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<span class="kw">take </span>
2
; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tbsp_1.html#D6" target="_self" onmouseover="rs('tbsp_1/D6')" onmouseout="rh()">TBSP_1:def 6</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"> (  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 &amp; (  for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="metric_1.html#K5" title="METRIC_1:func.5">DiscreteSpace</a> <font color="Maroon" title="c1">A</font></span>)</span> holds   <a href="metric_1.html#K1" title="METRIC_1:func.1">dist</a> (<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 ) )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c2">N</font> =  <a href="metric_1.html#K5" title="METRIC_1:func.5">DiscreteSpace</a> <font color="Maroon" title="c1">A</font>;<br/>
<span class="kw">thus </span><a NAME="E1:18_1_1"/>
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2
 ; <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>, <font color="Olive" title="b2">y</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="metric_1.html#K5" title="METRIC_1:func.5">DiscreteSpace</a> <font color="Maroon" title="c1">A</font></span>)</span> holds   <a href="metric_1.html#K1" title="METRIC_1:func.1">dist</a> (<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2</span><br/>

<span class="kw">let </span><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="metric_1.html#K5" title="METRIC_1:func.5">DiscreteSpace</a> <font color="Maroon" title="c1">A</font></span>)</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">  <a href="metric_1.html#K1" title="METRIC_1:func.1">dist</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2</span><br/>

<a NAME="E2:18_1_1"/><span class="lab"><font color="Green" title="E7">A1</font></span>: 
(  <a href="metric_1.html#K5" title="METRIC_1:func.5">DiscreteSpace</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="metric_1.html#G1" title="METRIC_1:aggr.1">MetrStruct</a>(# <font color="Maroon" title="c1">A</font>,<span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">discrete_dist</a> <font color="Maroon" title="c1">A</font></span>)</span> #) &amp;  <a href="metric_1.html#K3" title="METRIC_1:func.3">dist</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">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 <span class="p1">(<span class="default"><a href="metric_1.html#K5" title="METRIC_1:func.5">DiscreteSpace</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">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>, <a class="ref" href="metric_1.html#D11" onmouseover="rs('metric_1/D11')" onmouseout="rh()">METRIC_1:def 11</a></span>;<br/>
<a NAME="E3:18_1_1"/>
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y</font> or <font color="Maroon" title="c3">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">y</font> )
 
;<br/>
<a NAME="E4:18_1_1"/><span class="kw">then </span>
(  <a href="metric_1.html#K3" title="METRIC_1:func.3">dist</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or  <a href="metric_1.html#K3" title="METRIC_1:func.3">dist</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:18_1_1"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="ref" href="metric_1.html#D10" onmouseover="rs('metric_1/D10')" onmouseout="rh()">METRIC_1:def 10</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:18_1_1"/>
 <a href="metric_1.html#K3" title="METRIC_1:func.3">dist</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2
 ; <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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