</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c10">Xa</font>,<font color="Maroon" title="c12">Xc</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61_3_1_1_1_7"><span class="lab"><font color="Green" title="E54">A38</font></span></a>, <a class="txt" href="#E3:61_3_1_1_1_7"><span class="lab"><font color="Green" title="E55">A39</font></span></a>, <a class="ref" href="compl_sp.html#D10" target="_self" title="COMPL_SP:def.10">Def10</a></span>;<br/><span class="kw">hence </span><a NAME="E8:61_3_1_1_1_7"/>
 <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c9">c</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#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</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#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:61_3_1_1_1_7"><span class="lab"><font color="Green" title="E58">A42</font></span></a>, <a class="txt" href="#E4:61_3_1_1_1_7"><span class="lab"><font color="Green" title="E56">A40</font></span></a>, <a class="txt" href="#E5:61_3_1_1_1_7"><span class="lab"><font color="Green" title="E57">A41</font></span></a>, <a class="ref" href="compl_sp.html#D10" target="_self" title="COMPL_SP:def.10">Def10</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="61_3_1_1_1_8"><span class="kw">suppose </span></a><a NAME="E1:61_3_1_1_1_8"/>
( <font color="Maroon" title="c7">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c8">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c9">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">A</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c9">c</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#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</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#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font></span>)</span></span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">Xa</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x0</font>,<font color="Maroon" title="c7">a</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>, <font color="Maroon" title="c11">Xb</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x0</font>,<font color="Maroon" title="c8">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>, <font color="Maroon" title="c12">Xc</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x0</font>,<font color="Maroon" title="c9">c</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c3">X</font>,<span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c2">A</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:61"><span class="lab"><font color="Green" title="E51">A1</font></span></a>, <a class="ref" href="compl_sp.html#T38" target="_self" title="COMPL_SP:th.38">Th38</a></span>;<br/><a NAME="E3:61_3_1_1_1_8"/><span class="lab"><font color="Green" title="E54">A43</font></span>: 
 <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c9">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">X</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c10">Xa</font>,<font color="Maroon" title="c12">Xc</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="compl_sp.html#D10" target="_self" title="COMPL_SP:def.10">Def10</a></span>;<br/><a NAME="E4:61_3_1_1_1_8"/><span class="lab"><font color="Green" title="E55">A44</font></span>: 
 <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">X</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c10">Xa</font>,<font color="Maroon" title="c11">Xb</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="compl_sp.html#D10" target="_self" title="COMPL_SP:def.10">Def10</a></span>;<br/><a NAME="E5:61_3_1_1_1_8"/>
<span class="p1">(<span class="default"><a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">X</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c10">Xa</font>,<font color="Maroon" title="c12">Xc</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">X</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c10">Xa</font>,<font color="Maroon" title="c11">Xb</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">X</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c11">Xb</font>,<font color="Maroon" title="c12">Xc</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61_3"><span class="lab"><font color="Green" title="E53">A18</font></span></a>, <a class="ref" href="metric_1.html#D6" title="METRIC_1:def.6">METRIC_1:def 6</a></span>;<br/><span class="kw">hence </span><a NAME="E6:61_3_1_1_1_8"/>
 <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c9">c</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#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</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#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:61_3_1_1_1_8"><span class="lab"><font color="Green" title="E54">A43</font></span></a>, <a class="txt" href="#E4:61_3_1_1_1_8"><span class="lab"><font color="Green" title="E55">A44</font></span></a>, <a class="ref" href="compl_sp.html#D10" target="_self" title="COMPL_SP:def.10">Def10</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="E2:61_3_1"/>
 <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c9">c</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#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</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#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:61_3"/>
<font color="Maroon" title="c1">M</font> is  <a href="metric_1.html#V9" title="METRIC_1:attr.9">triangle</a> 
 <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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E6:61"/><span class="lab"><font color="Green" title="E53">A45</font></span>: 
(  <a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">X</font> is  <a href="metric_1.html#V3" title="METRIC_1:attr.3">discerning</a>  &amp;  <a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">X</font> is  <a href="metric_1.html#V2" title="METRIC_1:attr.2">Reflexive</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">M</font> is  <a href="metric_1.html#V7" title="METRIC_1:attr.7">discerning</a> </span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_4"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">a</font>, <font color="Maroon" title="c8">b</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b</font> )</span><br/><span class="kw">assume </span><a NAME="E1:61_4"/><span class="lab"><font color="Green" title="E54">A46</font></span>: 
 <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b</font></span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_4_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_4_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="E1:61_4_1_1"/>
( ( <font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c8">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> ) or ( <font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c8">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">A</font> ) or ( <font color="Maroon" title="c7">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c8">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> ) or ( <font color="Maroon" title="c7">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c8">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">A</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_4_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:61_4_1_1_1"/>
( <font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c8">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:61_4_1_1_1"/>
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="61_4_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:61_4_1_1_2"/><span class="lab"><font color="Green" title="E55">A47</font></span>: 
( <font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c8">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">A</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">Xa</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c7">a</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>, <font color="Maroon" title="c10">Xb</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x0</font>,<font color="Maroon" title="c8">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c3">X</font>,<span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c2">A</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:61"><span class="lab"><font color="Green" title="E51">A1</font></span></a>, <a class="ref" href="compl_sp.html#T38" target="_self" title="COMPL_SP:th.38">Th38</a></span>;<br/><a NAME="E3:61_4_1_1_2"/>
<font color="Maroon" title="c4">x0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:61"><span class="lab"><font color="Green" title="E51">A1</font></span></a></span>;<br/><a NAME="E4:61_4_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A48</font></span>: 
<span class="p1">(<span class="default"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c7">a</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#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">X</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c9">Xa</font>,<font color="Maroon" title="c10">Xb</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61_4_1_1_2"><span class="lab"><font color="Green" title="E55">A47</font></span></a>, <a class="ref" href="compl_sp.html#D10" target="_self" title="COMPL_SP:def.10">Def10</a></span>;<br/><a NAME="E5:61_4_1_1_2"/>
 <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:61"><span class="lab"><font color="Green" title="E52">A2</font></span></a>, <a class="txt" href="#E6:61"><span class="lab"><font color="Green" title="E53">A45</font></span></a>, <a class="ref" href="metric_1.html#T1" title="METRIC_1:th.1">METRIC_1:1</a></span>;<br/><a NAME="E6:61_4_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c9">Xa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Xb</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:61"><span class="lab"><font color="Green" title="E53">A45</font></span></a>, <a class="txt" href="#E1:61_4"><span class="lab"><font color="Green" title="E54">A46</font></span></a>, <a class="txt" href="#E4:61_4_1_1_2"><span class="lab"><font color="Green" title="E56">A48</font></span></a>, <a class="ref" href="metric_1.html#D4" title="METRIC_1:def.4">METRIC_1:def 4</a></span>;<br/><span class="kw">hence </span><a NAME="E7:61_4_1_1_2"/>
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="61_4_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:61_4_1_1_3"/><span class="lab"><font color="Green" title="E55">A49</font></span>: 
( <font color="Maroon" title="c7">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c8">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">Xa</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x0</font>,<font color="Maroon" title="c7">a</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>, <font color="Maroon" title="c10">Xb</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c8">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c3">X</font>,<span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c2">A</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:61"><span class="lab"><font color="Green" title="E51">A1</font></span></a>, <a class="ref" href="compl_sp.html#T38" target="_self" title="COMPL_SP:th.38">Th38</a></span>;<br/><a NAME="E3:61_4_1_1_3"/>
<font color="Maroon" title="c4">x0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:61"><span class="lab"><font color="Green" title="E51">A1</font></span></a></span>;<br/><a NAME="E4:61_4_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A50</font></span>: 
<span class="p1">(<span class="default"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</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#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c8">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">X</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c9">Xa</font>,<font color="Maroon" title="c10">Xb</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61_4_1_1_3"><span class="lab"><font color="Green" title="E55">A49</font></span></a>, <a class="ref" href="compl_sp.html#D10" target="_self" title="COMPL_SP:def.10">Def10</a></span>;<br/><a NAME="E5:61_4_1_1_3"/>
 <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c8">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:61"><span class="lab"><font color="Green" title="E52">A2</font></span></a>, <a class="txt" href="#E6:61"><span class="lab"><font color="Green" title="E53">A45</font></span></a>, <a class="ref" href="metric_1.html#T1" title="METRIC_1:th.1">METRIC_1:1</a></span>;<br/><a NAME="E6:61_4_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c9">Xa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Xb</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:61"><span class="lab"><font color="Green" title="E53">A45</font></span></a>, <a class="txt" href="#E1:61_4"><span class="lab"><font color="Green" title="E54">A46</font></span></a>, <a class="txt" href="#E4:61_4_1_1_3"><span class="lab"><font color="Green" title="E56">A50</font></span></a>, <a class="ref" href="metric_1.html#D4" title="METRIC_1:def.4">METRIC_1:def 4</a></span>;<br/><span class="kw">hence </span><a NAME="E7:61_4_1_1_3"/>
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="61_4_1_1_4"><span class="kw">suppose </span></a><a NAME="E1:61_4_1_1_4"/>
( <font color="Maroon" title="c7">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c8">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">A</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">Xa</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x0</font>,<font color="Maroon" title="c7">a</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>, <font color="Maroon" title="c10">Xb</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x0</font>,<font color="Maroon" title="c8">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c3">X</font>,<span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c2">A</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:61"><span class="lab"><font color="Green" title="E51">A1</font></span></a>, <a class="ref" href="compl_sp.html#T38" target="_self" title="COMPL_SP:th.38">Th38</a></span>;<br/><a NAME="E3:61_4_1_1_4"/>
 <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">X</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c9">Xa</font>,<font color="Maroon" title="c10">Xb</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="compl_sp.html#D10" target="_self" title="COMPL_SP:def.10">Def10</a></span>;<br/><a NAME="E4:61_4_1_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c9">Xa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Xb</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:61"><span class="lab"><font color="Green" title="E53">A45</font></span></a>, <a class="txt" href="#E1:61_4"><span class="lab"><font color="Green" title="E54">A46</font></span></a>, <a class="ref" href="metric_1.html#D4" title="METRIC_1:def.4">METRIC_1:def 4</a></span>;<br/><span class="kw">hence </span><a NAME="E5:61_4_1_1_4"/>
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="E3:61_4"/>
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E8:61"/>
<font color="Maroon" title="c1">M</font> is  <a href="metric_1.html#V7" title="METRIC_1:attr.7">discerning</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#T2" title="METRIC_1:th.2">METRIC_1:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
