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<span class="kw">let </span><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">b</font>, <font color="Maroon" title="c3">d</font>, <font color="Maroon" title="c4">e</font>, <font color="Maroon" title="c5">r3</font> be   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">PM</font>, <font color="Olive" title="b2">PM2</font> being   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="metric_1.html#L1" title="METRIC_1:struct.1">MetrStruct</a> <br/>  for <font color="Olive" title="b3">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b1">PM</font><br/>  for <font color="Olive" title="b4">x2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b2">PM2</font>  st <font color="Maroon" title="c3">d</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">a</font> &amp; <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font> &amp; <font color="Maroon" title="c2">b</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">e</font> &amp; <font color="Olive" title="b1">PM</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topmetr.html#K2" title="TOPMETR:func.2">Closed-Interval-MSpace</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>) &amp; <font color="Olive" title="b2">PM2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topmetr.html#K2" title="TOPMETR:func.2">Closed-Interval-MSpace</a> (<font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">e</font>) &amp; <font color="Olive" title="b3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">x2</font> holds <br/> <a href="metric_1.html#K8" title="METRIC_1:func.8">Ball</a> (<font color="Olive" title="b3">x</font>,<font color="Maroon" title="c5">r3</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="metric_1.html#K8" title="METRIC_1:func.8">Ball</a> (<font color="Olive" title="b4">x2</font>,<font color="Maroon" title="c5">r3</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c6">PM</font>, <font color="Maroon" title="c7">PM2</font> be   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="metric_1.html#L1" title="METRIC_1:struct.1">MetrStruct</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c6">PM</font><br/>  for <font color="Olive" title="b2">x2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">PM2</font>  st <font color="Maroon" title="c3">d</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">a</font> &amp; <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font> &amp; <font color="Maroon" title="c2">b</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">e</font> &amp; <font color="Maroon" title="c6">PM</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topmetr.html#K2" title="TOPMETR:func.2">Closed-Interval-MSpace</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>) &amp; <font color="Maroon" title="c7">PM2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topmetr.html#K2" title="TOPMETR:func.2">Closed-Interval-MSpace</a> (<font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">e</font>) &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">x2</font> holds <br/> <a href="metric_1.html#K8" title="METRIC_1:func.8">Ball</a> (<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c5">r3</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="metric_1.html#K8" title="METRIC_1:func.8">Ball</a> (<font color="Olive" title="b2">x2</font>,<font color="Maroon" title="c5">r3</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c8">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c6">PM</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">PM2</font>  st <font color="Maroon" title="c3">d</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">a</font> &amp; <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font> &amp; <font color="Maroon" title="c2">b</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">e</font> &amp; <font color="Maroon" title="c6">PM</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topmetr.html#K2" title="TOPMETR:func.2">Closed-Interval-MSpace</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>) &amp; <font color="Maroon" title="c7">PM2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topmetr.html#K2" title="TOPMETR:func.2">Closed-Interval-MSpace</a> (<font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">e</font>) &amp; <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x2</font> holds <br/> <a href="metric_1.html#K8" title="METRIC_1:func.8">Ball</a> (<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c5">r3</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="metric_1.html#K8" title="METRIC_1:func.8">Ball</a> (<font color="Olive" title="b1">x2</font>,<font color="Maroon" title="c5">r3</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c9">x2</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">PM2</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">d</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">a</font> &amp; <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font> &amp; <font color="Maroon" title="c2">b</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">e</font> &amp; <font color="Maroon" title="c6">PM</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topmetr.html#K2" title="TOPMETR:func.2">Closed-Interval-MSpace</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>) &amp; <font color="Maroon" title="c7">PM2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topmetr.html#K2" title="TOPMETR:func.2">Closed-Interval-MSpace</a> (<font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">e</font>) &amp; <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x2</font> implies  <a href="metric_1.html#K8" title="METRIC_1:func.8">Ball</a> (<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c5">r3</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="metric_1.html#K8" title="METRIC_1:func.8">Ball</a> (<font color="Maroon" title="c9">x2</font>,<font color="Maroon" title="c5">r3</font>) )</span><br/>







<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:3"/><span class="lab"><font color="Green" title="E3">A1</font></span>: 
<font color="Maroon" title="c3">d</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">a</font>
 <span class="kw">and </span><br/><a NAME="E2:3"/><span class="lab"><font color="Green" title="E4">A2</font></span>: 
<font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font>
 <span class="kw">and </span><br/><a NAME="E3:3"/><span class="lab"><font color="Green" title="E5">A3</font></span>: 
<font color="Maroon" title="c2">b</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">e</font>
 <span class="kw">and </span><br/><a NAME="E4:3"/><span class="lab"><font color="Green" title="E6">A4</font></span>: 
<font color="Maroon" title="c6">PM</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topmetr.html#K2" title="TOPMETR:func.2">Closed-Interval-MSpace</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>)
 <span class="kw">and </span><br/><a NAME="E5:3"/><span class="lab"><font color="Green" title="E7">A5</font></span>: 
<font color="Maroon" title="c7">PM2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topmetr.html#K2" title="TOPMETR:func.2">Closed-Interval-MSpace</a> (<font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">e</font>)
 <span class="kw">and </span><br/><a NAME="E6:3"/><span class="lab"><font color="Green" title="E8">A6</font></span>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x2</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="metric_1.html#K8" title="METRIC_1:func.8">Ball</a> (<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c5">r3</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="metric_1.html#K8" title="METRIC_1:func.8">Ball</a> (<font color="Maroon" title="c9">x2</font>,<font color="Maroon" title="c5">r3</font>)</span><br/>

<a NAME="E7:3"/>
<font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">e</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:3"><span class="lab"><font color="Green" title="E4">A2</font></span></a>, <a class="txt" href="#E3:3"><span class="lab"><font color="Green" title="E5">A3</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/>
<a NAME="E8:3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E9">A7</font></span>: 
<font color="Maroon" title="c1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">e</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:3"><span class="lab"><font color="Green" title="E3">A1</font></span></a>, <a class="ref" href="xxreal_1.html#T1" onmouseover="rs('xxreal_1/T1')" onmouseout="rh()">XXREAL_1:1</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c10">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c10">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="metric_1.html#K8" title="METRIC_1:func.8">Ball</a> (<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c5">r3</font>) or <font color="Maroon" title="c10">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="metric_1.html#K8" title="METRIC_1:func.8">Ball</a> (<font color="Maroon" title="c9">x2</font>,<font color="Maroon" title="c5">r3</font>) )</span><br/>

<span class="kw">assume </span><a NAME="E9:3"/>
<font color="Maroon" title="c10">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="metric_1.html#K8" title="METRIC_1:func.8">Ball</a> (<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c5">r3</font>)
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c10">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="metric_1.html#K8" title="METRIC_1:func.8">Ball</a> (<font color="Maroon" title="c9">x2</font>,<font color="Maroon" title="c5">r3</font>)</span><br/>

<a NAME="E10:3"/><span class="kw">then </span>
<font color="Maroon" title="c10">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">y</font> where <font color="Olive" title="b1">y</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c6">PM</font> :  <a href="metric_1.html#K1" title="METRIC_1:func.1">dist</a> (<font color="Maroon" title="c8">x</font>,<font color="Olive" title="b1">y</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">r3</font> </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#T17" onmouseover="rs('metric_1/T17')" onmouseout="rh()">METRIC_1:17</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c6">PM</font><span class="kw"> such that </span><br/><a NAME="E12:3"/><span class="lab"><font color="Green" title="E10">A8</font></span>: 
( <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">z</font> &amp;  <a href="metric_1.html#K1" title="METRIC_1:func.1">dist</a> (<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c11">y</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">r3</font> )
 ;<br/>
<a NAME="E13:3"/>
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">PM</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:3"><span class="lab"><font color="Green" title="E4">A2</font></span></a>, <a class="txt" href="#E4:3"><span class="lab"><font color="Green" title="E6">A4</font></span></a>, <a class="ref" href="topmetr.html#T10" onmouseover="rs('topmetr/T10')" onmouseout="rh()">TOPMETR:10</a></span>;<br/>
<a NAME="E14:3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E11">A9</font></span>: 
<font color="Maroon" title="c11">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
;<br/>
<a NAME="E15:3"/><span class="lab"><font color="Green" title="E12">A10</font></span>: 
<font color="Maroon" title="c3">d</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:3"><span class="lab"><font color="Green" title="E3">A1</font></span></a>, <a class="txt" href="#E2:3"><span class="lab"><font color="Green" title="E4">A2</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/>
<a NAME="E16:3"/><span class="kw">then </span>
<font color="Maroon" title="c2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">e</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:3"><span class="lab"><font color="Green" title="E5">A3</font></span></a>, <a class="ref" href="xxreal_1.html#T1" onmouseover="rs('xxreal_1/T1')" onmouseout="rh()">XXREAL_1:1</a></span>;<br/>
<a NAME="E17:3"/><span class="kw">then </span>
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">e</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:3"><span class="lab"><font color="Green" title="E9">A7</font></span></a>, <a class="ref" href="xxreal_2.html#D12" onmouseover="rs('xxreal_2/D12')" onmouseout="rh()">XXREAL_2:def 12</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">y3</font> = <font color="Maroon" title="c11">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">PM2</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:3"><span class="lab"><font color="Green" title="E5">A3</font></span></a>, <a class="txt" href="#E5:3"><span class="lab"><font color="Green" title="E7">A5</font></span></a>, <a class="txt" href="#E15:3"><span class="lab"><font color="Green" title="E12">A10</font></span></a>, <a class="txt" href="#E14:3"><span class="lab"><font color="Green" title="E11">A9</font></span></a>, <a class="ref" href="topmetr.html#T10" onmouseover="rs('topmetr/T10')" onmouseout="rh()">TOPMETR:10</a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/>
<a NAME="E19:3"/><span class="lab"><font color="Green" title="E13">A11</font></span>: 
 <a href="metric_1.html#K1" title="METRIC_1:func.1">dist</a> (<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c11">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <font color="Maroon" title="c6">PM</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c11">y</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#D1" onmouseover="rs('metric_1/D1')" onmouseout="rh()">METRIC_1:def 1</a></span>;<br/>
<a NAME="E20:3"/><span class="lab"><font color="Green" title="E14">A12</font></span>: 
 the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <font color="Maroon" title="c6">PM</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c11">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="metric_1.html#K6" title="METRIC_1:func.6">real_dist</a> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c11">y</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:3"><span class="lab"><font color="Green" title="E6">A4</font></span></a>, <a class="ref" href="metric_1.html#D13" onmouseover="rs('metric_1/D13')" onmouseout="rh()">METRIC_1:def 13</a>, <a class="ref" href="topmetr.html#D1" onmouseover="rs('topmetr/D1')" onmouseout="rh()">TOPMETR:def 1</a></span>;<br/>
<a href="metric_1.html#K6" title="METRIC_1:func.6">real_dist</a> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c11">y</font>) = 
 the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <font color="Maroon" title="c7">PM2</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c9">x2</font>,<font color="Maroon" title="c12">y3</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:3"><span class="lab"><font color="Green" title="E7">A5</font></span></a>, <a class="txt" href="#E6:3"><span class="lab"><font color="Green" title="E8">A6</font></span></a>, <a class="ref" href="metric_1.html#D13" onmouseover="rs('metric_1/D13')" onmouseout="rh()">METRIC_1:def 13</a>, <a class="ref" href="topmetr.html#D1" onmouseover="rs('topmetr/D1')" onmouseout="rh()">TOPMETR:def 1</a></span>
<br/>.= 
 <a href="metric_1.html#K1" title="METRIC_1:func.1">dist</a> (<font color="Maroon" title="c9">x2</font>,<font color="Maroon" title="c12">y3</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#D1" onmouseover="rs('metric_1/D1')" onmouseout="rh()">METRIC_1:def 1</a></span>
;<br/>
<a NAME="E22:3"/><span class="kw">then </span>
<font color="Maroon" title="c10">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">y2</font> where <font color="Olive" title="b1">y2</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">PM2</font> :  <a href="metric_1.html#K1" title="METRIC_1:func.1">dist</a> (<font color="Maroon" title="c9">x2</font>,<font color="Olive" title="b1">y2</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">r3</font> </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:3"><span class="lab"><font color="Green" title="E10">A8</font></span></a>, <a class="txt" href="#E20:3"><span class="lab"><font color="Green" title="E14">A12</font></span></a>, <a class="txt" href="#E19:3"><span class="lab"><font color="Green" title="E13">A11</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E23:3"/>
<font color="Maroon" title="c10">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="metric_1.html#K8" title="METRIC_1:func.8">Ball</a> (<font color="Maroon" title="c9">x2</font>,<font color="Maroon" title="c5">r3</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#T17" onmouseover="rs('metric_1/T17')" onmouseout="rh()">METRIC_1:17</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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