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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#V9" title="METRIC_1:attr.9">triangle</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">  <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#V9" title="METRIC_1:attr.9">triangle</a> </span><br/><span class="kw">let </span><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>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="metric_1.html#D5" onmouseover="rs('metric_1/D5')" onmouseout="rh()">METRIC_1:def 5</a>,<a class="ref" href="metric_1.html#D9" onmouseover="rs('metric_1/D9')" onmouseout="rh()">METRIC_1:def 9</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">  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="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="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>)</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</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="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font>)</span>)</span></span><br/>



<span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E2:25"/><span class="lab"><font color="Green" title="E18">A1</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="c6">x1</font>,<font color="Maroon" title="c8">x2</font></span><a href="topreal7.html#K2" title="TOPREAL7:func.2">]</a></span>
 <span class="kw">and </span><br/><a NAME="E3:25"/><span class="lab"><font color="Green" title="E19">A2</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E4:25"/><span class="lab"><font color="Green" title="E20">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> (<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="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="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c8">x2</font>,<font color="Maroon" title="c9">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/>
<span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E6:25"/><span class="lab"><font color="Green" title="E21">A4</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E7:25"/><span class="lab"><font color="Green" title="E22">A5</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E8:25"/><span class="lab"><font color="Green" title="E23">A6</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="c4">b</font>,<font color="Maroon" title="c5">c</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="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="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c12">m2</font>,<font color="Maroon" title="c13">n2</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="E9:25"/><span class="lab"><font color="Green" title="E24">A7</font></span>: 
( <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> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:25"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="txt" href="#E6:25"><span class="lab"><font color="Green" title="E21">A4</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">consider </span><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><span class="kw"> such that </span><br/><a NAME="E11:25"/><span class="lab"><font color="Green" title="E25">A8</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="c14">p1</font>,<font color="Maroon" title="c16">p2</font></span><a href="topreal7.html#K2" title="TOPREAL7:func.2">]</a></span>
 <span class="kw">and </span><br/><a NAME="E12:25"/><span class="lab"><font color="Green" title="E26">A9</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E13:25"/><span class="lab"><font color="Green" title="E27">A10</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="c5">c</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="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="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c16">p2</font>,<font color="Maroon" title="c17">q2</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="E14:25"/><span class="lab"><font color="Green" title="E28">A11</font></span>: 
( <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> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:25"><span class="lab"><font color="Green" title="E22">A5</font></span></a>, <a class="txt" href="#E12:25"><span class="lab"><font color="Green" title="E26">A9</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="E15:25"/>
 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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#D9" onmouseover="rs('metric_1/D9')" onmouseout="rh()">METRIC_1:def 9</a></span>;<br/>
<a NAME="E16:25"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A12</font></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="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="binop_1.html#K2" title="BINOP_1:func.2">.</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="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c9">y2</font>,<font color="Maroon" title="c17">q2</font>)</span>)</span>
 
;<br/>
<a NAME="E17:25"/>
 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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#D9" onmouseover="rs('metric_1/D9')" onmouseout="rh()">METRIC_1:def 9</a></span>;<br/>
<a NAME="E18:25"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A13</font></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="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="binop_1.html#K2" title="BINOP_1:func.2">.</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="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c7">y1</font>,<font color="Maroon" title="c15">q1</font>)</span>)</span>
 
;<br/>
<a NAME="E19:25"/>
( <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> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:25"><span class="lab"><font color="Green" title="E18">A1</font></span></a>, <a class="txt" href="#E11:25"><span class="lab"><font color="Green" title="E25">A8</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="E20:25"/>
 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="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="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>)</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</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="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:25"><span class="lab"><font color="Green" title="E20">A3</font></span></a>, <a class="txt" href="#E8:25"><span class="lab"><font color="Green" title="E23">A6</font></span></a>, <a class="txt" href="#E13:25"><span class="lab"><font color="Green" title="E27">A10</font></span></a>, <a class="txt" href="#E18:25"><span class="lab"><font color="Green" title="E30">A13</font></span></a>, <a class="txt" href="#E16:25"><span class="lab"><font color="Green" title="E29">A12</font></span></a>, <a class="txt" href="#E9:25"><span class="lab"><font color="Green" title="E24">A7</font></span></a>, <a class="txt" href="#E14:25"><span class="lab"><font color="Green" title="E28">A11</font></span></a>, <a class="ref" href="topreal7.html#T2" target="_self" onmouseover="rs('topreal7/T2')" onmouseout="rh()">Th2</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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