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<b>let </b><font color="Maroon" title="c1">T</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">FS2</font> being   <a href="seqfunc.html#M1" title="SEQFUNC:mode.1">Functional_Sequence</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</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>   st ( for <font color="Olive" title="b2">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   ex <font color="Olive" title="b3">pmet</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">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</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>  st <br/>( <font color="Olive" title="b1">FS2</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b2">n</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b3">pmet</font> &amp; <font color="Olive" title="b3">pmet</font> <a href="nagata_1.html#R1" title="NAGATA_1:pred.1">is_a_pseudometric_of</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> ) ) &amp; ( for <font color="Olive" title="b2">pq</font> being   <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">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> holds  <font color="Olive" title="b1">FS2</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Olive" title="b2">pq</font> is <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a> ) holds <br/>for <font color="Olive" title="b2">pmet</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">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</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>   st ( for <font color="Olive" title="b3">pq</font> being   <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">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> holds  <font color="Olive" title="b2">pmet</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">pq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">FS2</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Olive" title="b3">pq</font></span>)</span> ) holds <br/><font color="Olive" title="b2">pmet</font> <a href="nagata_1.html#R1" title="NAGATA_1:pred.1">is_a_pseudometric_of</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span><br/>

<b>set </b><font color="Maroon" title="c2">cT</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>;<br/>
<b>let </b><font color="Maroon" title="c3">FS2</font> be    <a href="seqfunc.html#M1" title="SEQFUNC:mode.1">Functional_Sequence</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   ex <font color="Olive" title="b2">pmet</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">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</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>  st <br/>( <font color="Maroon" title="c3">FS2</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b2">pmet</font> &amp; <font color="Olive" title="b2">pmet</font> <a href="nagata_1.html#R1" title="NAGATA_1:pred.1">is_a_pseudometric_of</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> ) ) &amp; ( for <font color="Olive" title="b1">pq</font> being   <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">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> holds  <font color="Maroon" title="c3">FS2</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Olive" title="b1">pq</font> is <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a> ) implies for <font color="Olive" title="b1">pmet</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">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</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>   st ( for <font color="Olive" title="b2">pq</font> being   <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">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> holds  <font color="Olive" title="b1">pmet</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">pq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">FS2</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Olive" title="b2">pq</font></span>)</span> ) holds <br/><font color="Olive" title="b1">pmet</font> <a href="nagata_1.html#R1" title="NAGATA_1:pred.1">is_a_pseudometric_of</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:15"/><i><font color="Green" title="E16">A1</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   ex <font color="Olive" title="b2">pmet</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">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</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>  st <br/>( <font color="Maroon" title="c3">FS2</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b2">pmet</font> &amp; <font color="Olive" title="b2">pmet</font> <a href="nagata_1.html#R1" title="NAGATA_1:pred.1">is_a_pseudometric_of</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> )
 <b>and </b><br/><a NAME="E2:15"/><i><font color="Green" title="E17">A2</font></i>: 
for <font color="Olive" title="b1">pq</font> being   <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">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> holds  <font color="Maroon" title="c3">FS2</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Olive" title="b1">pq</font> is <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">pmet</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">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</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>   st ( for <font color="Olive" title="b2">pq</font> being   <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">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> holds  <font color="Olive" title="b1">pmet</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">pq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">FS2</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Olive" title="b2">pq</font></span>)</span> ) holds <br/><font color="Olive" title="b1">pmet</font> <a href="nagata_1.html#R1" title="NAGATA_1:pred.1">is_a_pseudometric_of</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span><br/>

<b>let </b><font color="Maroon" title="c4">pmet</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">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( for <font color="Olive" title="b1">pq</font> being   <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">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> holds  <font color="Maroon" title="c4">pmet</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">pq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">FS2</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Olive" title="b1">pq</font></span>)</span> ) implies <font color="Maroon" title="c4">pmet</font> <a href="nagata_1.html#R1" title="NAGATA_1:pred.1">is_a_pseudometric_of</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> )</span><br/>

<b>assume </b><a NAME="E3:15"/><i><font color="Green" title="E18">A3</font></i>: 
for <font color="Olive" title="b1">pq</font> being   <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">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> holds  <font color="Maroon" title="c4">pmet</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">pq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">FS2</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Olive" title="b1">pq</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">pmet</font> <a href="nagata_1.html#R1" title="NAGATA_1:pred.1">is_a_pseudometric_of</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c6">b</font>, <font color="Maroon" title="c7">c</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">pmet</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c5">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="c4">pmet</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">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="c4">pmet</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">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="c4">pmet</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c6">b</font></span>)</span> )</span><br/><b>thus </b><a NAME="E1:15_1"/>
<font color="Maroon" title="c4">pmet</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c5">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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">pmet</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">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="c4">pmet</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">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="c4">pmet</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c6">b</font></span>)</span></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon" title="c8">aa</font> = <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c5">a</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>;<br/>
<b>set </b><font color="Maroon" title="c9">F</font> = <font color="Maroon" title="c3">FS2</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c5">a</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c10">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c3">FS2</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c5">a</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">FS2</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <span class="p3"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c5">a</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c10">n</font></span>)</span></span><br/><b>consider </b><font color="Maroon" title="c11">pmet1</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">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</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> <b> such that </b><br/><a NAME="E2:15_1_1_1"/><i><font color="Green" title="E19">A4</font></i>: 
( <font color="Maroon" title="c3">FS2</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c10">n</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c11">pmet1</font> &amp; <font color="Maroon" title="c11">pmet1</font> <a href="nagata_1.html#R1" title="NAGATA_1:pred.1">is_a_pseudometric_of</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> )
 <b>by </b><i><a class="txt" href="nagata_2.html#E1:15"><i><font color="Green" title="E16">A1</font></i></a></i>;<br/><a NAME="E3:15_1_1_1"/>
<font color="Maroon" title="c11">pmet1</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="nagata_2.html#E2:15_1_1_1"><i><font color="Green" title="E19">A4</font></i></a>, <a class="ref" href="nagata_1.html#T28" title="NAGATA_1:th.28">NAGATA_1:28</a></i>;<br/><b>hence </b><a NAME="E4:15_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">FS2</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c5">a</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">FS2</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <span class="p3"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c5">a</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c10">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="nagata_2.html#E2:15_1_1_1"><i><font color="Green" title="E19">A4</font></i></a>, <a class="ref" href="seqfunc.html#D11" title="SEQFUNC:def.11">SEQFUNC:def 11</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E2:15_1_1"/><b>then </b>
( <font color="Maroon" title="c3">FS2</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c5">a</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">FS2</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c5">a</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> &amp; <font color="Maroon" title="c3">FS2</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c5">a</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> is <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a> )
 
<b>by </b><i><a class="txt" href="nagata_2.html#E2:15"><i><font color="Green" title="E17">A2</font></i></a>, <a class="ref" href="seq_1.html#T13" title="SEQ_1:th.13">SEQ_1:13</a></i>;<br/>
<a NAME="E3:15_1_1"/><b>then </b>
 <a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">FS2</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c5">a</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">FS2</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <span class="p3"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c5">a</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="series_1.html#T13" title="SERIES_1:th.13">SERIES_1:13</a></i>;<br/>
<b>hence </b><a NAME="E4:15_1_1"/>
<font color="Maroon" title="c4">pmet</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="nagata_2.html#E3:15"><i><font color="Green" title="E18">A3</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><b>thus </b><a NAME="E2:15_1"/>
<font color="Maroon" title="c4">pmet</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">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="c4">pmet</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">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="c4">pmet</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c6">b</font></span>)</span>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_2"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon" title="c8">ab</font> = <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>;<br/>
<b>set </b><font color="Maroon" title="c9">cb</font> = <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c6">b</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>;<br/>
<b>set </b><font color="Maroon" title="c10">ac</font> = <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">c</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>;<br/>
<b>set </b><font color="Maroon" title="c11">Fac</font> = <font color="Maroon" title="c3">FS2</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">c</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>;<br/>
<b>set </b><font color="Maroon" title="c12">Fab</font> = <font color="Maroon" title="c3">FS2</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>;<br/>
<b>set </b><font color="Maroon" title="c13">Fcb</font> = <font color="Maroon" title="c3">FS2</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c6">b</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>;<br/>
<a NAME="E1:15_1_2"/>
( <font color="Maroon" title="c3">FS2</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> is <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a> &amp; <font color="Maroon" title="c3">FS2</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c6">b</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> is <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a> )
 
<b>by </b><i><a class="txt" href="nagata_2.html#E2:15"><i><font color="Green" title="E17">A2</font></i></a></i>;<br/>
<a NAME="E2:15_1_2"/><b>then </b><i><font color="Green" title="E19">A5</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c3">FS2</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">FS2</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c6">b</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> is <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a> &amp;  <a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">FS2</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <span class="p3"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">FS2</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <span class="p3"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c6">b</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">FS2</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <span class="p3"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">FS2</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <span class="p3"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c6">b</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="series_1.html#T10" title="SERIES_1:th.10">SERIES_1:10</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c14">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">FS2</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">c</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c14">n</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">FS2</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">c</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c14">n</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"><font color="Maroon" title="c3">FS2</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <span class="p3"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">FS2</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <span class="p3"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c6">b</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c14">n</font> )</span><br/><b>consider </b><font color="Maroon" title="c15">pmet1</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">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</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> <b> such that </b><br/><a NAME="E2:15_1_2_1"/><i><font color="Green" title="E20">A6</font></i>: 
( <font color="Maroon" title="c3">FS2</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c14">n</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c15">pmet1</font> &amp; <font color="Maroon" title="c15">pmet1</font> <a href="nagata_1.html#R1" title="NAGATA_1:pred.1">is_a_pseudometric_of</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> )
 <b>by </b><i><a class="txt" href="nagata_2.html#E1:15"><i><font color="Green" title="E16">A1</font></i></a></i>;<br/><a NAME="E3:15_1_2_1"/>
( <font color="Maroon" title="c15">pmet1</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">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="c15">pmet1</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">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="c15">pmet1</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c6">b</font></span>)</span> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">pmet1</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">c</font> &amp; <font color="Maroon" title="c15">pmet1</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">pmet1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c6">b</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> &amp; <font color="Maroon" title="c15">pmet1</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">pmet1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">c</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> )
 <b>by </b><i><a class="txt" href="nagata_2.html#E2:15_1_2_1"><i><font color="Green" title="E20">A6</font></i></a>, <a class="ref" href="nagata_1.html#T28" title="NAGATA_1:th.28">NAGATA_1:28</a>, <a class="ref" href="nagata_1.html#T29" title="NAGATA_1:th.29">NAGATA_1:29</a></i>;<br/><a NAME="E4:15_1_2_1"/><b>then </b>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">FS2</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c14">n</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">c</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">FS2</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c14">n</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">c</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <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"><font color="Maroon" title="c3">FS2</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c14">n</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></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"><font color="Maroon" title="c3">FS2</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c14">n</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c6">b</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">FS2</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c14">n</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">c</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">FS2</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">c</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c14">n</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">FS2</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c14">n</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">FS2</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c14">n</font> )
 <b>by </b><i><a class="txt" href="nagata_2.html#E2:15_1_2_1"><i><font color="Green" title="E20">A6</font></i></a>, <a class="ref" href="seqfunc.html#D11" title="SEQFUNC:def.11">SEQFUNC:def 11</a></i>;<br/><a NAME="E5:15_1_2_1"/><b>then </b>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">FS2</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">c</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c14">n</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">FS2</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">c</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c14">n</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"><font color="Maroon" title="c3">FS2</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <span class="p3"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c14">n</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"><font color="Maroon" title="c3">FS2</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <span class="p3"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c6">b</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c14">n</font></span>)</span> )
 <b>by </b><i><a class="ref" href="seqfunc.html#D11" title="SEQFUNC:def.11">SEQFUNC:def 11</a></i>;<br/><b>hence </b><a NAME="E6:15_1_2_1"/>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">FS2</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">c</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c14">n</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">FS2</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">c</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c14">n</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"><font color="Maroon" title="c3">FS2</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <span class="p3"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">FS2</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <span class="p3"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c6">b</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c14">n</font> )
 <b>by </b><i><a class="ref" href="seq_1.html#T11" title="SEQ_1:th.11">SEQ_1:11</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E4:15_1_2"/><b>then </b>
(  <a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">FS2</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">c</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">FS2</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <span class="p3"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">FS2</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <span class="p3"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c6">b</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span></span>)</span> &amp;  <a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">FS2</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">pmet</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> &amp;  <a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">FS2</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c6">b</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">pmet</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c6">b</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> &amp; <font color="Maroon" title="c4">pmet</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">pmet</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> &amp; <font color="Maroon" title="c4">pmet</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c6">b</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">pmet</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c6">b</font> )
 
<b>by </b><i><a class="txt" href="nagata_2.html#E3:15"><i><font color="Green" title="E18">A3</font></i></a>, <a class="txt" href="nagata_2.html#E2:15_1_2"><i><font color="Green" title="E19">A5</font></i></a>, <a class="ref" href="series_1.html#T24" title="SERIES_1:th.24">SERIES_1:24</a></i>;<br/>
<b>hence </b><a NAME="E5:15_1_2"/>
<font color="Maroon" title="c4">pmet</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">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="c4">pmet</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">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="c4">pmet</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c6">b</font></span>)</span>
 <b>by </b><i><a class="txt" href="nagata_2.html#E3:15"><i><font color="Green" title="E18">A3</font></i></a>, <a class="txt" href="nagata_2.html#E2:15_1_2"><i><font color="Green" title="E19">A5</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E5:15"/>
<font color="Maroon" title="c4">pmet</font> <a href="nagata_1.html#R1" title="NAGATA_1:pred.1">is_a_pseudometric_of</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>
 <b>by </b><i><a class="ref" href="nagata_1.html#T28" title="NAGATA_1:th.28">NAGATA_1:28</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
