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<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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#K2" title="MEASURE6:func.2">++</a> <font color="Maroon" title="c1">A</font></span>)</span></span><br/><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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#K2" title="MEASURE6:func.2">++</a> <font color="Maroon" title="c1">A</font></span>)</span></span><br/>



<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:68_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="68_1_1"><span class="kw">suppose </span></a><a NAME="E1:68_1_1"/><span class="lab"><font color="Green" title="E13">A1</font></span>: 
<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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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#K2" title="MEASURE6:func.2">++</a> <font color="Maroon" title="c1">A</font></span>)</span></span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:68_1_1_1"/>
not <font color="Maroon" title="c2">x</font> <a href="measure6.html#K2" title="MEASURE6:func.2">++</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">then </span><span class="kw">consider </span><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> <span class="kw"> such that </span><br/><a NAME="E3:68_1_1_1"/><span class="lab"><font color="Green" title="E14">A2</font></span>: 
<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#K2" title="MEASURE6:func.2">++</a> <font color="Maroon" title="c1">A</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="membered.html#T9" title="MEMBERED:th.9">MEMBERED:9</a></span>;<br/><span class="kw">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></span>;<br/><a NAME="E5:68_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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:68_1_1_1"><span class="lab"><font color="Green" title="E14">A2</font></span></a>, <a class="ref" href="measure6.html#D6" target="_self" title="MEASURE6:def.6">Def6</a></span>;<br/><span class="kw">hence </span><a NAME="E6:68_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:68_1_1"><span class="lab"><font color="Green" title="E13">A1</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:68_1_1"/>
 <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#K2" title="MEASURE6:func.2">++</a> <font color="Maroon" title="c1">A</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:68_1_1"><span class="lab"><font color="Green" title="E13">A1</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_1_2"><span class="kw">suppose </span></a><a NAME="E1:68_1_2"/><span class="lab"><font color="Green" title="E13">A3</font></span>: 
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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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#K2" title="MEASURE6:func.2">++</a> <font color="Maroon" title="c1">A</font></span>)</span></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><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> <span class="kw"> such that </span><br/><a NAME="E3:68_1_2"/><span class="lab"><font color="Green" title="E14">A4</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="membered.html#T9" title="MEMBERED:th.9">MEMBERED:9</a></span>;<br/><span class="kw">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></span>;<br/><a NAME="E5:68_1_2"/><span class="lab"><font color="Green" title="E15">A5</font></span>: 
<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#K2" title="MEASURE6:func.2">++</a> <font color="Maroon" title="c1">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:68_1_2"><span class="lab"><font color="Green" title="E14">A4</font></span></a>, <a class="ref" href="measure6.html#D6" target="_self" title="MEASURE6:def.6">Def6</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c5">y</font> = <font color="Maroon" title="c2">x</font> as   <a href="supinf_1.html#NM1" title="SUPINF_1:NM.1">R_eal</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#D1" title="XXREAL_0:def.1">XXREAL_0:def 1</a></span>;<br/><span class="kw">thus </span> <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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:68_1_2"><span class="lab"><font color="Green" title="E13">A3</font></span></a>, <a class="ref" href="measure5.html#D10" title="MEASURE5:def.10">MEASURE5:def 10</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_3.html#T34" title="XXREAL_3:th.34">XXREAL_3:34</a></span>
<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#K2" title="MEASURE6:func.2">++</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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="measure6.html#T67" target="_self" title="MEASURE6:th.67">Th67</a></span>
<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#K2" title="MEASURE6:func.2">++</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#K2" title="MEASURE6:func.2">++</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="measure6.html#T68" target="_self" title="MEASURE6:th.68">Th68</a></span>
<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#K2" title="MEASURE6:func.2">++</a> <font color="Maroon" title="c1">A</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:68_1_2"><span class="lab"><font color="Green" title="E15">A5</font></span></a>, <a class="ref" href="measure5.html#D10" title="MEASURE5:def.10">MEASURE5:def 10</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

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