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<span class="kw">given </span><font color="Maroon" title="c3">g1</font>, <font color="Maroon" title="c4">g2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">M</font><span class="kw"> such that </span><a NAME="E2:8_1_1"/><span class="lab"><font color="Green" title="E6">A2</font></span>: 
 for <font color="Olive" title="b1">r</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  holds <br/> ex <font color="Olive" title="b2">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/> for <font color="Olive" title="b3">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">m</font> holds <br/> <a href="metric_1.html#K3" title="METRIC_1:func.3">dist</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">S1</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b3">m</font></span>)</span>,<font color="Maroon" title="c3">g1</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">r</font>
 <span class="kw">and </span><br/><a NAME="E3:8_1_1"/><span class="lab"><font color="Green" title="E7">A3</font></span>: 
 for <font color="Olive" title="b1">r</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  holds <br/> ex <font color="Olive" title="b2">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/> for <font color="Olive" title="b3">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">m</font> holds <br/> <a href="metric_1.html#K3" title="METRIC_1:func.3">dist</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">S1</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b3">m</font></span>)</span>,<font color="Maroon" title="c4">g2</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">r</font>
 <span class="kw">and </span><br/><a NAME="E4:8_1_1"/><span class="lab"><font color="Green" title="E8">A4</font></span>: 
<font color="Maroon" title="c3">g1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">g2</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"> contradiction</span><br/>

<span class="kw">set </span><font color="Maroon" title="c5">a</font> = <span class="p1">(<span class="default"><a href="metric_1.html#K3" title="METRIC_1:func.3">dist</a> (<font color="Maroon" title="c3">g1</font>,<font color="Maroon" title="c4">g2</font>)</span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 4;<br/>
<a NAME="E5:8_1_1"/><span class="lab"><font color="Green" title="E9">A5</font></span>: 
 <a href="metric_1.html#K3" title="METRIC_1:func.3">dist</a> (<font color="Maroon" title="c3">g1</font>,<font color="Maroon" title="c4">g2</font>) <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<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="E6:8_1_1"/><span class="lab"><font color="Green" title="E10">A6</font></span>: 
 <a href="metric_1.html#K3" title="METRIC_1:func.3">dist</a> (<font color="Maroon" title="c3">g1</font>,<font color="Maroon" title="c4">g2</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:8_1_1"><span class="lab"><font color="Green" title="E8">A4</font></span></a>, <a class="ref" href="metric_1.html#T2" onmouseover="rs('metric_1/T2')" onmouseout="rh()">METRIC_1:2</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">n1</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E8:8_1_1"/><span class="lab"><font color="Green" title="E11">A7</font></span>: 
 for <font color="Olive" title="b1">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon" title="c6">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> holds <br/> <a href="metric_1.html#K3" title="METRIC_1:func.3">dist</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">S1</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">m</font></span>)</span>,<font color="Maroon" title="c3">g1</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="metric_1.html#K3" title="METRIC_1:func.3">dist</a> (<font color="Maroon" title="c3">g1</font>,<font color="Maroon" title="c4">g2</font>)</span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 4
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:8_1_1"><span class="lab"><font color="Green" title="E6">A2</font></span></a>, <a class="txt" href="#E5:8_1_1"><span class="lab"><font color="Green" title="E9">A5</font></span></a>, <a class="ref" href="xreal_1.html#T224" onmouseover="rs('xreal_1/T224')" onmouseout="rh()">XREAL_1:224</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">n2</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E10:8_1_1"/><span class="lab"><font color="Green" title="E12">A8</font></span>: 
 for <font color="Olive" title="b1">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon" title="c7">n2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> holds <br/> <a href="metric_1.html#K3" title="METRIC_1:func.3">dist</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">S1</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">m</font></span>)</span>,<font color="Maroon" title="c4">g2</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="metric_1.html#K3" title="METRIC_1:func.3">dist</a> (<font color="Maroon" title="c3">g1</font>,<font color="Maroon" title="c4">g2</font>)</span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 4
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:8_1_1"><span class="lab"><font color="Green" title="E7">A3</font></span></a>, <a class="txt" href="#E6:8_1_1"><span class="lab"><font color="Green" title="E10">A6</font></span></a>, <a class="txt" href="#E5:8_1_1"><span class="lab"><font color="Green" title="E9">A5</font></span></a>, <a class="ref" href="xreal_1.html#T224" onmouseover="rs('xreal_1/T224')" onmouseout="rh()">XREAL_1:224</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">k</font> = <font color="Maroon" title="c6">n1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c7">n2</font>;<br/>
<a NAME="E11:8_1_1"/><span class="lab"><font color="Green" title="E13">A9</font></span>: 
 <a href="metric_1.html#K3" title="METRIC_1:func.3">dist</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">S1</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">n1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c7">n2</font></span>)</span></span>)</span>,<font color="Maroon" title="c4">g2</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="metric_1.html#K3" title="METRIC_1:func.3">dist</a> (<font color="Maroon" title="c3">g1</font>,<font color="Maroon" title="c4">g2</font>)</span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 4
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:8_1_1"><span class="lab"><font color="Green" title="E12">A8</font></span></a>, <a class="ref" href="nat_1.html#T12" onmouseover="rs('nat_1/T12')" onmouseout="rh()">NAT_1:12</a></span>;<br/>
<a NAME="E12:8_1_1"/><span class="lab"><font color="Green" title="E14">A10</font></span>: 
 <a href="metric_1.html#K3" title="METRIC_1:func.3">dist</a> (<font color="Maroon" title="c3">g1</font>,<font color="Maroon" title="c4">g2</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K3" title="METRIC_1:func.3">dist</a> (<font color="Maroon" title="c3">g1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">S1</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">n1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c7">n2</font></span>)</span></span>)</span>)</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K3" title="METRIC_1:func.3">dist</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">S1</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">n1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c7">n2</font></span>)</span></span>)</span>,<font color="Maroon" title="c4">g2</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#T4" onmouseover="rs('metric_1/T4')" onmouseout="rh()">METRIC_1:4</a></span>;<br/>
<a NAME="E13:8_1_1"/>
 <a href="metric_1.html#K3" title="METRIC_1:func.3">dist</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">S1</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">n1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c7">n2</font></span>)</span></span>)</span>,<font color="Maroon" title="c3">g1</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="metric_1.html#K3" title="METRIC_1:func.3">dist</a> (<font color="Maroon" title="c3">g1</font>,<font color="Maroon" title="c4">g2</font>)</span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 4
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:8_1_1"><span class="lab"><font color="Green" title="E11">A7</font></span></a>, <a class="ref" href="nat_1.html#T12" onmouseover="rs('nat_1/T12')" onmouseout="rh()">NAT_1:12</a></span>;<br/>
<a NAME="E14:8_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="metric_1.html#K3" title="METRIC_1:func.3">dist</a> (<font color="Maroon" title="c3">g1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">S1</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">n1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c7">n2</font></span>)</span></span>)</span>)</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K3" title="METRIC_1:func.3">dist</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">S1</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">n1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c7">n2</font></span>)</span></span>)</span>,<font color="Maroon" title="c4">g2</font>)</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K3" title="METRIC_1:func.3">dist</a> (<font color="Maroon" title="c3">g1</font>,<font color="Maroon" title="c4">g2</font>)</span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 4</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K3" title="METRIC_1:func.3">dist</a> (<font color="Maroon" title="c3">g1</font>,<font color="Maroon" title="c4">g2</font>)</span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 4</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:8_1_1"><span class="lab"><font color="Green" title="E13">A9</font></span></a>, <a class="ref" href="xreal_1.html#T8" onmouseover="rs('xreal_1/T8')" onmouseout="rh()">XREAL_1:8</a></span>;<br/>
<a NAME="E15:8_1_1"/><span class="kw">then </span>
 <a href="metric_1.html#K3" title="METRIC_1:func.3">dist</a> (<font color="Maroon" title="c3">g1</font>,<font color="Maroon" title="c4">g2</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="metric_1.html#K3" title="METRIC_1:func.3">dist</a> (<font color="Maroon" title="c3">g1</font>,<font color="Maroon" title="c4">g2</font>)</span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:8_1_1"><span class="lab"><font color="Green" title="E14">A10</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E16:8_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:8_1_1"><span class="lab"><font color="Green" title="E10">A6</font></span></a>, <a class="txt" href="#E5:8_1_1"><span class="lab"><font color="Green" title="E9">A5</font></span></a>, <a class="ref" href="xreal_1.html#T216" onmouseover="rs('xreal_1/T216')" onmouseout="rh()">XREAL_1:216</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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