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<b>let </b><font color="Maroon" title="c1">M</font>, <font color="Maroon" title="c2">N</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="metric_1.html#L1" title="METRIC_1:struct.1">MetrStruct</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">m1</font>, <font color="Olive" title="b2">m2</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font><br/> for <font color="Olive" title="b3">n1</font>, <font color="Olive" title="b4">n2</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">N</font> holds   <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <span class="p1"><a href="topreal7.html#K2" title="TOPREAL7:func.2">[</a><span class="default"><font color="Olive" title="b1">m1</font>,<font color="Olive" title="b3">n1</font></span><a href="topreal7.html#K2" title="TOPREAL7:func.2">]</a></span>,<span class="p1"><a href="topreal7.html#K2" title="TOPREAL7:func.2">[</a><span class="default"><font color="Olive" title="b2">m2</font>,<font color="Olive" title="b4">n2</font></span><a href="topreal7.html#K2" title="TOPREAL7:func.2">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="square_1.html#K2" title="SQUARE_1:func.2">max</a> <span class="p1">(<span class="default"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Olive" title="b1">m1</font>,<font color="Olive" title="b2">m2</font></span>)</span>,<span class="p1">(<span class="default"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Olive" title="b3">n1</font>,<font color="Olive" title="b4">n2</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c3">m1</font>, <font color="Maroon" title="c4">m2</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</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">n1</font>, <font color="Olive" title="b2">n2</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">N</font> holds   <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <span class="p1"><a href="topreal7.html#K2" title="TOPREAL7:func.2">[</a><span class="default"><font color="Maroon" title="c3">m1</font>,<font color="Olive" title="b1">n1</font></span><a href="topreal7.html#K2" title="TOPREAL7:func.2">]</a></span>,<span class="p1"><a href="topreal7.html#K2" title="TOPREAL7:func.2">[</a><span class="default"><font color="Maroon" title="c4">m2</font>,<font color="Olive" title="b2">n2</font></span><a href="topreal7.html#K2" title="TOPREAL7:func.2">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="square_1.html#K2" title="SQUARE_1:func.2">max</a> <span class="p1">(<span class="default"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c3">m1</font>,<font color="Maroon" title="c4">m2</font></span>)</span>,<span class="p1">(<span class="default"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Olive" title="b1">n1</font>,<font color="Olive" title="b2">n2</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c5">n1</font>, <font color="Maroon" title="c6">n2</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">N</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <span class="p1"><a href="topreal7.html#K2" title="TOPREAL7:func.2">[</a><span class="default"><font color="Maroon" title="c3">m1</font>,<font color="Maroon" title="c5">n1</font></span><a href="topreal7.html#K2" title="TOPREAL7:func.2">]</a></span>,<span class="p1"><a href="topreal7.html#K2" title="TOPREAL7:func.2">[</a><span class="default"><font color="Maroon" title="c4">m2</font>,<font color="Maroon" title="c6">n2</font></span><a href="topreal7.html#K2" title="TOPREAL7:func.2">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="square_1.html#K2" title="SQUARE_1:func.2">max</a> <span class="p1">(<span class="default"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c3">m1</font>,<font color="Maroon" title="c4">m2</font></span>)</span>,<span class="p1">(<span class="default"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c5">n1</font>,<font color="Maroon" title="c6">n2</font></span>)</span></span><br/>





<b>consider </b><font color="Maroon" title="c7">x1</font>, <font color="Maroon" title="c8">y1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font>, <font color="Maroon" title="c9">x2</font>, <font color="Maroon" title="c10">y2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">N</font><b> such that </b><br/><a NAME="E2:22"/><i><font color="Green" title="E14">A1</font></i>: 
( <span class="p1"><a href="topreal7.html#K2" title="TOPREAL7:func.2">[</a><span class="default"><font color="Maroon" title="c3">m1</font>,<font color="Maroon" title="c5">n1</font></span><a href="topreal7.html#K2" title="TOPREAL7:func.2">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="topreal7.html#K2" title="TOPREAL7:func.2">[</a><span class="default"><font color="Maroon" title="c7">x1</font>,<font color="Maroon" title="c9">x2</font></span><a href="topreal7.html#K2" title="TOPREAL7:func.2">]</a></span> &amp; <span class="p1"><a href="topreal7.html#K2" title="TOPREAL7:func.2">[</a><span class="default"><font color="Maroon" title="c4">m2</font>,<font color="Maroon" title="c6">n2</font></span><a href="topreal7.html#K2" title="TOPREAL7:func.2">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="topreal7.html#K2" title="TOPREAL7:func.2">[</a><span class="default"><font color="Maroon" title="c8">y1</font>,<font color="Maroon" title="c10">y2</font></span><a href="topreal7.html#K2" title="TOPREAL7:func.2">]</a></span> )
 <b>and </b><br/><a NAME="E3:22"/><i><font color="Green" title="E15">A2</font></i>: 
the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <span class="p1">(<span class="default"><a href="topreal7.html#K1" title="TOPREAL7:func.1">max-Prod2</a> <font color="Maroon" title="c1">M</font>,<font color="Maroon" title="c2">N</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <span class="p1"><a href="topreal7.html#K2" title="TOPREAL7:func.2">[</a><span class="default"><font color="Maroon" title="c3">m1</font>,<font color="Maroon" title="c5">n1</font></span><a href="topreal7.html#K2" title="TOPREAL7:func.2">]</a></span>,<span class="p1"><a href="topreal7.html#K2" title="TOPREAL7:func.2">[</a><span class="default"><font color="Maroon" title="c4">m2</font>,<font color="Maroon" title="c6">n2</font></span><a href="topreal7.html#K2" title="TOPREAL7:func.2">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="square_1.html#K2" title="SQUARE_1:func.2">max</a> <span class="p1">(<span class="default">the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <font color="Maroon" title="c1">M</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c7">x1</font>,<font color="Maroon" title="c8">y1</font></span>)</span>,<span class="p1">(<span class="default">the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <font color="Maroon" title="c2">N</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c9">x2</font>,<font color="Maroon" title="c10">y2</font></span>)</span>
 <b>by </b><i><a class="ref" href="topreal7.html#D1" target="_self" title="TOPREAL7:def.1">Def1</a></i>;<br/>
<a NAME="E4:22"/>
( <font color="Maroon" title="c3">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x1</font> &amp; <font color="Maroon" title="c5">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x2</font> &amp; <font color="Maroon" title="c4">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">y1</font> &amp; <font color="Maroon" title="c6">n2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">y2</font> )
 
<b>by </b><i><a class="txt" href="#E2:22"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/>
<b>hence </b><a NAME="E5:22"/>
 <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <span class="p1"><a href="topreal7.html#K2" title="TOPREAL7:func.2">[</a><span class="default"><font color="Maroon" title="c3">m1</font>,<font color="Maroon" title="c5">n1</font></span><a href="topreal7.html#K2" title="TOPREAL7:func.2">]</a></span>,<span class="p1"><a href="topreal7.html#K2" title="TOPREAL7:func.2">[</a><span class="default"><font color="Maroon" title="c4">m2</font>,<font color="Maroon" title="c6">n2</font></span><a href="topreal7.html#K2" title="TOPREAL7:func.2">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="square_1.html#K2" title="SQUARE_1:func.2">max</a> <span class="p1">(<span class="default"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c3">m1</font>,<font color="Maroon" title="c4">m2</font></span>)</span>,<span class="p1">(<span class="default"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c5">n1</font>,<font color="Maroon" title="c6">n2</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E3:22"><i><font color="Green" title="E15">A2</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/>


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