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<span class="kw">let </span><font color="Maroon" title="c1">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> holds <br/> ( <font color="Olive" title="b1">f</font> <a href="nagata_1.html#R1" title="NAGATA_1:pred.1">is_a_pseudometric_of</a> <font color="Maroon" title="c1">X</font> iff  for <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">b</font>, <font color="Olive" title="b4">c</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> holds <br/> ( <font color="Olive" title="b1">f</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> (<font color="Olive" title="b2">a</font>,<font color="Olive" title="b2">a</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b1">f</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> (<font color="Olive" title="b2">a</font>,<font color="Olive" title="b3">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> (<font color="Olive" title="b3">b</font>,<font color="Olive" title="b2">a</font>) &amp; <font color="Olive" title="b1">f</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> (<font color="Olive" title="b2">a</font>,<font color="Olive" title="b4">c</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> (<font color="Olive" title="b2">a</font>,<font color="Olive" title="b3">b</font>)</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> (<font color="Olive" title="b3">b</font>,<font color="Olive" title="b4">c</font>)</span>)</span> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">f</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">f</font> <a href="nagata_1.html#R1" title="NAGATA_1:pred.1">is_a_pseudometric_of</a> <font color="Maroon" title="c1">X</font> iff  for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> holds <br/> ( <font color="Maroon" title="c2">f</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> (<font color="Olive" title="b1">a</font>,<font color="Olive" title="b1">a</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c2">f</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> (<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> (<font color="Olive" title="b2">b</font>,<font color="Olive" title="b1">a</font>) &amp; <font color="Maroon" title="c2">f</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> (<font color="Olive" title="b1">a</font>,<font color="Olive" title="b3">c</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> (<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>)</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> (<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font>)</span>)</span> ) )</span><br/>



<a NAME="E1:53"/>
( <font color="Maroon" title="c2">f</font> <a href="nagata_1.html#R1" title="NAGATA_1:pred.1">is_a_pseudometric_of</a> <font color="Maroon" title="c1">X</font> iff ( <font color="Maroon" title="c2">f</font> is  <a href="metric_1.html#V2" title="METRIC_1:attr.2">Reflexive</a>  &amp; <font color="Maroon" title="c2">f</font> is  <a href="metric_1.html#V4" title="METRIC_1:attr.4">symmetric</a>  &amp; <font color="Maroon" title="c2">f</font> is  <a href="metric_1.html#V5" title="METRIC_1:attr.5">triangle</a>  ) )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nagata_1.html#D10" target="_self" title="NAGATA_1:def.10">Def10</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:53"/>
( <font color="Maroon" title="c2">f</font> <a href="nagata_1.html#R1" title="NAGATA_1:pred.1">is_a_pseudometric_of</a> <font color="Maroon" title="c1">X</font> iff  for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> holds <br/> ( <font color="Maroon" title="c2">f</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> (<font color="Olive" title="b1">a</font>,<font color="Olive" title="b1">a</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c2">f</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> (<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> (<font color="Olive" title="b2">b</font>,<font color="Olive" title="b1">a</font>) &amp; <font color="Maroon" title="c2">f</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> (<font color="Olive" title="b1">a</font>,<font color="Olive" title="b3">c</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> (<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>)</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> (<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font>)</span>)</span> ) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#D2" title="METRIC_1:def.2">METRIC_1:def 2</a>, <a class="ref" href="metric_1.html#D4" title="METRIC_1:def.4">METRIC_1:def 4</a>, <a class="ref" href="metric_1.html#D5" title="METRIC_1:def.5">METRIC_1:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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