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<div class="add">

<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">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 <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> holds <br/>for <font color="Olive" title="b2">x</font>, <font color="Olive" title="b3">y</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font><br/> for <font color="Olive" title="b4">A</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> holds   <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="nagata_2.html#K3" title="NAGATA_2:func.3">inf</a> <font color="Olive" title="b1">pmet</font>,<font color="Olive" title="b4">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">x</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="nagata_2.html#K3" title="NAGATA_2:func.3">inf</a> <font color="Olive" title="b1">pmet</font>,<font color="Olive" title="b4">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">y</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">pmet</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Olive" title="b2">x</font>,<font color="Olive" title="b3">y</font></span><br/><b>let </b><font color="Maroon" title="c2">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"> ( <font color="Maroon" title="c2">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> implies 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 <font color="Maroon" title="c1">T</font><br/> for <font color="Olive" title="b3">A</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> holds   <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="nagata_2.html#K3" title="NAGATA_2:func.3">inf</a> <font color="Maroon" title="c2">pmet</font>,<font color="Olive" title="b3">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="nagata_2.html#K3" title="NAGATA_2:func.3">inf</a> <font color="Maroon" title="c2">pmet</font>,<font color="Olive" title="b3">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">y</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">pmet</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font> )</span><br/>



<b>assume </b><a NAME="E1:11"/><i><font color="Green" title="E12">A1</font></i>: 
<font color="Maroon" title="c2">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>
 ; <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">x</font>, <font color="Olive" title="b2">y</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font><br/> for <font color="Olive" title="b3">A</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> holds   <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="nagata_2.html#K3" title="NAGATA_2:func.3">inf</a> <font color="Maroon" title="c2">pmet</font>,<font color="Olive" title="b3">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="nagata_2.html#K3" title="NAGATA_2:func.3">inf</a> <font color="Maroon" title="c2">pmet</font>,<font color="Olive" title="b3">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">y</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">pmet</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><br/>

<b>let </b><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 <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"> for <font color="Olive" title="b1">A</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> holds   <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="nagata_2.html#K3" title="NAGATA_2:func.3">inf</a> <font color="Maroon" title="c2">pmet</font>,<font color="Olive" title="b1">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="nagata_2.html#K3" title="NAGATA_2:func.3">inf</a> <font color="Maroon" title="c2">pmet</font>,<font color="Olive" title="b1">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">y</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">pmet</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span><br/><b>let </b><font color="Maroon" title="c5">A</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</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">  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="nagata_2.html#K3" title="NAGATA_2:func.3">inf</a> <font color="Maroon" title="c2">pmet</font>,<font color="Maroon" title="c5">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="nagata_2.html#K3" title="NAGATA_2:func.3">inf</a> <font color="Maroon" title="c2">pmet</font>,<font color="Maroon" title="c5">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">y</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">pmet</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span><br/>



<a NAME="E2:11"/>
for <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">y1</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font> holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nagata_2.html#K3" title="NAGATA_2:func.3">inf</a> <font color="Maroon" title="c2">pmet</font>,<font color="Maroon" title="c5">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">y1</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nagata_2.html#K3" title="NAGATA_2:func.3">inf</a> <font color="Maroon" title="c2">pmet</font>,<font color="Maroon" title="c5">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x1</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">pmet</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Olive" title="b1">x1</font>,<font color="Olive" title="b2">y1</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">x1</font>, <font color="Maroon" title="c7">y1</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"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nagata_2.html#K3" title="NAGATA_2:func.3">inf</a> <font color="Maroon" title="c2">pmet</font>,<font color="Maroon" title="c5">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">y1</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nagata_2.html#K3" title="NAGATA_2:func.3">inf</a> <font color="Maroon" title="c2">pmet</font>,<font color="Maroon" title="c5">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x1</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">pmet</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c6">x1</font>,<font color="Maroon" title="c7">y1</font></span>)</span></span><br/>

<a NAME="E1:11_1"/><i><font color="Green" title="E13">A2</font></i>: 
( not <span class="p1">(<span class="default"><a href="nagata_2.html#K2" title="NAGATA_2:func.2">dist</a> <font color="Maroon" title="c2">pmet</font>,<font color="Maroon" title="c6">x1</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c5">A</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; <span class="p1">(<span class="default"><a href="nagata_2.html#K2" title="NAGATA_2:func.2">dist</a> <font color="Maroon" title="c2">pmet</font>,<font color="Maroon" title="c6">x1</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c5">A</font> is <a href="xxreal_2.html#V3" title="XXREAL_2:attr.3">bounded_below</a> &amp; not <span class="p1">(<span class="default"><a href="nagata_2.html#K2" title="NAGATA_2:func.2">dist</a> <font color="Maroon" title="c2">pmet</font>,<font color="Maroon" title="c7">y1</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c5">A</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; <span class="p1">(<span class="default"><a href="nagata_2.html#K2" title="NAGATA_2:func.2">dist</a> <font color="Maroon" title="c2">pmet</font>,<font color="Maroon" title="c7">y1</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c5">A</font> is <a href="xxreal_2.html#V3" title="XXREAL_2:attr.3">bounded_below</a> &amp; <font color="Maroon" title="c2">pmet</font> is <a href="seq_2.html#V2" title="SEQ_2:attr.2">bounded_below</a> )
 
<b>by </b><i><a class="txt" href="#E1:11"><i><font color="Green" title="E12">A1</font></i></a>, <a class="txt" href="nagata_2.html#E8"><i><font color="Green" title="E8">Lm1</font></i></a></i>;<br/>
<a NAME="E2:11_1"/>
for <font color="Olive" title="b1">rn</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b1">rn</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="nagata_2.html#K2" title="NAGATA_2:func.2">dist</a> <font color="Maroon" title="c2">pmet</font>,<font color="Maroon" title="c7">y1</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c5">A</font> holds <br/><font color="Olive" title="b1">rn</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#NK3" title="PSCOMP_1:NK.3">inf</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="nagata_2.html#K2" title="NAGATA_2:func.2">dist</a> <font color="Maroon" title="c2">pmet</font>,<font color="Maroon" title="c6">x1</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">pmet</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c6">x1</font>,<font color="Maroon" title="c7">y1</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">rn</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c8">rn</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="nagata_2.html#K2" title="NAGATA_2:func.2">dist</a> <font color="Maroon" title="c2">pmet</font>,<font color="Maroon" title="c7">y1</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c5">A</font> implies <font color="Maroon" title="c8">rn</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#NK3" title="PSCOMP_1:NK.3">inf</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="nagata_2.html#K2" title="NAGATA_2:func.2">dist</a> <font color="Maroon" title="c2">pmet</font>,<font color="Maroon" title="c6">x1</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">pmet</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c6">x1</font>,<font color="Maroon" title="c7">y1</font></span>)</span> )</span><br/>

<b>assume </b><a NAME="E1:11_1_1"/>
<font color="Maroon" title="c8">rn</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="nagata_2.html#K2" title="NAGATA_2:func.2">dist</a> <font color="Maroon" title="c2">pmet</font>,<font color="Maroon" title="c7">y1</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c5">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">rn</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#NK3" title="PSCOMP_1:NK.3">inf</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="nagata_2.html#K2" title="NAGATA_2:func.2">dist</a> <font color="Maroon" title="c2">pmet</font>,<font color="Maroon" title="c6">x1</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">pmet</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c6">x1</font>,<font color="Maroon" title="c7">y1</font></span>)</span></span><br/>

<b>then consider </b><font color="Maroon" title="c9">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:11_1_1"/><i><font color="Green" title="E14">A3</font></i>: 
( <font color="Maroon" title="c9">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="nagata_2.html#K2" title="NAGATA_2:func.2">dist</a> <font color="Maroon" title="c2">pmet</font>,<font color="Maroon" title="c7">y1</font></span>)</span> &amp; <font color="Maroon" title="c9">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A</font> &amp; <span class="p1">(<span class="default"><a href="nagata_2.html#K2" title="NAGATA_2:func.2">dist</a> <font color="Maroon" title="c2">pmet</font>,<font color="Maroon" title="c7">y1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">rn</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">z</font> = <font color="Maroon" title="c9">z</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font> <b>by </b><i><a class="txt" href="#E3:11_1_1"><i><font color="Green" title="E14">A3</font></i></a></i>;<br/>
<a NAME="E5:11_1_1"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="nagata_2.html#K2" title="NAGATA_2:func.2">dist</a> <font color="Maroon" title="c2">pmet</font>,<font color="Maroon" title="c6">x1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E6:11_1_1"/><b>then </b>
( <font color="Maroon" title="c2">pmet</font> is <a href="metric_1.html#V5" title="METRIC_1:attr.5">triangle</a> &amp; <span class="p1">(<span class="default"><a href="nagata_2.html#K2" title="NAGATA_2:func.2">dist</a> <font color="Maroon" title="c2">pmet</font>,<font color="Maroon" title="c6">x1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="nagata_2.html#K2" title="NAGATA_2:func.2">dist</a> <font color="Maroon" title="c2">pmet</font>,<font color="Maroon" title="c6">x1</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c5">A</font> &amp; <span class="p1">(<span class="default"><a href="nagata_2.html#K2" title="NAGATA_2:func.2">dist</a> <font color="Maroon" title="c2">pmet</font>,<font color="Maroon" title="c6">x1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">pmet</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c6">x1</font>,<font color="Maroon" title="c10">z</font> )
 
<b>by </b><i><a class="txt" href="#E1:11"><i><font color="Green" title="E12">A1</font></i></a>, <a class="txt" href="#E3:11_1_1"><i><font color="Green" title="E14">A3</font></i></a>, <a class="ref" href="nagata_2.html#D2" target="_self" title="NAGATA_2:def.2">Def2</a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a>, <a class="ref" href="nagata_1.html#D10" title="NAGATA_1:def.10">NAGATA_1:def 10</a></i>;<br/>
<a NAME="E7:11_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">pmet</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c6">x1</font>,<font color="Maroon" title="c7">y1</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">pmet</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c7">y1</font>,<font color="Maroon" title="c10">z</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">pmet</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c6">x1</font>,<font color="Maroon" title="c10">z</font> &amp; <font color="Maroon" title="c2">pmet</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c6">x1</font>,<font color="Maroon" title="c10">z</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="pscomp_1.html#NK3" title="PSCOMP_1:NK.3">inf</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nagata_2.html#K2" title="NAGATA_2:func.2">dist</a> <font color="Maroon" title="c2">pmet</font>,<font color="Maroon" title="c6">x1</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c5">A</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="#E1:11_1"><i><font color="Green" title="E13">A2</font></i></a>, <a class="ref" href="metric_1.html#D6" title="METRIC_1:def.6">METRIC_1:def 6</a>, <a class="ref" href="seq_4.html#D5" title="SEQ_4:def.5">SEQ_4:def 5</a></i>;<br/>
<a NAME="E8:11_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">pmet</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c6">x1</font>,<font color="Maroon" title="c7">y1</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">pmet</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c7">y1</font>,<font color="Maroon" title="c10">z</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#NK3" title="PSCOMP_1:NK.3">inf</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="nagata_2.html#K2" title="NAGATA_2:func.2">dist</a> <font color="Maroon" title="c2">pmet</font>,<font color="Maroon" title="c6">x1</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E9:11_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">pmet</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c7">y1</font>,<font color="Maroon" title="c10">z</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#NK3" title="PSCOMP_1:NK.3">inf</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="nagata_2.html#K2" title="NAGATA_2:func.2">dist</a> <font color="Maroon" title="c2">pmet</font>,<font color="Maroon" title="c6">x1</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">pmet</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c6">x1</font>,<font color="Maroon" title="c7">y1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T23" title="XREAL_1:th.23">XREAL_1:23</a></i>;<br/>
<b>hence </b><a NAME="E10:11_1_1"/>
<font color="Maroon" title="c8">rn</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#NK3" title="PSCOMP_1:NK.3">inf</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="nagata_2.html#K2" title="NAGATA_2:func.2">dist</a> <font color="Maroon" title="c2">pmet</font>,<font color="Maroon" title="c6">x1</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">pmet</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c6">x1</font>,<font color="Maroon" title="c7">y1</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E3:11_1_1"><i><font color="Green" title="E14">A3</font></i></a>, <a class="ref" href="nagata_2.html#D2" target="_self" title="NAGATA_2:def.2">Def2</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="E3:11_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="pscomp_1.html#NK3" title="PSCOMP_1:NK.3">inf</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="nagata_2.html#K2" title="NAGATA_2:func.2">dist</a> <font color="Maroon" title="c2">pmet</font>,<font color="Maroon" title="c7">y1</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#NK3" title="PSCOMP_1:NK.3">inf</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="nagata_2.html#K2" title="NAGATA_2:func.2">dist</a> <font color="Maroon" title="c2">pmet</font>,<font color="Maroon" title="c6">x1</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">pmet</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c6">x1</font>,<font color="Maroon" title="c7">y1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E1:11_1"><i><font color="Green" title="E13">A2</font></i></a>, <a class="ref" href="seq_4.html#T60" title="SEQ_4:th.60">SEQ_4:60</a></i>;<br/>
<a NAME="E4:11_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="pscomp_1.html#NK3" title="PSCOMP_1:NK.3">inf</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="nagata_2.html#K2" title="NAGATA_2:func.2">dist</a> <font color="Maroon" title="c2">pmet</font>,<font color="Maroon" title="c7">y1</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#NK3" title="PSCOMP_1:NK.3">inf</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="nagata_2.html#K2" title="NAGATA_2:func.2">dist</a> <font color="Maroon" title="c2">pmet</font>,<font color="Maroon" title="c6">x1</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">pmet</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c6">x1</font>,<font color="Maroon" title="c7">y1</font></span>)</span> &amp;  <a href="pscomp_1.html#NK3" title="PSCOMP_1:NK.3">inf</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nagata_2.html#K2" title="NAGATA_2:func.2">dist</a> <font color="Maroon" title="c2">pmet</font>,<font color="Maroon" title="c7">y1</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="nagata_2.html#K3" title="NAGATA_2:func.3">inf</a> <font color="Maroon" title="c2">pmet</font>,<font color="Maroon" title="c5">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">y1</font> )
 
<b>by </b><i><a class="ref" href="nagata_2.html#D3" target="_self" title="NAGATA_2:def.3">Def3</a>, <a class="ref" href="xreal_1.html#T19" title="XREAL_1:th.19">XREAL_1:19</a></i>;<br/>
<b>hence </b><a NAME="E5:11_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nagata_2.html#K3" title="NAGATA_2:func.3">inf</a> <font color="Maroon" title="c2">pmet</font>,<font color="Maroon" title="c5">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">y1</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nagata_2.html#K3" title="NAGATA_2:func.3">inf</a> <font color="Maroon" title="c2">pmet</font>,<font color="Maroon" title="c5">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x1</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">pmet</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c6">x1</font>,<font color="Maroon" title="c7">y1</font></span>)</span>
 <b>by </b><i><a class="ref" href="nagata_2.html#D3" target="_self" title="NAGATA_2:def.3">Def3</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="E3:11"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nagata_2.html#K3" title="NAGATA_2:func.3">inf</a> <font color="Maroon" title="c2">pmet</font>,<font color="Maroon" title="c5">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nagata_2.html#K3" title="NAGATA_2:func.3">inf</a> <font color="Maroon" title="c2">pmet</font>,<font color="Maroon" title="c5">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">pmet</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nagata_2.html#K3" title="NAGATA_2:func.3">inf</a> <font color="Maroon" title="c2">pmet</font>,<font color="Maroon" title="c5">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nagata_2.html#K3" title="NAGATA_2:func.3">inf</a> <font color="Maroon" title="c2">pmet</font>,<font color="Maroon" title="c5">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">pmet</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c3">x</font></span>)</span> &amp; <font color="Maroon" title="c2">pmet</font> is <a href="metric_1.html#V4" title="METRIC_1:attr.4">symmetric</a> )
 
<b>by </b><i><a class="txt" href="#E1:11"><i><font color="Green" title="E12">A1</font></i></a>, <a class="ref" href="nagata_1.html#D10" title="NAGATA_1:def.10">NAGATA_1:def 10</a></i>;<br/>
<a NAME="E4:11"/><b>then </b>
(  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="nagata_2.html#K3" title="NAGATA_2:func.3">inf</a> <font color="Maroon" title="c2">pmet</font>,<font color="Maroon" title="c5">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="nagata_2.html#K3" title="NAGATA_2:func.3">inf</a> <font color="Maroon" title="c2">pmet</font>,<font color="Maroon" title="c5">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">y</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">pmet</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nagata_2.html#K3" title="NAGATA_2:func.3">inf</a> <font color="Maroon" title="c2">pmet</font>,<font color="Maroon" title="c5">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nagata_2.html#K3" title="NAGATA_2:func.3">inf</a> <font color="Maroon" title="c2">pmet</font>,<font color="Maroon" title="c5">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">pmet</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="metric_1.html#D5" title="METRIC_1:def.5">METRIC_1:def 5</a></i>;<br/>
<a NAME="E5:11"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nagata_2.html#K3" title="NAGATA_2:func.3">inf</a> <font color="Maroon" title="c2">pmet</font>,<font color="Maroon" title="c5">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nagata_2.html#K3" title="NAGATA_2:func.3">inf</a> <font color="Maroon" title="c2">pmet</font>,<font color="Maroon" title="c5">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">pmet</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nagata_2.html#K3" title="NAGATA_2:func.3">inf</a> <font color="Maroon" title="c2">pmet</font>,<font color="Maroon" title="c5">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nagata_2.html#K3" title="NAGATA_2:func.3">inf</a> <font color="Maroon" title="c2">pmet</font>,<font color="Maroon" title="c5">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">pmet</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T26" title="XREAL_1:th.26">XREAL_1:26</a></i>;<br/>
<b>hence </b><a NAME="E6:11"/>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="nagata_2.html#K3" title="NAGATA_2:func.3">inf</a> <font color="Maroon" title="c2">pmet</font>,<font color="Maroon" title="c5">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="nagata_2.html#K3" title="NAGATA_2:func.3">inf</a> <font color="Maroon" title="c2">pmet</font>,<font color="Maroon" title="c5">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">y</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">pmet</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font>
 <b>by </b><i><a class="ref" href="absvalue.html#T12" title="ABSVALUE:th.12">ABSVALUE:12</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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