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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#V3" title="STRUCT_0:attr.3">empty</a> <a href="metric_1.html#V9" title="METRIC_1:attr.9">triangle</a>  <a href="metric_1.html#L1" title="METRIC_1:struct.1">MetrStruct</a> ;<br/><b>let </b><font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">b</font>, <font color="Maroon" title="c5">c</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</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>; <i><font color="Red">:: according to </font></i><a class="ref" href="metric_1.html#D6" title="METRIC_1:def.6">METRIC_1:def 6</a>,<a class="ref" href="metric_1.html#D10" title="METRIC_1:def.10">METRIC_1:def 10</a><br/>



<b>consider </b><font color="Maroon" title="c6">x1</font>, <font color="Maroon" title="c7">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="c8">x2</font>, <font color="Maroon" title="c9">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="E2:30"/><i><font color="Green" title="E20">A1</font></i>: 
( <font color="Maroon" title="c3">a</font> <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="c6">x1</font>,<font color="Maroon" title="c8">x2</font></span><a href="topreal7.html#K2" title="TOPREAL7:func.2">]</a></span> &amp; <font color="Maroon" title="c4">b</font> <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">y1</font>,<font color="Maroon" title="c9">y2</font></span><a href="topreal7.html#K2" title="TOPREAL7:func.2">]</a></span> )
 <b>and </b><br/><a NAME="E3:30"/><i><font color="Green" title="E21">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> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font> <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="c6">x1</font>,<font color="Maroon" title="c7">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="c8">x2</font>,<font color="Maroon" title="c9">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/>
<b>consider </b><font color="Maroon" title="c10">m1</font>, <font color="Maroon" title="c11">n1</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="c12">m2</font>, <font color="Maroon" title="c13">n2</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="E5:30"/><i><font color="Green" title="E22">A3</font></i>: 
( <font color="Maroon" title="c4">b</font> <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="c10">m1</font>,<font color="Maroon" title="c12">m2</font></span><a href="topreal7.html#K2" title="TOPREAL7:func.2">]</a></span> &amp; <font color="Maroon" title="c5">c</font> <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="c11">n1</font>,<font color="Maroon" title="c13">n2</font></span><a href="topreal7.html#K2" title="TOPREAL7:func.2">]</a></span> )
 <b>and </b><br/><a NAME="E6:30"/><i><font color="Green" title="E23">A4</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> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font> <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="c10">m1</font>,<font color="Maroon" title="c11">n1</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="c12">m2</font>,<font color="Maroon" title="c13">n2</font></span>)</span>
 <b>by </b><i><a class="ref" href="topreal7.html#D1" target="_self" title="TOPREAL7:def.1">Def1</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c14">p1</font>, <font color="Maroon" title="c15">q1</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="c16">p2</font>, <font color="Maroon" title="c17">q2</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="E8:30"/><i><font color="Green" title="E24">A5</font></i>: 
( <font color="Maroon" title="c3">a</font> <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="c14">p1</font>,<font color="Maroon" title="c16">p2</font></span><a href="topreal7.html#K2" title="TOPREAL7:func.2">]</a></span> &amp; <font color="Maroon" title="c5">c</font> <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="c15">q1</font>,<font color="Maroon" title="c17">q2</font></span><a href="topreal7.html#K2" title="TOPREAL7:func.2">]</a></span> )
 <b>and </b><br/><a NAME="E9:30"/><i><font color="Green" title="E25">A6</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> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">c</font> <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="c14">p1</font>,<font color="Maroon" title="c15">q1</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="c16">p2</font>,<font color="Maroon" title="c17">q2</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="E10:30"/>
( the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <font color="Maroon" title="c1">M</font> is <a href="metric_1.html#V5" title="METRIC_1:attr.5">triangle</a> &amp; the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <font color="Maroon" title="c2">N</font> is <a href="metric_1.html#V5" title="METRIC_1:attr.5">triangle</a> )
 
<b>by </b><i><a class="ref" href="metric_1.html#D10" title="METRIC_1:def.10">METRIC_1:def 10</a></i>;<br/>
<a NAME="E11:30"/><b>then </b><i><font color="Green" title="E26">A7</font></i>: 
( 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="c14">p1</font>,<font color="Maroon" title="c15">q1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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="c14">p1</font>,<font color="Maroon" title="c7">y1</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</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">y1</font>,<font color="Maroon" title="c15">q1</font></span>)</span> &amp; 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="c16">p2</font>,<font color="Maroon" title="c17">q2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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="c2">N</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c16">p2</font>,<font color="Maroon" title="c9">y2</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</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="c2">N</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c9">y2</font>,<font color="Maroon" title="c17">q2</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="metric_1.html#D6" title="METRIC_1:def.6">METRIC_1:def 6</a></i>;<br/>
<a NAME="E12:30"/>
( <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">p1</font> &amp; <font color="Maroon" title="c8">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">p2</font> &amp; <font color="Maroon" title="c7">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">m1</font> &amp; <font color="Maroon" title="c9">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">m2</font> &amp; <font color="Maroon" title="c15">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">n1</font> &amp; <font color="Maroon" title="c17">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">n2</font> )
 
<b>by </b><i><a class="txt" href="topreal7.html#E2:30"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="topreal7.html#E5:30"><i><font color="Green" title="E22">A3</font></i></a>, <a class="txt" href="topreal7.html#E8:30"><i><font color="Green" title="E24">A5</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="E13:30"/>
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> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <span class="p2">(<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> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default">the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <span class="p2">(<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> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font></span>)</span>
 <b>by </b><i><a class="txt" href="topreal7.html#E3:30"><i><font color="Green" title="E21">A2</font></i></a>, <a class="txt" href="topreal7.html#E6:30"><i><font color="Green" title="E23">A4</font></i></a>, <a class="txt" href="topreal7.html#E9:30"><i><font color="Green" title="E25">A6</font></i></a>, <a class="txt" href="topreal7.html#E11:30"><i><font color="Green" title="E26">A7</font></i></a>, <a class="ref" href="topreal7.html#T3" target="_self" title="TOPREAL7:th.3">Th3</a></i>;<br/>


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