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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#NM1" title="METRIC_1:NM.1">MetrSpace</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">  <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>) is  <a href="metric_1.html#V7" title="METRIC_1:attr.7">discerning</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">b</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>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="metric_1.html#D3" onmouseover="rs('metric_1/D3')" onmouseout="rh()">METRIC_1:def 3</a>,<a class="ref" href="metric_1.html#D7" onmouseover="rs('metric_1/D7')" onmouseout="rh()">METRIC_1:def 7</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"> (  not  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#K1" title="BINOP_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="numbers.html#K5" title="NUMBERS:func.5">0</a>  or <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> )</span><br/>



<span class="kw">assume </span><a NAME="E1:22"/><span class="lab"><font color="Green" title="E16">A1</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> (<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="numbers.html#K5" title="NUMBERS:func.5">0</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"> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font></span><br/>

<a NAME="E2:22"/><span class="lab"><font color="Green" title="E17">A2</font></span>: 
 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#V3" title="METRIC_1:attr.3">discerning</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#D7" onmouseover="rs('metric_1/D7')" onmouseout="rh()">METRIC_1:def 7</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c5">x1</font>, <font color="Maroon" title="c6">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="c7">x2</font>, <font color="Maroon" title="c8">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="E4:22"/><span class="lab"><font color="Green" title="E18">A3</font></span>: 
( <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="c5">x1</font>,<font color="Maroon" title="c7">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="c6">y1</font>,<font color="Maroon" title="c8">y2</font></span><a href="topreal7.html#K2" title="TOPREAL7:func.2">]</a></span> )
 <span class="kw">and </span><br/><a NAME="E5:22"/><span class="lab"><font color="Green" title="E19">A4</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> (<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="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="c5">x1</font>,<font color="Maroon" title="c6">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="c7">x2</font>,<font color="Maroon" title="c8">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="E6:22"/>
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="metric_1.html#K3" title="METRIC_1:func.3">dist</a> (<font color="Maroon" title="c5">x1</font>,<font color="Maroon" title="c6">y1</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#T5" onmouseover="rs('metric_1/T5')" onmouseout="rh()">METRIC_1:5</a></span>;<br/>
<a NAME="E7:22"/><span class="kw">then </span>
 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="c5">x1</font>,<font color="Maroon" title="c6">y1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22"><span class="lab"><font color="Green" title="E16">A1</font></span></a>, <a class="txt" href="#E5:22"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="ref" href="xxreal_0.html#T49" onmouseover="rs('xxreal_0/T49')" onmouseout="rh()">XXREAL_0:49</a></span>;<br/>
<a NAME="E8:22"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A5</font></span>: 
(  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#V3" title="METRIC_1:attr.3">discerning</a>  &amp; <font color="Maroon" title="c5">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y1</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:22"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="ref" href="metric_1.html#D7" onmouseover="rs('metric_1/D7')" onmouseout="rh()">METRIC_1:def 7</a></span>;<br/>
<a NAME="E9:22"/>
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="metric_1.html#K3" title="METRIC_1:func.3">dist</a> (<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c8">y2</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#T5" onmouseover="rs('metric_1/T5')" onmouseout="rh()">METRIC_1:5</a></span>;<br/>
<a NAME="E10:22"/><span class="kw">then </span>
 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="c7">x2</font>,<font color="Maroon" title="c8">y2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22"><span class="lab"><font color="Green" title="E16">A1</font></span></a>, <a class="txt" href="#E5:22"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="ref" href="xxreal_0.html#T49" onmouseover="rs('xxreal_0/T49')" onmouseout="rh()">XXREAL_0:49</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:22"/>
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:22"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="txt" href="#E8:22"><span class="lab"><font color="Green" title="E20">A5</font></span></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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