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<span class="kw">let </span><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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">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#K1" title="METRIC_1:func.1">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="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<span class="p1">(<span class="default"><a href="metric_1.html#K1" title="METRIC_1:func.1">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#K1" title="METRIC_1:func.1">dist</a> (<font color="Olive" title="b3">n1</font>,<font color="Olive" title="b4">n2</font>)</span>)</span>)</span><br/><span class="kw">let </span><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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">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#K1" title="METRIC_1:func.1">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="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<span class="p1">(<span class="default"><a href="metric_1.html#K1" title="METRIC_1:func.1">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#K1" title="METRIC_1:func.1">dist</a> (<font color="Olive" title="b1">n1</font>,<font color="Olive" title="b2">n2</font>)</span>)</span>)</span><br/><span class="kw">let </span><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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="metric_1.html#K1" title="METRIC_1:func.1">dist</a> (<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="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<span class="p1">(<span class="default"><a href="metric_1.html#K1" title="METRIC_1:func.1">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#K1" title="METRIC_1:func.1">dist</a> (<font color="Maroon" title="c5">n1</font>,<font color="Maroon" title="c6">n2</font>)</span>)</span>)</span><br/>





<span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E2:18"/><span class="lab"><font color="Green" title="E14">A1</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E3:18"/><span class="lab"><font color="Green" title="E15">A2</font></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> <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>
 <span class="kw">and </span><br/><a NAME="E4:18"/><span class="lab"><font color="Green" title="E16">A3</font></span>: 
 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="binop_1.html#K2" title="BINOP_1:func.2">.</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="xxreal_0.html#K4" title="XXREAL_0:func.4">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="binop_1.html#K2" title="BINOP_1:func.2">.</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="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c9">x2</font>,<font color="Maroon" title="c10">y2</font>)</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="topreal7.html#D1" target="_self" onmouseover="rs('topreal7/D1')" onmouseout="rh()">Def1</a></span>;<br/>
<a NAME="E5:18"/><span class="lab"><font color="Green" title="E17">A4</font></span>: 
<font color="Maroon" title="c4">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">y1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:18"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="ref" href="xtuple_0.html#T1" onmouseover="rs('xtuple_0/T1')" onmouseout="rh()">XTUPLE_0:1</a></span>;<br/>
<a NAME="E6:18"/>
( <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> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:18"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="ref" href="xtuple_0.html#T1" onmouseover="rs('xtuple_0/T1')" onmouseout="rh()">XTUPLE_0:1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:18"/>
 <a href="metric_1.html#K1" title="METRIC_1:func.1">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="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<span class="p1">(<span class="default"><a href="metric_1.html#K1" title="METRIC_1:func.1">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#K1" title="METRIC_1:func.1">dist</a> (<font color="Maroon" title="c5">n1</font>,<font color="Maroon" title="c6">n2</font>)</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:18"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="txt" href="#E4:18"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="txt" href="#E5:18"><span class="lab"><font color="Green" title="E17">A4</font></span></a>, <a class="ref" href="xtuple_0.html#T1" onmouseover="rs('xtuple_0/T1')" onmouseout="rh()">XTUPLE_0:1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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