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<b>let </b><font color="Maroon" title="c1">A</font> be   <a href="measure5.html#NM1" title="MEASURE5:NM.1">Interval</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">x</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  holds   <a href="measure5.html#K2" title="MEASURE5:func.2">diameter</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure5.html#K2" title="MEASURE5:func.2">diameter</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="measure6.html#K4" title="MEASURE6:func.4">++</a> <font color="Maroon" title="c1">A</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c2">x</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="measure5.html#K2" title="MEASURE5:func.2">diameter</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure5.html#K2" title="MEASURE5:func.2">diameter</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="measure6.html#K4" title="MEASURE6:func.4">++</a> <font color="Maroon" title="c1">A</font></span>)</span></span><br/>



<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:52_1"/>
( <font color="Maroon" title="c1">A</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> or  not <font color="Maroon" title="c1">A</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_1"><b>suppose </b></a><a NAME="E1:52_1_1"/><i><font color="Green" title="E33">S</font></i>: 
<font color="Maroon" title="c1">A</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="measure5.html#K2" title="MEASURE5:func.2">diameter</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure5.html#K2" title="MEASURE5:func.2">diameter</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="measure6.html#K4" title="MEASURE6:func.4">++</a> <font color="Maroon" title="c1">A</font></span>)</span></span><br/><div class="add"><div><i><font color="Green" title="E34">X</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:52_1_1_1"/>
not <font color="Maroon" title="c2">x</font> <a href="measure6.html#K4" title="MEASURE6:func.4">++</a> <font color="Maroon" title="c1">A</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><b>then consider </b><font color="Maroon" title="c3">y</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E3:52_1_1_1"/><i><font color="Green" title="E35">W</font></i>: 
<font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">x</font> <a href="measure6.html#K4" title="MEASURE6:func.4">++</a> <font color="Maroon" title="c1">A</font>
 <b>by </b><i><a class="ref" href="membered.html#T9" title="MEMBERED:th.9">MEMBERED:9</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c4">y</font> = <font color="Maroon" title="c3">y</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/><a NAME="E5:52_1_1_1"/>
 ex <font color="Olive" title="b1">z</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Olive" title="b1">z</font> )
 <b>by </b><i><a class="txt" href="#E3:52_1_1_1"><i><font color="Green" title="E35">W</font></i></a>, <a class="ref" href="measure6.html#D6" target="_self" title="MEASURE6:def.6">Def6</a></i>;<br/><b>hence </b><a NAME="E6:52_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:52_1_1"><i><font color="Green" title="E33">S</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>thus </b> <a href="measure5.html#K2" title="MEASURE5:func.2">diameter</a> <font color="Maroon" title="c1">A</font> = 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="txt" href="#E1:52_1_1"><i><font color="Green" title="E33">S</font></i></a>, <a class="ref" href="measure5.html#T60" title="MEASURE5:th.60">MEASURE5:60</a></i>
<br/>.= 
 <a href="measure5.html#K2" title="MEASURE5:func.2">diameter</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="measure6.html#K4" title="MEASURE6:func.4">++</a> <font color="Maroon" title="c1">A</font></span>)</span>
<b>by </b><i><a class="txt" href="#E2:52_1_1"><i><font color="Green" title="E34">X</font></i></a>, <a class="ref" href="measure5.html#T60" title="MEASURE5:th.60">MEASURE5:60</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_2"><b>suppose </b></a><a NAME="E1:52_1_2"/><i><font color="Green" title="E33">S</font></i>: 
not <font color="Maroon" title="c1">A</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="measure5.html#K2" title="MEASURE5:func.2">diameter</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure5.html#K2" title="MEASURE5:func.2">diameter</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="measure6.html#K4" title="MEASURE6:func.4">++</a> <font color="Maroon" title="c1">A</font></span>)</span></span><br/><div class="add"><b>then consider </b><font color="Maroon" title="c3">y</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E3:52_1_2"/><i><font color="Green" title="E34">W</font></i>: 
<font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 <b>by </b><i><a class="ref" href="membered.html#T9" title="MEMBERED:th.9">MEMBERED:9</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c4">y</font> = <font color="Maroon" title="c3">y</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/><a NAME="E5:52_1_2"/>
<font color="Maroon" title="c2">x</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">x</font> <a href="measure6.html#K4" title="MEASURE6:func.4">++</a> <font color="Maroon" title="c1">A</font>
 <b>by </b><i><a class="txt" href="#E3:52_1_2"><i><font color="Green" title="E34">W</font></i></a>, <a class="ref" href="measure6.html#D6" target="_self" title="MEASURE6:def.6">Def6</a></i>;<br/><a NAME="E6:52_1_2"/><b>then </b><i><font color="Green" title="E35">X</font></i>: 
not <font color="Maroon" title="c2">x</font> <a href="measure6.html#K4" title="MEASURE6:func.4">++</a> <font color="Maroon" title="c1">A</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/><b>reconsider </b><font color="Maroon" title="c5">y</font> = <font color="Maroon" title="c2">x</font> as   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> <b>by </b><i><a class="ref" href="xxreal_0.html#D1" title="XXREAL_0:def.1">XXREAL_0:def 1</a></i>;<br/><a NAME="E8:52_1_2"/><i><font color="Green" title="E36">Y</font></i>: 
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 <b>by </b><i><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/><b>thus </b> <a href="measure5.html#K2" title="MEASURE5:func.2">diameter</a> <font color="Maroon" title="c1">A</font> = 
<span class="p1">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <font color="Maroon" title="c1">A</font></span>)</span>
<b>by </b><i><a class="txt" href="#E1:52_1_2"><i><font color="Green" title="E33">S</font></i></a>, <a class="ref" href="measure5.html#D10" title="MEASURE5:def.10">MEASURE5:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="measure6.html#T10" target="_self" title="MEASURE6:th.10">Th10</a>, <a class="txt" href="#E8:52_1_2"><i><font color="Green" title="E36">Y</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="measure6.html#K4" title="MEASURE6:func.4">++</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="measure6.html#T67" target="_self" title="MEASURE6:th.67">Tx1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="measure6.html#K4" title="MEASURE6:func.4">++</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="measure6.html#K4" title="MEASURE6:func.4">++</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="measure6.html#T68" target="_self" title="MEASURE6:th.68">Tx2</a></i>
<br/>.= 
 <a href="measure5.html#K2" title="MEASURE5:func.2">diameter</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="measure6.html#K4" title="MEASURE6:func.4">++</a> <font color="Maroon" title="c1">A</font></span>)</span>
<b>by </b><i><a class="txt" href="#E6:52_1_2"><i><font color="Green" title="E35">X</font></i></a>, <a class="ref" href="measure5.html#D10" title="MEASURE5:def.10">MEASURE5:def 10</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

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