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<span class="kw">let </span><font color="Maroon" title="c1">A</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="measure5.html#NM1" title="MEASURE5:NM.1">Interval</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">  for <font color="Olive" title="b1">x</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Maroon" title="c1">A</font> is  <a href="measure5.html#V1" title="MEASURE5:attr.1">open_interval</a>  holds <br/><font color="Olive" title="b1">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> is  <a href="measure5.html#V1" title="MEASURE5:attr.1">open_interval</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">x</font> be   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</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="c2">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Maroon" title="c1">A</font> is  <a href="measure5.html#V1" title="MEASURE5:attr.1">open_interval</a>  implies <font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> is  <a href="measure5.html#V1" title="MEASURE5:attr.1">open_interval</a>  )</span><br/>



<span class="kw">assume </span><a NAME="E1:6"/><span class="lab"><font color="Green" title="E7">A1</font></span>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</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"> (  not <font color="Maroon" title="c1">A</font> is  <a href="measure5.html#V1" title="MEASURE5:attr.1">open_interval</a>  or <font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> is  <a href="measure5.html#V1" title="MEASURE5:attr.1">open_interval</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E2:6"/><span class="lab"><font color="Green" title="E8">A2</font></span>: 
<font color="Maroon" title="c1">A</font> is  <a href="measure5.html#V1" title="MEASURE5:attr.1">open_interval</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="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> is  <a href="measure5.html#V1" title="MEASURE5:attr.1">open_interval</a> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">b</font> being   <a href="supinf_1.html#NM1" title="SUPINF_1:NM.1">R_eal</a><span class="kw"> such that </span><br/><a NAME="E4:6"/><span class="lab"><font color="Green" title="E9">A3</font></span>: 
<font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">].</a><span class="default"><font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.[</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="measure5.html#D2" onmouseover="rs('measure5/D2')" onmouseout="rh()">MEASURE5:def 2</a></span>;<br/>
<a NAME="E5:6"/><span class="lab"><font color="Green" title="E10">A4</font></span>: 
<font color="Maroon" title="c3">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:6"><span class="lab"><font color="Green" title="E9">A3</font></span></a>, <a class="ref" href="xxreal_1.html#T28" onmouseover="rs('xxreal_1/T28')" onmouseout="rh()">XXREAL_1:28</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="6_1"><span class="kw">now </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"> ( ( <font color="Maroon" title="c2">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> is  <a href="measure5.html#V1" title="MEASURE5:attr.1">open_interval</a>  ) or ( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> is  <a href="measure5.html#V1" title="MEASURE5:attr.1">open_interval</a>  ) or (  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">x</font> &amp; <font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> is  <a href="measure5.html#V1" title="MEASURE5:attr.1">open_interval</a>  ) )</span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:6_1_1"/>
( <font color="Maroon" title="c2">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">x</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_1_1"><span class="kw">case </span></a><a NAME="E1:6_1_1_1"/><span class="lab"><font color="Green" title="E11">A5</font></span>: 
<font color="Maroon" title="c2">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> is  <a href="measure5.html#V1" title="MEASURE5:attr.1">open_interval</a> </span><br/><div class="add"><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="6_1_1_1_1"><span class="kw">now </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"> ( ( <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  &amp; <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  &amp; <font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> is  <a href="measure5.html#V1" title="MEASURE5:attr.1">open_interval</a>  ) or ( <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  &amp; <font color="Maroon" title="c4">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  &amp; <font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> is  <a href="measure5.html#V1" title="MEASURE5:attr.1">open_interval</a>  ) or ( <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  &amp; <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  &amp; <font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> is  <a href="measure5.html#V1" title="MEASURE5:attr.1">open_interval</a>  ) or ( <font color="Maroon" title="c3">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  &amp; <font color="Maroon" title="c4">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  &amp; <font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> is  <a href="measure5.html#V1" title="MEASURE5:attr.1">open_interval</a>  ) or ( <font color="Maroon" title="c3">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  &amp; <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  &amp; <font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> is  <a href="measure5.html#V1" title="MEASURE5:attr.1">open_interval</a>  ) or ( <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  &amp; <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  &amp; <font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> is  <a href="measure5.html#V1" title="MEASURE5:attr.1">open_interval</a>  ) )</span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_1_1_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:6_1_1_1_1_1"/>
( ( <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  &amp; <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  ) or ( <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  &amp; <font color="Maroon" title="c4">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  ) or ( <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  &amp; <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  ) or ( <font color="Maroon" title="c3">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  &amp; <font color="Maroon" title="c4">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  ) or ( <font color="Maroon" title="c3">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  &amp; <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  ) or ( <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  &amp; <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:6"><span class="lab"><font color="Green" title="E10">A4</font></span></a>, <a class="ref" href="urysohn2.html#T5" target="_self" onmouseover="rs('urysohn2/T5')" onmouseout="rh()">Th5</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_1_1_1_1_1"><span class="kw">case </span></a><a NAME="E1:6_1_1_1_1_1_1"/>
( <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  &amp; <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</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="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> is  <a href="measure5.html#V1" title="MEASURE5:attr.1">open_interval</a> </span><br/><div class="add"><a NAME="E2:6_1_1_1_1_1_1"/><span class="kw">then </span>
<span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">].</a><span class="default"><font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.[</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><a NAME="E3:6_1_1_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:6"><span class="lab"><font color="Green" title="E9">A3</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:6_1_1_1_1_1_1"/>
<font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> is  <a href="measure5.html#V1" title="MEASURE5:attr.1">open_interval</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"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_1_1_1_1_2"><span class="kw">case </span></a><a NAME="E1:6_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E12">A6</font></span>: 
( <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  &amp; <font color="Maroon" title="c4">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</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="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> is  <a href="measure5.html#V1" title="MEASURE5:attr.1">open_interval</a> </span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">s</font> = <font color="Maroon" title="c4">b</font> as   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> ;<br/><span class="kw">set </span><font color="Maroon" title="c6">c</font> =  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> ;<br/><a NAME="E3:6_1_1_1_1_1_2"/>
<font color="Maroon" title="c2">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">s</font> is   <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" onmouseover="rs('xxreal_0/D1')" onmouseout="rh()">XXREAL_0:def 1</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">d</font> being   <a href="supinf_1.html#NM1" title="SUPINF_1:NM.1">R_eal</a><span class="kw"> such that </span><br/><a NAME="E5:6_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E13">A7</font></span>: 
<font color="Maroon" title="c7">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">s</font>
 ;<br/><a NAME="E6:6_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E14">A8</font></span>: 
<span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">].</a><span class="default"><font color="Maroon" title="c7">d</font>,<a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a></span><a href="measure5.html#K1" title="MEASURE5:func.1">.[</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="6_1_1_1_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">q</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</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 <font color="Maroon" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">].</a><span class="default"><font color="Maroon" title="c7">d</font>,<a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a></span><a href="measure5.html#K1" title="MEASURE5:func.1">.[</a></span> or <font color="Maroon" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:6_1_1_1_1_1_2_1"/><span class="lab"><font color="Green" title="E15">A9</font></span>: 
<font color="Maroon" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">].</a><span class="default"><font color="Maroon" title="c7">d</font>,<a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a></span><a href="measure5.html#K1" title="MEASURE5:func.1">.[</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"> <font color="Maroon" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">q</font> = <font color="Maroon" title="c8">q</font> as   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">q2</font> = <font color="Maroon" title="c9">q</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c2">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" onmouseover="rs('xreal_0/D1')" onmouseout="rh()">XREAL_0:def 1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">q1</font> = <font color="Maroon" title="c9">q</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" onmouseover="rs('xxreal_0/D1')" onmouseout="rh()">XXREAL_0:def 1</a></span>;<br/>
<a NAME="E5:6_1_1_1_1_1_2_1"/><span class="lab"><font color="Green" title="E16">A10</font></span>: 
<font color="Maroon" title="c7">d</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">q1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:6_1_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E15">A9</font></span></a>, <a class="ref" href="measure5.html#D1" onmouseover="rs('measure5/D1')" onmouseout="rh()">MEASURE5:def 1</a></span>;<br/>
<a NAME="E6:6_1_1_1_1_1_2_1"/><span class="lab"><font color="Green" title="E17">A11</font></span>: 
<font color="Maroon" title="c10">q2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="6_1_1_1_1_1_2_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c12">q3</font> = <font color="Maroon" title="c10">q2</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" onmouseover="rs('xxreal_0/D1')" onmouseout="rh()">XXREAL_0:def 1</a></span>;<br/>
<a NAME="E2:6_1_1_1_1_1_2_1_1"/>
<font color="Maroon" title="c2">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c10">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:6"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="ref" href="xcmplx_1.html#T87" onmouseover="rs('xcmplx_1/T87')" onmouseout="rh()">XCMPLX_1:87</a></span>;<br/>
<a NAME="E3:6_1_1_1_1_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E18">A12</font></span>: 
<font color="Maroon" title="c12">q3</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:6_1_1_1"><span class="lab"><font color="Green" title="E11">A5</font></span></a>, <a class="txt" href="#E5:6_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E13">A7</font></span></a>, <a class="txt" href="#E5:6_1_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E16">A10</font></span></a>, <a class="ref" href="xreal_1.html#T65" onmouseover="rs('xreal_1/T65')" onmouseout="rh()">XREAL_1:65</a></span>;<br/>
<a NAME="E4:6_1_1_1_1_1_2_1_1"/>
<font color="Maroon" title="c3">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">q3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:6_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E12">A6</font></span></a>, <a class="ref" href="xxreal_0.html#T12" onmouseover="rs('xxreal_0/T12')" onmouseout="rh()">XXREAL_0:12</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:6_1_1_1_1_1_2_1_1"/>
<font color="Maroon" title="c10">q2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:6"><span class="lab"><font color="Green" title="E9">A3</font></span></a>, <a class="txt" href="#E3:6_1_1_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E18">A12</font></span></a>, <a class="ref" href="measure5.html#D1" onmouseover="rs('measure5/D1')" onmouseout="rh()">MEASURE5:def 1</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/>


</div><span class="kw">end;</span></div>
<a NAME="E7:6_1_1_1_1_1_2_1"/>
<font color="Maroon" title="c9">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">q</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c2">x</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:6"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="ref" href="xcmplx_1.html#T87" onmouseover="rs('xcmplx_1/T87')" onmouseout="rh()">XCMPLX_1:87</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:6_1_1_1_1_1_2_1"/>
<font color="Maroon" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:6_1_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E17">A11</font></span></a>, <a class="ref" href="member_1.html#T193" onmouseover="rs('member_1/T193')" onmouseout="rh()">MEMBER_1:193</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/>


</div><span class="kw">end;</span></div><a NAME="E7:6_1_1_1_1_1_2"/>
<font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">].</a><span class="default"><font color="Maroon" title="c7">d</font>,<a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a></span><a href="measure5.html#K1" title="MEASURE5:func.1">.[</a></span>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="6_1_1_1_1_1_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">q</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</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 <font color="Maroon" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> or <font color="Maroon" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">].</a><span class="default"><font color="Maroon" title="c7">d</font>,<a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a></span><a href="measure5.html#K1" title="MEASURE5:func.1">.[</a></span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:6_1_1_1_1_1_2_2"/><span class="lab"><font color="Green" title="E15">A13</font></span>: 
<font color="Maroon" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</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"> <font color="Maroon" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">].</a><span class="default"><font color="Maroon" title="c7">d</font>,<a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a></span><a href="measure5.html#K1" title="MEASURE5:func.1">.[</a></span></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">q</font> = <font color="Maroon" title="c8">q</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c10">z2</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E4:6_1_1_1_1_1_2_2"/><span class="lab"><font color="Green" title="E16">A14</font></span>: 
<font color="Maroon" title="c10">z2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 <span class="kw">and </span><br/><a NAME="E5:6_1_1_1_1_1_2_2"/><span class="lab"><font color="Green" title="E17">A15</font></span>: 
<font color="Maroon" title="c9">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c10">z2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:6_1_1_1_1_1_2_2"><span class="lab"><font color="Green" title="E15">A13</font></span></a>, <a class="ref" href="integra2.html#T39" onmouseover="rs('integra2/T39')" onmouseout="rh()">INTEGRA2:39</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">q</font> = <font color="Maroon" title="c9">q</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" onmouseover="rs('xxreal_0/D1')" onmouseout="rh()">XXREAL_0:def 1</a></span>;<br/>
<a NAME="E7:6_1_1_1_1_1_2_2"/><span class="lab"><font color="Green" title="E18">A16</font></span>: 
<font color="Maroon" title="c11">q</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T9" onmouseover="rs('xxreal_0/T9')" onmouseout="rh()">XXREAL_0:9</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c12">z2</font> = <font color="Maroon" title="c10">z2</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" onmouseover="rs('xxreal_0/D1')" onmouseout="rh()">XXREAL_0:def 1</a></span>;<br/>
<a NAME="E9:6_1_1_1_1_1_2_2"/>
<font color="Maroon" title="c12">z2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:6"><span class="lab"><font color="Green" title="E9">A3</font></span></a>, <a class="txt" href="#E4:6_1_1_1_1_1_2_2"><span class="lab"><font color="Green" title="E16">A14</font></span></a>, <a class="ref" href="measure5.html#D1" onmouseover="rs('measure5/D1')" onmouseout="rh()">MEASURE5:def 1</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">r</font>, <font color="Maroon" title="c14">o</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E11:6_1_1_1_1_1_2_2"/><span class="lab"><font color="Green" title="E19">A17</font></span>: 
( <font color="Maroon" title="c13">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">z2</font> &amp; <font color="Maroon" title="c14">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> )
 <span class="kw">and </span><br/><a NAME="E12:6_1_1_1_1_1_2_2"/>
<font color="Maroon" title="c13">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">o</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:6_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E12">A6</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c15">o1</font> = <font color="Maroon" title="c2">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c14">o</font>, <font color="Maroon" title="c16">r1</font> = <font color="Maroon" title="c2">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c13">r</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" onmouseover="rs('xxreal_0/D1')" onmouseout="rh()">XXREAL_0:def 1</a></span>;<br/>
<a NAME="E14:6_1_1_1_1_1_2_2"/>
<font color="Maroon" title="c13">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">o</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:6"><span class="lab"><font color="Green" title="E9">A3</font></span></a>, <a class="txt" href="#E4:6_1_1_1_1_1_2_2"><span class="lab"><font color="Green" title="E16">A14</font></span></a>, <a class="txt" href="#E11:6_1_1_1_1_1_2_2"><span class="lab"><font color="Green" title="E19">A17</font></span></a>, <a class="ref" href="measure5.html#D1" onmouseover="rs('measure5/D1')" onmouseout="rh()">MEASURE5:def 1</a></span>;<br/>
<a NAME="E15:6_1_1_1_1_1_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c15">o1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">r1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:6_1_1_1"><span class="lab"><font color="Green" title="E11">A5</font></span></a>, <a class="ref" href="xreal_1.html#T69" onmouseover="rs('xreal_1/T69')" onmouseout="rh()">XREAL_1:69</a></span>;<br/>
<span class="kw">hence </span><a NAME="E16:6_1_1_1_1_1_2_2"/>
<font color="Maroon" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">].</a><span class="default"><font color="Maroon" title="c7">d</font>,<a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a></span><a href="measure5.html#K1" title="MEASURE5:func.1">.[</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:6_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E13">A7</font></span></a>, <a class="txt" href="#E5:6_1_1_1_1_1_2_2"><span class="lab"><font color="Green" title="E17">A15</font></span></a>, <a class="txt" href="#E11:6_1_1_1_1_1_2_2"><span class="lab"><font color="Green" title="E19">A17</font></span></a>, <a class="txt" href="#E7:6_1_1_1_1_1_2_2"><span class="lab"><font color="Green" title="E18">A16</font></span></a>, <a class="ref" href="measure5.html#D1" onmouseover="rs('measure5/D1')" onmouseout="rh()">MEASURE5:def 1</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/>


</div><span class="kw">end;</span></div><a NAME="E8:6_1_1_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">].</a><span class="default"><font color="Maroon" title="c7">d</font>,<a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a></span><a href="measure5.html#K1" title="MEASURE5:func.1">.[</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:6_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E14">A8</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E9:6_1_1_1_1_1_2"/>
<font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> is  <a href="measure5.html#V1" title="MEASURE5:attr.1">open_interval</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="measure5.html#D2" onmouseover="rs('measure5/D2')" onmouseout="rh()">MEASURE5:def 2</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/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_1_1_1_1_3"><span class="kw">case </span></a><a NAME="E1:6_1_1_1_1_1_3"/>
( <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  &amp; <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</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="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> is  <a href="measure5.html#V1" title="MEASURE5:attr.1">open_interval</a> </span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:6_1_1_1_1_1_3"/>
<font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> is  <a href="measure5.html#V1" title="MEASURE5:attr.1">open_interval</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:6"><span class="lab"><font color="Green" title="E8">A2</font></span></a>, <a class="txt" href="#E4:6"><span class="lab"><font color="Green" title="E9">A3</font></span></a>, <a class="txt" href="#E1:6_1_1_1"><span class="lab"><font color="Green" title="E11">A5</font></span></a>, <a class="ref" href="urysohn2.html#T2" target="_self" onmouseover="rs('urysohn2/T2')" onmouseout="rh()">Th2</a>, <a class="ref" href="xxreal_1.html#T224" onmouseover="rs('xxreal_1/T224')" onmouseout="rh()">XXREAL_1:224</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/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_1_1_1_1_4"><span class="kw">case </span></a><a NAME="E1:6_1_1_1_1_1_4"/><span class="lab"><font color="Green" title="E12">A18</font></span>: 
( <font color="Maroon" title="c3">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  &amp; <font color="Maroon" title="c4">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</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="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> is  <a href="measure5.html#V1" title="MEASURE5:attr.1">open_interval</a> </span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">s</font> = <font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c6">r</font> = <font color="Maroon" title="c4">b</font> as   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c7">d</font> = <font color="Maroon" title="c2">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">s</font>, <font color="Maroon" title="c8">g</font> = <font color="Maroon" title="c2">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">r</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" onmouseover="rs('xxreal_0/D1')" onmouseout="rh()">XXREAL_0:def 1</a></span>;<br/><a NAME="E4:6_1_1_1_1_1_4"/><span class="lab"><font color="Green" title="E13">A19</font></span>: 
<span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">].</a><span class="default"><font color="Maroon" title="c8">g</font>,<font color="Maroon" title="c7">d</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.[</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="6_1_1_1_1_1_4_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">q</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</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 <font color="Maroon" title="c9">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">].</a><span class="default"><font color="Maroon" title="c8">g</font>,<font color="Maroon" title="c7">d</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.[</a></span> or <font color="Maroon" title="c9">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:6_1_1_1_1_1_4_1"/><span class="lab"><font color="Green" title="E14">A20</font></span>: 
<font color="Maroon" title="c9">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">].</a><span class="default"><font color="Maroon" title="c8">g</font>,<font color="Maroon" title="c7">d</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.[</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"> <font color="Maroon" title="c9">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">q</font> = <font color="Maroon" title="c9">q</font> as   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c11">q2</font> = <font color="Maroon" title="c10">q</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c2">x</font>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c12">q1</font> = <font color="Maroon" title="c10">q</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" onmouseover="rs('xxreal_0/D1')" onmouseout="rh()">XXREAL_0:def 1</a></span>;<br/>
<a NAME="E4:6_1_1_1_1_1_4_1"/><span class="lab"><font color="Green" title="E15">A21</font></span>: 
<font color="Maroon" title="c12">q1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">d</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:6_1_1_1_1_1_4_1"><span class="lab"><font color="Green" title="E14">A20</font></span></a>, <a class="ref" href="measure5.html#D1" onmouseover="rs('measure5/D1')" onmouseout="rh()">MEASURE5:def 1</a></span>;<br/>
<a NAME="E5:6_1_1_1_1_1_4_1"/><span class="lab"><font color="Green" title="E16">A22</font></span>: 
<font color="Maroon" title="c8">g</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">q1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:6_1_1_1_1_1_4_1"><span class="lab"><font color="Green" title="E14">A20</font></span></a>, <a class="ref" href="measure5.html#D1" onmouseover="rs('measure5/D1')" onmouseout="rh()">MEASURE5:def 1</a></span>;<br/>
<a NAME="E6:6_1_1_1_1_1_4_1"/><span class="lab"><font color="Green" title="E17">A23</font></span>: 
<font color="Maroon" title="c10">q</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c2">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="6_1_1_1_1_1_4_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c13">q3</font> = <font color="Maroon" title="c10">q</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <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" onmouseover="rs('xxreal_0/D1')" onmouseout="rh()">XXREAL_0:def 1</a></span>;<br/>
<a NAME="E2:6_1_1_1_1_1_4_1_1"/>
<font color="Maroon" title="c2">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">q</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">q</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:6"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="ref" href="xcmplx_1.html#T87" onmouseover="rs('xcmplx_1/T87')" onmouseout="rh()">XCMPLX_1:87</a></span>;<br/>
<a NAME="E3:6_1_1_1_1_1_4_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E18">A24</font></span>: 
<font color="Maroon" title="c3">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">q3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:6_1_1_1"><span class="lab"><font color="Green" title="E11">A5</font></span></a>, <a class="txt" href="#E4:6_1_1_1_1_1_4_1"><span class="lab"><font color="Green" title="E15">A21</font></span></a>, <a class="ref" href="xreal_1.html#T65" onmouseover="rs('xreal_1/T65')" onmouseout="rh()">XREAL_1:65</a></span>;<br/>
<a NAME="E4:6_1_1_1_1_1_4_1_1"/>
<font color="Maroon" title="c10">q</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c2">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">r</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c2">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:6_1_1_1"><span class="lab"><font color="Green" title="E11">A5</font></span></a>, <a class="txt" href="#E5:6_1_1_1_1_1_4_1"><span class="lab"><font color="Green" title="E16">A22</font></span></a>, <a class="ref" href="xreal_1.html#T75" onmouseover="rs('xreal_1/T75')" onmouseout="rh()">XREAL_1:75</a></span>;<br/>
<a NAME="E5:6_1_1_1_1_1_4_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c13">q3</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:6_1_1_1"><span class="lab"><font color="Green" title="E11">A5</font></span></a>, <a class="ref" href="xcmplx_1.html#T89" onmouseover="rs('xcmplx_1/T89')" onmouseout="rh()">XCMPLX_1:89</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:6_1_1_1_1_1_4_1_1"/>
<font color="Maroon" title="c10">q</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c2">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:6"><span class="lab"><font color="Green" title="E9">A3</font></span></a>, <a class="txt" href="#E3:6_1_1_1_1_1_4_1_1"><span class="lab"><font color="Green" title="E18">A24</font></span></a>, <a class="ref" href="measure5.html#D1" onmouseover="rs('measure5/D1')" onmouseout="rh()">MEASURE5:def 1</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/>


</div><span class="kw">end;</span></div>
<a NAME="E7:6_1_1_1_1_1_4_1"/>
<font color="Maroon" title="c10">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">q</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c2">x</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:6"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="ref" href="xcmplx_1.html#T87" onmouseover="rs('xcmplx_1/T87')" onmouseout="rh()">XCMPLX_1:87</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:6_1_1_1_1_1_4_1"/>
<font color="Maroon" title="c9">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:6_1_1_1_1_1_4_1"><span class="lab"><font color="Green" title="E17">A23</font></span></a>, <a class="ref" href="member_1.html#T193" onmouseover="rs('member_1/T193')" onmouseout="rh()">MEMBER_1:193</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/>


</div><span class="kw">end;</span></div><a NAME="E5:6_1_1_1_1_1_4"/>
<font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">].</a><span class="default"><font color="Maroon" title="c8">g</font>,<font color="Maroon" title="c7">d</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.[</a></span>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="6_1_1_1_1_1_4_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">q</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</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 <font color="Maroon" title="c9">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> or <font color="Maroon" title="c9">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">].</a><span class="default"><font color="Maroon" title="c8">g</font>,<font color="Maroon" title="c7">d</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.[</a></span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:6_1_1_1_1_1_4_2"/><span class="lab"><font color="Green" title="E14">A25</font></span>: 
<font color="Maroon" title="c9">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</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"> <font color="Maroon" title="c9">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">].</a><span class="default"><font color="Maroon" title="c8">g</font>,<font color="Maroon" title="c7">d</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.[</a></span></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">q</font> = <font color="Maroon" title="c9">q</font> as   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c11">z2</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E4:6_1_1_1_1_1_4_2"/><span class="lab"><font color="Green" title="E15">A26</font></span>: 
<font color="Maroon" title="c11">z2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 <span class="kw">and </span><br/><a NAME="E5:6_1_1_1_1_1_4_2"/><span class="lab"><font color="Green" title="E16">A27</font></span>: 
<font color="Maroon" title="c10">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c11">z2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:6_1_1_1_1_1_4_2"><span class="lab"><font color="Green" title="E14">A25</font></span></a>, <a class="ref" href="integra2.html#T39" onmouseover="rs('integra2/T39')" onmouseout="rh()">INTEGRA2:39</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c12">z2</font> = <font color="Maroon" title="c11">z2</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" onmouseover="rs('xxreal_0/D1')" onmouseout="rh()">XXREAL_0:def 1</a></span>;<br/>
<a NAME="E7:6_1_1_1_1_1_4_2"/>
<font color="Maroon" title="c3">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">z2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:6"><span class="lab"><font color="Green" title="E9">A3</font></span></a>, <a class="txt" href="#E4:6_1_1_1_1_1_4_2"><span class="lab"><font color="Green" title="E15">A26</font></span></a>, <a class="ref" href="measure5.html#D1" onmouseover="rs('measure5/D1')" onmouseout="rh()">MEASURE5:def 1</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">1o</font>, <font color="Maroon" title="c14">1ra</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E9:6_1_1_1_1_1_4_2"/><span class="lab"><font color="Green" title="E17">A28</font></span>: 
( <font color="Maroon" title="c13">1o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> &amp; <font color="Maroon" title="c14">1ra</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">z2</font> )
 <span class="kw">and </span><br/><a NAME="E10:6_1_1_1_1_1_4_2"/>
<font color="Maroon" title="c13">1o</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">1ra</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:6_1_1_1_1_1_4"><span class="lab"><font color="Green" title="E12">A18</font></span></a></span>;<br/>
<a NAME="E11:6_1_1_1_1_1_4_2"/>
<font color="Maroon" title="c12">z2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:6"><span class="lab"><font color="Green" title="E9">A3</font></span></a>, <a class="txt" href="#E4:6_1_1_1_1_1_4_2"><span class="lab"><font color="Green" title="E15">A26</font></span></a>, <a class="ref" href="measure5.html#D1" onmouseover="rs('measure5/D1')" onmouseout="rh()">MEASURE5:def 1</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c15">2o</font>, <font color="Maroon" title="c16">2r</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E13:6_1_1_1_1_1_4_2"/><span class="lab"><font color="Green" title="E18">A29</font></span>: 
( <font color="Maroon" title="c15">2o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">z2</font> &amp; <font color="Maroon" title="c16">2r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> )
 <span class="kw">and </span><br/><a NAME="E14:6_1_1_1_1_1_4_2"/>
<font color="Maroon" title="c15">2o</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">2r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:6_1_1_1_1_1_4"><span class="lab"><font color="Green" title="E12">A18</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c17">1o1</font> = <font color="Maroon" title="c2">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c13">1o</font>, <font color="Maroon" title="c18">1r1</font> = <font color="Maroon" title="c2">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c14">1ra</font>, <font color="Maroon" title="c19">2o1</font> = <font color="Maroon" title="c2">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c15">2o</font>, <font color="Maroon" title="c20">2r1</font> = <font color="Maroon" title="c2">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c16">2r</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" onmouseover="rs('xxreal_0/D1')" onmouseout="rh()">XXREAL_0:def 1</a></span>;<br/>
<a NAME="E16:6_1_1_1_1_1_4_2"/>
<font color="Maroon" title="c15">2o</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">2r</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:6"><span class="lab"><font color="Green" title="E9">A3</font></span></a>, <a class="txt" href="#E4:6_1_1_1_1_1_4_2"><span class="lab"><font color="Green" title="E15">A26</font></span></a>, <a class="txt" href="#E13:6_1_1_1_1_1_4_2"><span class="lab"><font color="Green" title="E18">A29</font></span></a>, <a class="ref" href="measure5.html#D1" onmouseover="rs('measure5/D1')" onmouseout="rh()">MEASURE5:def 1</a></span>;<br/>
<a NAME="E17:6_1_1_1_1_1_4_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E19">A30</font></span>: 
<font color="Maroon" title="c20">2r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c19">2o1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:6_1_1_1"><span class="lab"><font color="Green" title="E11">A5</font></span></a>, <a class="ref" href="xreal_1.html#T69" onmouseover="rs('xreal_1/T69')" onmouseout="rh()">XREAL_1:69</a></span>;<br/>
<a NAME="E18:6_1_1_1_1_1_4_2"/>
<font color="Maroon" title="c13">1o</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">1ra</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:6"><span class="lab"><font color="Green" title="E9">A3</font></span></a>, <a class="txt" href="#E4:6_1_1_1_1_1_4_2"><span class="lab"><font color="Green" title="E15">A26</font></span></a>, <a class="txt" href="#E9:6_1_1_1_1_1_4_2"><span class="lab"><font color="Green" title="E17">A28</font></span></a>, <a class="ref" href="measure5.html#D1" onmouseover="rs('measure5/D1')" onmouseout="rh()">MEASURE5:def 1</a></span>;<br/>
<a NAME="E19:6_1_1_1_1_1_4_2"/><span class="kw">then </span>
<font color="Maroon" title="c18">1r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c17">1o1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:6_1_1_1"><span class="lab"><font color="Green" title="E11">A5</font></span></a>, <a class="ref" href="xreal_1.html#T69" onmouseover="rs('xreal_1/T69')" onmouseout="rh()">XREAL_1:69</a></span>;<br/>
<span class="kw">hence </span><a NAME="E20:6_1_1_1_1_1_4_2"/>
<font color="Maroon" title="c9">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">].</a><span class="default"><font color="Maroon" title="c8">g</font>,<font color="Maroon" title="c7">d</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.[</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:6_1_1_1_1_1_4_2"><span class="lab"><font color="Green" title="E16">A27</font></span></a>, <a class="txt" href="#E9:6_1_1_1_1_1_4_2"><span class="lab"><font color="Green" title="E17">A28</font></span></a>, <a class="txt" href="#E13:6_1_1_1_1_1_4_2"><span class="lab"><font color="Green" title="E18">A29</font></span></a>, <a class="txt" href="#E17:6_1_1_1_1_1_4_2"><span class="lab"><font color="Green" title="E19">A30</font></span></a>, <a class="ref" href="measure5.html#D1" onmouseover="rs('measure5/D1')" onmouseout="rh()">MEASURE5:def 1</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/>


</div><span class="kw">end;</span></div><a NAME="E6:6_1_1_1_1_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">].</a><span class="default"><font color="Maroon" title="c8">g</font>,<font color="Maroon" title="c7">d</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.[</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:6_1_1_1_1_1_4"><span class="lab"><font color="Green" title="E13">A19</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E7:6_1_1_1_1_1_4"/>
<font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> is  <a href="measure5.html#V1" title="MEASURE5:attr.1">open_interval</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="measure5.html#D2" onmouseover="rs('measure5/D2')" onmouseout="rh()">MEASURE5:def 2</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/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_1_1_1_1_5"><span class="kw">case </span></a><a NAME="E1:6_1_1_1_1_1_5"/><span class="lab"><font color="Green" title="E12">A31</font></span>: 
( <font color="Maroon" title="c3">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  &amp; <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</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="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> is  <a href="measure5.html#V1" title="MEASURE5:attr.1">open_interval</a> </span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">s</font> = <font color="Maroon" title="c3">a</font> as   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> ;<br/><span class="kw">set </span><font color="Maroon" title="c6">c</font> =  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c7">d</font> = <font color="Maroon" title="c2">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">s</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" onmouseover="rs('xxreal_0/D1')" onmouseout="rh()">XXREAL_0:def 1</a></span>;<br/><a NAME="E4:6_1_1_1_1_1_5"/><span class="lab"><font color="Green" title="E13">A32</font></span>: 
<span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">].</a><span class="default"><a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>,<font color="Maroon" title="c7">d</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.[</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="6_1_1_1_1_1_5_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">q</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</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 <font color="Maroon" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">].</a><span class="default"><a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>,<font color="Maroon" title="c7">d</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.[</a></span> or <font color="Maroon" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:6_1_1_1_1_1_5_1"/><span class="lab"><font color="Green" title="E14">A33</font></span>: 
<font color="Maroon" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">].</a><span class="default"><a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>,<font color="Maroon" title="c7">d</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.[</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"> <font color="Maroon" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">q</font> = <font color="Maroon" title="c8">q</font> as   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">q2</font> = <font color="Maroon" title="c9">q</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c2">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" onmouseover="rs('xreal_0/D1')" onmouseout="rh()">XREAL_0:def 1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">q1</font> = <font color="Maroon" title="c9">q</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" onmouseover="rs('xxreal_0/D1')" onmouseout="rh()">XXREAL_0:def 1</a></span>;<br/>
<a NAME="E5:6_1_1_1_1_1_5_1"/><span class="lab"><font color="Green" title="E15">A34</font></span>: 
<font color="Maroon" title="c9">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">q</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c2">x</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:6"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="ref" href="xcmplx_1.html#T87" onmouseover="rs('xcmplx_1/T87')" onmouseout="rh()">XCMPLX_1:87</a></span>;<br/>
<a NAME="E6:6_1_1_1_1_1_5_1"/>
<font color="Maroon" title="c10">q2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="6_1_1_1_1_1_5_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c12">q3</font> = <font color="Maroon" title="c10">q2</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" onmouseover="rs('xxreal_0/D1')" onmouseout="rh()">XXREAL_0:def 1</a></span>;<br/>
<a NAME="E2:6_1_1_1_1_1_5_1_1"/>
<font color="Maroon" title="c11">q1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">d</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:6_1_1_1_1_1_5_1"><span class="lab"><font color="Green" title="E14">A33</font></span></a>, <a class="ref" href="measure5.html#D1" onmouseover="rs('measure5/D1')" onmouseout="rh()">MEASURE5:def 1</a></span>;<br/>
<a NAME="E3:6_1_1_1_1_1_5_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c10">q2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:6_1_1_1_1_1_5_1"><span class="lab"><font color="Green" title="E14">A33</font></span></a>, <a class="txt" href="#E5:6_1_1_1_1_1_5_1"><span class="lab"><font color="Green" title="E15">A34</font></span></a>, <a class="ref" href="measure5.html#D1" onmouseover="rs('measure5/D1')" onmouseout="rh()">MEASURE5:def 1</a></span>;<br/>
<a NAME="E4:6_1_1_1_1_1_5_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E16">A35</font></span>: 
<font color="Maroon" title="c3">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">q3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:6_1_1_1"><span class="lab"><font color="Green" title="E11">A5</font></span></a>, <a class="ref" href="xreal_1.html#T65" onmouseover="rs('xreal_1/T65')" onmouseout="rh()">XREAL_1:65</a></span>;<br/>
<a NAME="E5:6_1_1_1_1_1_5_1_1"/>
<font color="Maroon" title="c12">q3</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:6_1_1_1_1_1_5"><span class="lab"><font color="Green" title="E12">A31</font></span></a>, <a class="ref" href="xxreal_0.html#T9" onmouseover="rs('xxreal_0/T9')" onmouseout="rh()">XXREAL_0:9</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:6_1_1_1_1_1_5_1_1"/>
<font color="Maroon" title="c10">q2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:6"><span class="lab"><font color="Green" title="E9">A3</font></span></a>, <a class="txt" href="#E4:6_1_1_1_1_1_5_1_1"><span class="lab"><font color="Green" title="E16">A35</font></span></a>, <a class="ref" href="measure5.html#D1" onmouseover="rs('measure5/D1')" onmouseout="rh()">MEASURE5:def 1</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/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E7:6_1_1_1_1_1_5_1"/>
<font color="Maroon" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:6_1_1_1_1_1_5_1"><span class="lab"><font color="Green" title="E15">A34</font></span></a>, <a class="ref" href="member_1.html#T193" onmouseover="rs('member_1/T193')" onmouseout="rh()">MEMBER_1:193</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/>


</div><span class="kw">end;</span></div><a NAME="E5:6_1_1_1_1_1_5"/>
<font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">].</a><span class="default"><a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>,<font color="Maroon" title="c7">d</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.[</a></span>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="6_1_1_1_1_1_5_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">q</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</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 <font color="Maroon" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> or <font color="Maroon" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">].</a><span class="default"><a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>,<font color="Maroon" title="c7">d</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.[</a></span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:6_1_1_1_1_1_5_2"/><span class="lab"><font color="Green" title="E14">A36</font></span>: 
<font color="Maroon" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</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"> <font color="Maroon" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">].</a><span class="default"><a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>,<font color="Maroon" title="c7">d</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.[</a></span></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">q</font> = <font color="Maroon" title="c8">q</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c10">z2</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E4:6_1_1_1_1_1_5_2"/><span class="lab"><font color="Green" title="E15">A37</font></span>: 
<font color="Maroon" title="c10">z2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 <span class="kw">and </span><br/><a NAME="E5:6_1_1_1_1_1_5_2"/><span class="lab"><font color="Green" title="E16">A38</font></span>: 
<font color="Maroon" title="c9">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c10">z2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:6_1_1_1_1_1_5_2"><span class="lab"><font color="Green" title="E14">A36</font></span></a>, <a class="ref" href="integra2.html#T39" onmouseover="rs('integra2/T39')" onmouseout="rh()">INTEGRA2:39</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">z2</font> = <font color="Maroon" title="c10">z2</font>, <font color="Maroon" title="c12">q</font> = <font color="Maroon" title="c9">q</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" onmouseover="rs('xxreal_0/D1')" onmouseout="rh()">XXREAL_0:def 1</a></span>;<br/>
<a NAME="E7:6_1_1_1_1_1_5_2"/>
<font color="Maroon" title="c3">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">z2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:6"><span class="lab"><font color="Green" title="E9">A3</font></span></a>, <a class="txt" href="#E4:6_1_1_1_1_1_5_2"><span class="lab"><font color="Green" title="E15">A37</font></span></a>, <a class="ref" href="measure5.html#D1" onmouseover="rs('measure5/D1')" onmouseout="rh()">MEASURE5:def 1</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">o</font>, <font color="Maroon" title="c14">r</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E9:6_1_1_1_1_1_5_2"/><span class="lab"><font color="Green" title="E17">A39</font></span>: 
( <font color="Maroon" title="c13">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> &amp; <font color="Maroon" title="c14">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">z2</font> )
 <span class="kw">and </span><br/><a NAME="E10:6_1_1_1_1_1_5_2"/>
<font color="Maroon" title="c13">o</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:6_1_1_1_1_1_5"><span class="lab"><font color="Green" title="E12">A31</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c15">o1</font> = <font color="Maroon" title="c2">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c13">o</font>, <font color="Maroon" title="c16">r1</font> = <font color="Maroon" title="c2">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c14">r</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" onmouseover="rs('xxreal_0/D1')" onmouseout="rh()">XXREAL_0:def 1</a></span>;<br/>
<a NAME="E12:6_1_1_1_1_1_5_2"/><span class="lab"><font color="Green" title="E18">A40</font></span>: 
 <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">q</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T12" onmouseover="rs('xxreal_0/T12')" onmouseout="rh()">XXREAL_0:12</a></span>;<br/>
<a NAME="E13:6_1_1_1_1_1_5_2"/>
<font color="Maroon" title="c13">o</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:6"><span class="lab"><font color="Green" title="E9">A3</font></span></a>, <a class="txt" href="#E4:6_1_1_1_1_1_5_2"><span class="lab"><font color="Green" title="E15">A37</font></span></a>, <a class="txt" href="#E9:6_1_1_1_1_1_5_2"><span class="lab"><font color="Green" title="E17">A39</font></span></a>, <a class="ref" href="measure5.html#D1" onmouseover="rs('measure5/D1')" onmouseout="rh()">MEASURE5:def 1</a></span>;<br/>
<a NAME="E14:6_1_1_1_1_1_5_2"/><span class="kw">then </span>
<font color="Maroon" title="c16">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">o1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:6_1_1_1"><span class="lab"><font color="Green" title="E11">A5</font></span></a>, <a class="ref" href="xreal_1.html#T69" onmouseover="rs('xreal_1/T69')" onmouseout="rh()">XREAL_1:69</a></span>;<br/>
<span class="kw">hence </span><a NAME="E15:6_1_1_1_1_1_5_2"/>
<font color="Maroon" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">].</a><span class="default"><a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>,<font color="Maroon" title="c7">d</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.[</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:6_1_1_1_1_1_5_2"><span class="lab"><font color="Green" title="E16">A38</font></span></a>, <a class="txt" href="#E9:6_1_1_1_1_1_5_2"><span class="lab"><font color="Green" title="E17">A39</font></span></a>, <a class="txt" href="#E12:6_1_1_1_1_1_5_2"><span class="lab"><font color="Green" title="E18">A40</font></span></a>, <a class="ref" href="measure5.html#D1" onmouseover="rs('measure5/D1')" onmouseout="rh()">MEASURE5:def 1</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/>


</div><span class="kw">end;</span></div><a NAME="E6:6_1_1_1_1_1_5"/><span class="kw">then </span>
<font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">].</a><span class="default"><a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>,<font color="Maroon" title="c7">d</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.[</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:6_1_1_1_1_1_5"><span class="lab"><font color="Green" title="E13">A32</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E7:6_1_1_1_1_1_5"/>
<font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> is  <a href="measure5.html#V1" title="MEASURE5:attr.1">open_interval</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="measure5.html#D2" onmouseover="rs('measure5/D2')" onmouseout="rh()">MEASURE5:def 2</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/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_1_1_1_1_6"><span class="kw">case </span></a><a NAME="E1:6_1_1_1_1_1_6"/>
( <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  &amp; <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</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="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> is  <a href="measure5.html#V1" title="MEASURE5:attr.1">open_interval</a> </span><br/><div class="add"><a NAME="E2:6_1_1_1_1_1_6"/><span class="kw">then </span>
<span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">].</a><span class="default"><font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.[</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><a NAME="E3:6_1_1_1_1_1_6"/><span class="kw">then </span>
<font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:6"><span class="lab"><font color="Green" title="E9">A3</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:6_1_1_1_1_1_6"/>
<font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> is  <a href="measure5.html#V1" title="MEASURE5:attr.1">open_interval</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"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:6_1_1_1"/>
<font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> is  <a href="measure5.html#V1" title="MEASURE5:attr.1">open_interval</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"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_1_2"><span class="kw">case </span></a><a NAME="E1:6_1_1_2"/>
<font color="Maroon" title="c2">x</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="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> is  <a href="measure5.html#V1" title="MEASURE5:attr.1">open_interval</a> </span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:6_1_1_2"/>
<font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> is  <a href="measure5.html#V1" title="MEASURE5:attr.1">open_interval</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:6"><span class="lab"><font color="Green" title="E7">A1</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/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_1_3"><span class="kw">case </span></a><a NAME="E1:6_1_1_3"/><span class="lab"><font color="Green" title="E11">A41</font></span>: 
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">x</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"> <font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> is  <a href="measure5.html#V1" title="MEASURE5:attr.1">open_interval</a> </span><br/><div class="add"><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="6_1_1_3_1"><span class="kw">now </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"> ( ( <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  &amp; <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  &amp; <font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> is  <a href="measure5.html#V1" title="MEASURE5:attr.1">open_interval</a>  ) or ( <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  &amp; <font color="Maroon" title="c4">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  &amp; <font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> is  <a href="measure5.html#V1" title="MEASURE5:attr.1">open_interval</a>  ) or ( <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  &amp; <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  &amp; <font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> is  <a href="measure5.html#V1" title="MEASURE5:attr.1">open_interval</a>  ) or ( <font color="Maroon" title="c3">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  &amp; <font color="Maroon" title="c4">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  &amp; <font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> is  <a href="measure5.html#V1" title="MEASURE5:attr.1">open_interval</a>  ) or ( <font color="Maroon" title="c3">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  &amp; <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  &amp; <font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> is  <a href="measure5.html#V1" title="MEASURE5:attr.1">open_interval</a>  ) or ( <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  &amp; <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  &amp; <font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> is  <a href="measure5.html#V1" title="MEASURE5:attr.1">open_interval</a>  ) )</span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_1_3_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:6_1_1_3_1_1"/>
( ( <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  &amp; <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  ) or ( <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  &amp; <font color="Maroon" title="c4">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  ) or ( <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  &amp; <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  ) or ( <font color="Maroon" title="c3">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  &amp; <font color="Maroon" title="c4">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  ) or ( <font color="Maroon" title="c3">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  &amp; <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  ) or ( <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  &amp; <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:6"><span class="lab"><font color="Green" title="E10">A4</font></span></a>, <a class="ref" href="urysohn2.html#T5" target="_self" onmouseover="rs('urysohn2/T5')" onmouseout="rh()">Th5</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_1_3_1_1_1"><span class="kw">case </span></a><a NAME="E1:6_1_1_3_1_1_1"/>
( <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  &amp; <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</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="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> is  <a href="measure5.html#V1" title="MEASURE5:attr.1">open_interval</a> </span><br/><div class="add"><a NAME="E2:6_1_1_3_1_1_1"/><span class="kw">then </span>
<span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">].</a><span class="default"><font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.[</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><a NAME="E3:6_1_1_3_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:6"><span class="lab"><font color="Green" title="E9">A3</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:6_1_1_3_1_1_1"/>
<font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> is  <a href="measure5.html#V1" title="MEASURE5:attr.1">open_interval</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"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_1_3_1_1_2"><span class="kw">case </span></a><a NAME="E1:6_1_1_3_1_1_2"/><span class="lab"><font color="Green" title="E12">A42</font></span>: 
( <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  &amp; <font color="Maroon" title="c4">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</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="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> is  <a href="measure5.html#V1" title="MEASURE5:attr.1">open_interval</a> </span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">s</font> = <font color="Maroon" title="c4">b</font> as   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> ;<br/><span class="kw">set </span><font color="Maroon" title="c6">c</font> =  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c7">d</font> = <font color="Maroon" title="c2">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">s</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" onmouseover="rs('xxreal_0/D1')" onmouseout="rh()">XXREAL_0:def 1</a></span>;<br/><a NAME="E4:6_1_1_3_1_1_2"/><span class="lab"><font color="Green" title="E13">A43</font></span>: 
<span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">].</a><span class="default"><a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>,<font color="Maroon" title="c7">d</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.[</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="6_1_1_3_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">q</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</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 <font color="Maroon" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">].</a><span class="default"><a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>,<font color="Maroon" title="c7">d</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.[</a></span> or <font color="Maroon" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:6_1_1_3_1_1_2_1"/><span class="lab"><font color="Green" title="E14">A44</font></span>: 
<font color="Maroon" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">].</a><span class="default"><a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>,<font color="Maroon" title="c7">d</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.[</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"> <font color="Maroon" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">q</font> = <font color="Maroon" title="c8">q</font> as   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">q2</font> = <font color="Maroon" title="c9">q</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c2">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" onmouseover="rs('xreal_0/D1')" onmouseout="rh()">XREAL_0:def 1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">q1</font> = <font color="Maroon" title="c9">q</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" onmouseover="rs('xxreal_0/D1')" onmouseout="rh()">XXREAL_0:def 1</a></span>;<br/>
<a NAME="E5:6_1_1_3_1_1_2_1"/><span class="lab"><font color="Green" title="E15">A45</font></span>: 
<font color="Maroon" title="c11">q1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">d</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:6_1_1_3_1_1_2_1"><span class="lab"><font color="Green" title="E14">A44</font></span></a>, <a class="ref" href="measure5.html#D1" onmouseover="rs('measure5/D1')" onmouseout="rh()">MEASURE5:def 1</a></span>;<br/>
<a NAME="E6:6_1_1_3_1_1_2_1"/><span class="lab"><font color="Green" title="E16">A46</font></span>: 
<font color="Maroon" title="c10">q2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="6_1_1_3_1_1_2_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c12">q3</font> = <font color="Maroon" title="c10">q2</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" onmouseover="rs('xxreal_0/D1')" onmouseout="rh()">XXREAL_0:def 1</a></span>;<br/>
<a NAME="E2:6_1_1_3_1_1_2_1_1"/>
<font color="Maroon" title="c2">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c10">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:6"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="ref" href="xcmplx_1.html#T87" onmouseover="rs('xcmplx_1/T87')" onmouseout="rh()">XCMPLX_1:87</a></span>;<br/>
<a NAME="E3:6_1_1_3_1_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E17">A47</font></span>: 
<font color="Maroon" title="c12">q3</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:6_1_1_3"><span class="lab"><font color="Green" title="E11">A41</font></span></a>, <a class="txt" href="#E5:6_1_1_3_1_1_2_1"><span class="lab"><font color="Green" title="E15">A45</font></span></a>, <a class="ref" href="xreal_1.html#T64" onmouseover="rs('xreal_1/T64')" onmouseout="rh()">XREAL_1:64</a></span>;<br/>
<a NAME="E4:6_1_1_3_1_1_2_1_1"/>
<font color="Maroon" title="c3">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">q3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:6_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E12">A42</font></span></a>, <a class="ref" href="xxreal_0.html#T12" onmouseover="rs('xxreal_0/T12')" onmouseout="rh()">XXREAL_0:12</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:6_1_1_3_1_1_2_1_1"/>
<font color="Maroon" title="c10">q2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:6"><span class="lab"><font color="Green" title="E9">A3</font></span></a>, <a class="txt" href="#E3:6_1_1_3_1_1_2_1_1"><span class="lab"><font color="Green" title="E17">A47</font></span></a>, <a class="ref" href="measure5.html#D1" onmouseover="rs('measure5/D1')" onmouseout="rh()">MEASURE5:def 1</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/>


</div><span class="kw">end;</span></div>
<a NAME="E7:6_1_1_3_1_1_2_1"/>
<font color="Maroon" title="c9">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">q</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c2">x</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:6"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="ref" href="xcmplx_1.html#T87" onmouseover="rs('xcmplx_1/T87')" onmouseout="rh()">XCMPLX_1:87</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:6_1_1_3_1_1_2_1"/>
<font color="Maroon" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:6_1_1_3_1_1_2_1"><span class="lab"><font color="Green" title="E16">A46</font></span></a>, <a class="ref" href="member_1.html#T193" onmouseover="rs('member_1/T193')" onmouseout="rh()">MEMBER_1:193</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/>


</div><span class="kw">end;</span></div><a NAME="E5:6_1_1_3_1_1_2"/>
<font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">].</a><span class="default"><a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>,<font color="Maroon" title="c7">d</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.[</a></span>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="6_1_1_3_1_1_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">q</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</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 <font color="Maroon" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> or <font color="Maroon" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">].</a><span class="default"><a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>,<font color="Maroon" title="c7">d</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.[</a></span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:6_1_1_3_1_1_2_2"/><span class="lab"><font color="Green" title="E14">A48</font></span>: 
<font color="Maroon" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</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"> <font color="Maroon" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">].</a><span class="default"><a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>,<font color="Maroon" title="c7">d</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.[</a></span></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">q</font> = <font color="Maroon" title="c8">q</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c10">z2</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E4:6_1_1_3_1_1_2_2"/><span class="lab"><font color="Green" title="E15">A49</font></span>: 
<font color="Maroon" title="c10">z2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 <span class="kw">and </span><br/><a NAME="E5:6_1_1_3_1_1_2_2"/><span class="lab"><font color="Green" title="E16">A50</font></span>: 
<font color="Maroon" title="c9">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c10">z2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:6_1_1_3_1_1_2_2"><span class="lab"><font color="Green" title="E14">A48</font></span></a>, <a class="ref" href="integra2.html#T39" onmouseover="rs('integra2/T39')" onmouseout="rh()">INTEGRA2:39</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">z2</font> = <font color="Maroon" title="c10">z2</font>, <font color="Maroon" title="c12">q</font> = <font color="Maroon" title="c9">q</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" onmouseover="rs('xxreal_0/D1')" onmouseout="rh()">XXREAL_0:def 1</a></span>;<br/>
<a NAME="E7:6_1_1_3_1_1_2_2"/>
<font color="Maroon" title="c11">z2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:6"><span class="lab"><font color="Green" title="E9">A3</font></span></a>, <a class="txt" href="#E4:6_1_1_3_1_1_2_2"><span class="lab"><font color="Green" title="E15">A49</font></span></a>, <a class="ref" href="measure5.html#D1" onmouseover="rs('measure5/D1')" onmouseout="rh()">MEASURE5:def 1</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">r</font>, <font color="Maroon" title="c14">o</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E9:6_1_1_3_1_1_2_2"/><span class="lab"><font color="Green" title="E17">A51</font></span>: 
( <font color="Maroon" title="c13">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">z2</font> &amp; <font color="Maroon" title="c14">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> )
 <span class="kw">and </span><br/><a NAME="E10:6_1_1_3_1_1_2_2"/>
<font color="Maroon" title="c13">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">o</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:6_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E12">A42</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c15">o1</font> = <font color="Maroon" title="c2">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c14">o</font>, <font color="Maroon" title="c16">r1</font> = <font color="Maroon" title="c2">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c13">r</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" onmouseover="rs('xxreal_0/D1')" onmouseout="rh()">XXREAL_0:def 1</a></span>;<br/>
<a NAME="E12:6_1_1_3_1_1_2_2"/><span class="lab"><font color="Green" title="E18">A52</font></span>: 
 <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">q</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T12" onmouseover="rs('xxreal_0/T12')" onmouseout="rh()">XXREAL_0:12</a></span>;<br/>
<a NAME="E13:6_1_1_3_1_1_2_2"/>
<font color="Maroon" title="c13">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">o</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:6"><span class="lab"><font color="Green" title="E9">A3</font></span></a>, <a class="txt" href="#E4:6_1_1_3_1_1_2_2"><span class="lab"><font color="Green" title="E15">A49</font></span></a>, <a class="txt" href="#E9:6_1_1_3_1_1_2_2"><span class="lab"><font color="Green" title="E17">A51</font></span></a>, <a class="ref" href="measure5.html#D1" onmouseover="rs('measure5/D1')" onmouseout="rh()">MEASURE5:def 1</a></span>;<br/>
<a NAME="E14:6_1_1_3_1_1_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c16">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">o1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:6_1_1_3"><span class="lab"><font color="Green" title="E11">A41</font></span></a>, <a class="ref" href="xreal_1.html#T68" onmouseover="rs('xreal_1/T68')" onmouseout="rh()">XREAL_1:68</a></span>;<br/>
<span class="kw">hence </span><a NAME="E15:6_1_1_3_1_1_2_2"/>
<font color="Maroon" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">].</a><span class="default"><a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>,<font color="Maroon" title="c7">d</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.[</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:6_1_1_3_1_1_2_2"><span class="lab"><font color="Green" title="E16">A50</font></span></a>, <a class="txt" href="#E9:6_1_1_3_1_1_2_2"><span class="lab"><font color="Green" title="E17">A51</font></span></a>, <a class="txt" href="#E12:6_1_1_3_1_1_2_2"><span class="lab"><font color="Green" title="E18">A52</font></span></a>, <a class="ref" href="measure5.html#D1" onmouseover="rs('measure5/D1')" onmouseout="rh()">MEASURE5:def 1</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/>


</div><span class="kw">end;</span></div><a NAME="E6:6_1_1_3_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">].</a><span class="default"><a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>,<font color="Maroon" title="c7">d</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.[</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:6_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E13">A43</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E7:6_1_1_3_1_1_2"/>
<font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> is  <a href="measure5.html#V1" title="MEASURE5:attr.1">open_interval</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="measure5.html#D2" onmouseover="rs('measure5/D2')" onmouseout="rh()">MEASURE5:def 2</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/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_1_3_1_1_3"><span class="kw">case </span></a><a NAME="E1:6_1_1_3_1_1_3"/>
( <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  &amp; <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</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="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> is  <a href="measure5.html#V1" title="MEASURE5:attr.1">open_interval</a> </span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:6_1_1_3_1_1_3"/>
<font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> is  <a href="measure5.html#V1" title="MEASURE5:attr.1">open_interval</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:6"><span class="lab"><font color="Green" title="E8">A2</font></span></a>, <a class="txt" href="#E4:6"><span class="lab"><font color="Green" title="E9">A3</font></span></a>, <a class="txt" href="#E1:6_1_1_3"><span class="lab"><font color="Green" title="E11">A41</font></span></a>, <a class="ref" href="urysohn2.html#T2" target="_self" onmouseover="rs('urysohn2/T2')" onmouseout="rh()">Th2</a>, <a class="ref" href="xxreal_1.html#T224" onmouseover="rs('xxreal_1/T224')" onmouseout="rh()">XXREAL_1:224</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/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_1_3_1_1_4"><span class="kw">case </span></a><a NAME="E1:6_1_1_3_1_1_4"/><span class="lab"><font color="Green" title="E12">A53</font></span>: 
( <font color="Maroon" title="c3">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  &amp; <font color="Maroon" title="c4">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</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="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> is  <a href="measure5.html#V1" title="MEASURE5:attr.1">open_interval</a> </span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">s</font> = <font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c6">r</font> = <font color="Maroon" title="c4">b</font> as   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c7">d</font> = <font color="Maroon" title="c2">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">s</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" onmouseover="rs('xxreal_0/D1')" onmouseout="rh()">XXREAL_0:def 1</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c8">g</font> = <font color="Maroon" title="c2">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">r</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" onmouseover="rs('xxreal_0/D1')" onmouseout="rh()">XXREAL_0:def 1</a></span>;<br/><a NAME="E5:6_1_1_3_1_1_4"/><span class="lab"><font color="Green" title="E13">A54</font></span>: 
<span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">].</a><span class="default"><font color="Maroon" title="c7">d</font>,<font color="Maroon" title="c8">g</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.[</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="6_1_1_3_1_1_4_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">q</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</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 <font color="Maroon" title="c9">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">].</a><span class="default"><font color="Maroon" title="c7">d</font>,<font color="Maroon" title="c8">g</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.[</a></span> or <font color="Maroon" title="c9">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:6_1_1_3_1_1_4_1"/><span class="lab"><font color="Green" title="E14">A55</font></span>: 
<font color="Maroon" title="c9">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">].</a><span class="default"><font color="Maroon" title="c7">d</font>,<font color="Maroon" title="c8">g</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.[</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"> <font color="Maroon" title="c9">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">q</font> = <font color="Maroon" title="c9">q</font> as   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c11">q2</font> = <font color="Maroon" title="c10">q</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c2">x</font>;<br/>
<a NAME="E3:6_1_1_3_1_1_4_1"/>
<font color="Maroon" title="c10">q</font> is   <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" onmouseover="rs('xxreal_0/D1')" onmouseout="rh()">XXREAL_0:def 1</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">q1</font> being   <a href="supinf_1.html#NM1" title="SUPINF_1:NM.1">R_eal</a><span class="kw"> such that </span><br/><a NAME="E5:6_1_1_3_1_1_4_1"/><span class="lab"><font color="Green" title="E15">A56</font></span>: 
<font color="Maroon" title="c12">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">q</font>
 ;<br/>
<a NAME="E6:6_1_1_3_1_1_4_1"/><span class="lab"><font color="Green" title="E16">A57</font></span>: 
<font color="Maroon" title="c12">q1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:6_1_1_3_1_1_4_1"><span class="lab"><font color="Green" title="E14">A55</font></span></a>, <a class="txt" href="#E5:6_1_1_3_1_1_4_1"><span class="lab"><font color="Green" title="E15">A56</font></span></a>, <a class="ref" href="measure5.html#D1" onmouseover="rs('measure5/D1')" onmouseout="rh()">MEASURE5:def 1</a></span>;<br/>
<a NAME="E7:6_1_1_3_1_1_4_1"/><span class="lab"><font color="Green" title="E17">A58</font></span>: 
<font color="Maroon" title="c7">d</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">q1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:6_1_1_3_1_1_4_1"><span class="lab"><font color="Green" title="E14">A55</font></span></a>, <a class="txt" href="#E5:6_1_1_3_1_1_4_1"><span class="lab"><font color="Green" title="E15">A56</font></span></a>, <a class="ref" href="measure5.html#D1" onmouseover="rs('measure5/D1')" onmouseout="rh()">MEASURE5:def 1</a></span>;<br/>
<a NAME="E8:6_1_1_3_1_1_4_1"/><span class="lab"><font color="Green" title="E18">A59</font></span>: 
<font color="Maroon" title="c10">q</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c2">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="6_1_1_3_1_1_4_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c13">q3</font> = <font color="Maroon" title="c10">q</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <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" onmouseover="rs('xxreal_0/D1')" onmouseout="rh()">XXREAL_0:def 1</a></span>;<br/>
<a NAME="E2:6_1_1_3_1_1_4_1_1"/>
<font color="Maroon" title="c2">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">q</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">q</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:6"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="ref" href="xcmplx_1.html#T87" onmouseover="rs('xcmplx_1/T87')" onmouseout="rh()">XCMPLX_1:87</a></span>;<br/>
<a NAME="E3:6_1_1_3_1_1_4_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E19">A60</font></span>: 
<font color="Maroon" title="c3">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">q3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:6_1_1_3"><span class="lab"><font color="Green" title="E11">A41</font></span></a>, <a class="txt" href="#E5:6_1_1_3_1_1_4_1"><span class="lab"><font color="Green" title="E15">A56</font></span></a>, <a class="txt" href="#E7:6_1_1_3_1_1_4_1"><span class="lab"><font color="Green" title="E17">A58</font></span></a>, <a class="ref" href="xreal_1.html#T64" onmouseover="rs('xreal_1/T64')" onmouseout="rh()">XREAL_1:64</a></span>;<br/>
<a NAME="E4:6_1_1_3_1_1_4_1_1"/>
<font color="Maroon" title="c10">q</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c2">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">r</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c2">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:6_1_1_3"><span class="lab"><font color="Green" title="E11">A41</font></span></a>, <a class="txt" href="#E5:6_1_1_3_1_1_4_1"><span class="lab"><font color="Green" title="E15">A56</font></span></a>, <a class="txt" href="#E6:6_1_1_3_1_1_4_1"><span class="lab"><font color="Green" title="E16">A57</font></span></a>, <a class="ref" href="xreal_1.html#T74" onmouseover="rs('xreal_1/T74')" onmouseout="rh()">XREAL_1:74</a></span>;<br/>
<a NAME="E5:6_1_1_3_1_1_4_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c13">q3</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:6_1_1_3"><span class="lab"><font color="Green" title="E11">A41</font></span></a>, <a class="ref" href="xcmplx_1.html#T89" onmouseover="rs('xcmplx_1/T89')" onmouseout="rh()">XCMPLX_1:89</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:6_1_1_3_1_1_4_1_1"/>
<font color="Maroon" title="c10">q</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c2">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:6"><span class="lab"><font color="Green" title="E9">A3</font></span></a>, <a class="txt" href="#E3:6_1_1_3_1_1_4_1_1"><span class="lab"><font color="Green" title="E19">A60</font></span></a>, <a class="ref" href="measure5.html#D1" onmouseover="rs('measure5/D1')" onmouseout="rh()">MEASURE5:def 1</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/>


</div><span class="kw">end;</span></div>
<a NAME="E9:6_1_1_3_1_1_4_1"/>
<font color="Maroon" title="c10">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">q</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c2">x</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:6"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="ref" href="xcmplx_1.html#T87" onmouseover="rs('xcmplx_1/T87')" onmouseout="rh()">XCMPLX_1:87</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:6_1_1_3_1_1_4_1"/>
<font color="Maroon" title="c9">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:6_1_1_3_1_1_4_1"><span class="lab"><font color="Green" title="E18">A59</font></span></a>, <a class="ref" href="member_1.html#T193" onmouseover="rs('member_1/T193')" onmouseout="rh()">MEMBER_1:193</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/>


</div><span class="kw">end;</span></div><a NAME="E6:6_1_1_3_1_1_4"/>
<font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">].</a><span class="default"><font color="Maroon" title="c7">d</font>,<font color="Maroon" title="c8">g</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.[</a></span>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="6_1_1_3_1_1_4_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">q</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</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 <font color="Maroon" title="c9">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> or <font color="Maroon" title="c9">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">].</a><span class="default"><font color="Maroon" title="c7">d</font>,<font color="Maroon" title="c8">g</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.[</a></span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:6_1_1_3_1_1_4_2"/><span class="lab"><font color="Green" title="E14">A61</font></span>: 
<font color="Maroon" title="c9">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</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"> <font color="Maroon" title="c9">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">].</a><span class="default"><font color="Maroon" title="c7">d</font>,<font color="Maroon" title="c8">g</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.[</a></span></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">q</font> = <font color="Maroon" title="c9">q</font> as   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c11">z2</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E4:6_1_1_3_1_1_4_2"/><span class="lab"><font color="Green" title="E15">A62</font></span>: 
<font color="Maroon" title="c11">z2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 <span class="kw">and </span><br/><a NAME="E5:6_1_1_3_1_1_4_2"/><span class="lab"><font color="Green" title="E16">A63</font></span>: 
<font color="Maroon" title="c10">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c11">z2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:6_1_1_3_1_1_4_2"><span class="lab"><font color="Green" title="E14">A61</font></span></a>, <a class="ref" href="integra2.html#T39" onmouseover="rs('integra2/T39')" onmouseout="rh()">INTEGRA2:39</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c12">z2</font> = <font color="Maroon" title="c11">z2</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" onmouseover="rs('xxreal_0/D1')" onmouseout="rh()">XXREAL_0:def 1</a></span>;<br/>
<a NAME="E7:6_1_1_3_1_1_4_2"/>
<font color="Maroon" title="c12">z2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:6"><span class="lab"><font color="Green" title="E9">A3</font></span></a>, <a class="txt" href="#E4:6_1_1_3_1_1_4_2"><span class="lab"><font color="Green" title="E15">A62</font></span></a>, <a class="ref" href="measure5.html#D1" onmouseover="rs('measure5/D1')" onmouseout="rh()">MEASURE5:def 1</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">2o</font>, <font color="Maroon" title="c14">2r</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E9:6_1_1_3_1_1_4_2"/><span class="lab"><font color="Green" title="E17">A64</font></span>: 
( <font color="Maroon" title="c13">2o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">z2</font> &amp; <font color="Maroon" title="c14">2r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> )
 <span class="kw">and </span><br/><a NAME="E10:6_1_1_3_1_1_4_2"/>
<font color="Maroon" title="c13">2o</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">2r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:6_1_1_3_1_1_4"><span class="lab"><font color="Green" title="E12">A53</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c15">2o1</font> = <font color="Maroon" title="c2">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c13">2o</font>, <font color="Maroon" title="c16">2r1</font> = <font color="Maroon" title="c2">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c14">2r</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" onmouseover="rs('xxreal_0/D1')" onmouseout="rh()">XXREAL_0:def 1</a></span>;<br/>
<a NAME="E12:6_1_1_3_1_1_4_2"/>
<font color="Maroon" title="c13">2o</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">2r</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:6"><span class="lab"><font color="Green" title="E9">A3</font></span></a>, <a class="txt" href="#E4:6_1_1_3_1_1_4_2"><span class="lab"><font color="Green" title="E15">A62</font></span></a>, <a class="txt" href="#E9:6_1_1_3_1_1_4_2"><span class="lab"><font color="Green" title="E17">A64</font></span></a>, <a class="ref" href="measure5.html#D1" onmouseover="rs('measure5/D1')" onmouseout="rh()">MEASURE5:def 1</a></span>;<br/>
<a NAME="E13:6_1_1_3_1_1_4_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E18">A65</font></span>: 
<font color="Maroon" title="c15">2o1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">2r1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:6_1_1_3"><span class="lab"><font color="Green" title="E11">A41</font></span></a>, <a class="ref" href="xreal_1.html#T68" onmouseover="rs('xreal_1/T68')" onmouseout="rh()">XREAL_1:68</a></span>;<br/>
<a NAME="E14:6_1_1_3_1_1_4_2"/>
<font color="Maroon" title="c3">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">z2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:6"><span class="lab"><font color="Green" title="E9">A3</font></span></a>, <a class="txt" href="#E4:6_1_1_3_1_1_4_2"><span class="lab"><font color="Green" title="E15">A62</font></span></a>, <a class="ref" href="measure5.html#D1" onmouseover="rs('measure5/D1')" onmouseout="rh()">MEASURE5:def 1</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c17">1o</font>, <font color="Maroon" title="c18">1ra</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E16:6_1_1_3_1_1_4_2"/><span class="lab"><font color="Green" title="E19">A66</font></span>: 
( <font color="Maroon" title="c17">1o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> &amp; <font color="Maroon" title="c18">1ra</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">z2</font> )
 <span class="kw">and </span><br/><a NAME="E17:6_1_1_3_1_1_4_2"/>
<font color="Maroon" title="c17">1o</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">1ra</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:6_1_1_3_1_1_4"><span class="lab"><font color="Green" title="E12">A53</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c19">1o1</font> = <font color="Maroon" title="c2">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c17">1o</font>, <font color="Maroon" title="c20">1r1</font> = <font color="Maroon" title="c2">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c18">1ra</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" onmouseover="rs('xxreal_0/D1')" onmouseout="rh()">XXREAL_0:def 1</a></span>;<br/>
<a NAME="E19:6_1_1_3_1_1_4_2"/>
<font color="Maroon" title="c17">1o</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c18">1ra</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:6"><span class="lab"><font color="Green" title="E9">A3</font></span></a>, <a class="txt" href="#E4:6_1_1_3_1_1_4_2"><span class="lab"><font color="Green" title="E15">A62</font></span></a>, <a class="txt" href="#E16:6_1_1_3_1_1_4_2"><span class="lab"><font color="Green" title="E19">A66</font></span></a>, <a class="ref" href="measure5.html#D1" onmouseover="rs('measure5/D1')" onmouseout="rh()">MEASURE5:def 1</a></span>;<br/>
<a NAME="E20:6_1_1_3_1_1_4_2"/><span class="kw">then </span>
<font color="Maroon" title="c19">1o1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c20">1r1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:6_1_1_3"><span class="lab"><font color="Green" title="E11">A41</font></span></a>, <a class="ref" href="xreal_1.html#T68" onmouseover="rs('xreal_1/T68')" onmouseout="rh()">XREAL_1:68</a></span>;<br/>
<span class="kw">hence </span><a NAME="E21:6_1_1_3_1_1_4_2"/>
<font color="Maroon" title="c9">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">].</a><span class="default"><font color="Maroon" title="c7">d</font>,<font color="Maroon" title="c8">g</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.[</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:6_1_1_3_1_1_4_2"><span class="lab"><font color="Green" title="E16">A63</font></span></a>, <a class="txt" href="#E16:6_1_1_3_1_1_4_2"><span class="lab"><font color="Green" title="E19">A66</font></span></a>, <a class="txt" href="#E9:6_1_1_3_1_1_4_2"><span class="lab"><font color="Green" title="E17">A64</font></span></a>, <a class="txt" href="#E13:6_1_1_3_1_1_4_2"><span class="lab"><font color="Green" title="E18">A65</font></span></a>, <a class="ref" href="measure5.html#D1" onmouseover="rs('measure5/D1')" onmouseout="rh()">MEASURE5:def 1</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/>


</div><span class="kw">end;</span></div><a NAME="E7:6_1_1_3_1_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">].</a><span class="default"><font color="Maroon" title="c7">d</font>,<font color="Maroon" title="c8">g</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.[</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:6_1_1_3_1_1_4"><span class="lab"><font color="Green" title="E13">A54</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E8:6_1_1_3_1_1_4"/>
<font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> is  <a href="measure5.html#V1" title="MEASURE5:attr.1">open_interval</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="measure5.html#D2" onmouseover="rs('measure5/D2')" onmouseout="rh()">MEASURE5:def 2</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/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_1_3_1_1_5"><span class="kw">case </span></a><a NAME="E1:6_1_1_3_1_1_5"/><span class="lab"><font color="Green" title="E12">A67</font></span>: 
( <font color="Maroon" title="c3">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  &amp; <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</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="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> is  <a href="measure5.html#V1" title="MEASURE5:attr.1">open_interval</a> </span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">s</font> = <font color="Maroon" title="c3">a</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  ;<br/><span class="kw">set </span><font color="Maroon" title="c6">c</font> =  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c7">d</font> = <font color="Maroon" title="c2">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">s</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" onmouseover="rs('xxreal_0/D1')" onmouseout="rh()">XXREAL_0:def 1</a></span>;<br/><a NAME="E4:6_1_1_3_1_1_5"/><span class="lab"><font color="Green" title="E13">A68</font></span>: 
<font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">].</a><span class="default"><font color="Maroon" title="c7">d</font>,<a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a></span><a href="measure5.html#K1" title="MEASURE5:func.1">.[</a></span>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="6_1_1_3_1_1_5_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">q</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</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 <font color="Maroon" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> or <font color="Maroon" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">].</a><span class="default"><font color="Maroon" title="c7">d</font>,<a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a></span><a href="measure5.html#K1" title="MEASURE5:func.1">.[</a></span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:6_1_1_3_1_1_5_1"/><span class="lab"><font color="Green" title="E14">A69</font></span>: 
<font color="Maroon" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</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"> <font color="Maroon" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">].</a><span class="default"><font color="Maroon" title="c7">d</font>,<a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a></span><a href="measure5.html#K1" title="MEASURE5:func.1">.[</a></span></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">q</font> = <font color="Maroon" title="c8">q</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c10">z2</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E4:6_1_1_3_1_1_5_1"/><span class="lab"><font color="Green" title="E15">A70</font></span>: 
<font color="Maroon" title="c10">z2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 <span class="kw">and </span><br/><a NAME="E5:6_1_1_3_1_1_5_1"/><span class="lab"><font color="Green" title="E16">A71</font></span>: 
<font color="Maroon" title="c9">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c10">z2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:6_1_1_3_1_1_5_1"><span class="lab"><font color="Green" title="E14">A69</font></span></a>, <a class="ref" href="integra2.html#T39" onmouseover="rs('integra2/T39')" onmouseout="rh()">INTEGRA2:39</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">q</font> = <font color="Maroon" title="c9">q</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" onmouseover="rs('xxreal_0/D1')" onmouseout="rh()">XXREAL_0:def 1</a></span>;<br/>
<a NAME="E7:6_1_1_3_1_1_5_1"/><span class="lab"><font color="Green" title="E17">A72</font></span>: 
<font color="Maroon" title="c11">q</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T9" onmouseover="rs('xxreal_0/T9')" onmouseout="rh()">XXREAL_0:9</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c12">z2</font> = <font color="Maroon" title="c10">z2</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" onmouseover="rs('xxreal_0/D1')" onmouseout="rh()">XXREAL_0:def 1</a></span>;<br/>
<a NAME="E9:6_1_1_3_1_1_5_1"/>
<font color="Maroon" title="c3">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">z2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:6"><span class="lab"><font color="Green" title="E9">A3</font></span></a>, <a class="txt" href="#E4:6_1_1_3_1_1_5_1"><span class="lab"><font color="Green" title="E15">A70</font></span></a>, <a class="ref" href="measure5.html#D1" onmouseover="rs('measure5/D1')" onmouseout="rh()">MEASURE5:def 1</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">o</font>, <font color="Maroon" title="c14">r</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E11:6_1_1_3_1_1_5_1"/><span class="lab"><font color="Green" title="E18">A73</font></span>: 
( <font color="Maroon" title="c13">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> &amp; <font color="Maroon" title="c14">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">z2</font> )
 <span class="kw">and </span><br/><a NAME="E12:6_1_1_3_1_1_5_1"/>
<font color="Maroon" title="c13">o</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:6_1_1_3_1_1_5"><span class="lab"><font color="Green" title="E12">A67</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c15">o1</font> = <font color="Maroon" title="c2">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c13">o</font>, <font color="Maroon" title="c16">r1</font> = <font color="Maroon" title="c2">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c14">r</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" onmouseover="rs('xxreal_0/D1')" onmouseout="rh()">XXREAL_0:def 1</a></span>;<br/>
<a NAME="E14:6_1_1_3_1_1_5_1"/>
<font color="Maroon" title="c13">o</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:6"><span class="lab"><font color="Green" title="E9">A3</font></span></a>, <a class="txt" href="#E4:6_1_1_3_1_1_5_1"><span class="lab"><font color="Green" title="E15">A70</font></span></a>, <a class="txt" href="#E11:6_1_1_3_1_1_5_1"><span class="lab"><font color="Green" title="E18">A73</font></span></a>, <a class="ref" href="measure5.html#D1" onmouseover="rs('measure5/D1')" onmouseout="rh()">MEASURE5:def 1</a></span>;<br/>
<a NAME="E15:6_1_1_3_1_1_5_1"/><span class="kw">then </span>
<font color="Maroon" title="c15">o1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">r1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:6_1_1_3"><span class="lab"><font color="Green" title="E11">A41</font></span></a>, <a class="ref" href="xreal_1.html#T68" onmouseover="rs('xreal_1/T68')" onmouseout="rh()">XREAL_1:68</a></span>;<br/>
<span class="kw">hence </span><a NAME="E16:6_1_1_3_1_1_5_1"/>
<font color="Maroon" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">].</a><span class="default"><font color="Maroon" title="c7">d</font>,<a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a></span><a href="measure5.html#K1" title="MEASURE5:func.1">.[</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:6_1_1_3_1_1_5_1"><span class="lab"><font color="Green" title="E16">A71</font></span></a>, <a class="txt" href="#E11:6_1_1_3_1_1_5_1"><span class="lab"><font color="Green" title="E18">A73</font></span></a>, <a class="txt" href="#E7:6_1_1_3_1_1_5_1"><span class="lab"><font color="Green" title="E17">A72</font></span></a>, <a class="ref" href="measure5.html#D1" onmouseover="rs('measure5/D1')" onmouseout="rh()">MEASURE5:def 1</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/>


</div><span class="kw">end;</span></div><a NAME="E5:6_1_1_3_1_1_5"/>
<span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">].</a><span class="default"><font color="Maroon" title="c7">d</font>,<a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a></span><a href="measure5.html#K1" title="MEASURE5:func.1">.[</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="6_1_1_3_1_1_5_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">q</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</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 <font color="Maroon" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">].</a><span class="default"><font color="Maroon" title="c7">d</font>,<a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a></span><a href="measure5.html#K1" title="MEASURE5:func.1">.[</a></span> or <font color="Maroon" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:6_1_1_3_1_1_5_2"/><span class="lab"><font color="Green" title="E14">A74</font></span>: 
<font color="Maroon" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">].</a><span class="default"><font color="Maroon" title="c7">d</font>,<a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a></span><a href="measure5.html#K1" title="MEASURE5:func.1">.[</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"> <font color="Maroon" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">q</font> = <font color="Maroon" title="c8">q</font> as   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">q2</font> = <font color="Maroon" title="c9">q</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c2">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" onmouseover="rs('xreal_0/D1')" onmouseout="rh()">XREAL_0:def 1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">q1</font> = <font color="Maroon" title="c9">q</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" onmouseover="rs('xxreal_0/D1')" onmouseout="rh()">XXREAL_0:def 1</a></span>;<br/>
<a NAME="E5:6_1_1_3_1_1_5_2"/><span class="lab"><font color="Green" title="E15">A75</font></span>: 
<font color="Maroon" title="c9">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">q</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c2">x</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:6"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="ref" href="xcmplx_1.html#T87" onmouseover="rs('xcmplx_1/T87')" onmouseout="rh()">XCMPLX_1:87</a></span>;<br/>
<a NAME="E6:6_1_1_3_1_1_5_2"/><span class="lab"><font color="Green" title="E16">A76</font></span>: 
<font color="Maroon" title="c7">d</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">q1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:6_1_1_3_1_1_5_2"><span class="lab"><font color="Green" title="E14">A74</font></span></a>, <a class="ref" href="measure5.html#D1" onmouseover="rs('measure5/D1')" onmouseout="rh()">MEASURE5:def 1</a></span>;<br/>
<a NAME="E7:6_1_1_3_1_1_5_2"/>
<font color="Maroon" title="c10">q2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="6_1_1_3_1_1_5_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c12">q3</font> = <font color="Maroon" title="c10">q2</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" onmouseover="rs('xxreal_0/D1')" onmouseout="rh()">XXREAL_0:def 1</a></span>;<br/>
<a NAME="E2:6_1_1_3_1_1_5_2_1"/>
( <font color="Maroon" title="c3">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">q3</font> &amp; <font color="Maroon" title="c12">q3</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">b</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:6_1_1_3"><span class="lab"><font color="Green" title="E11">A41</font></span></a>, <a class="txt" href="#E1:6_1_1_3_1_1_5"><span class="lab"><font color="Green" title="E12">A67</font></span></a>, <a class="txt" href="#E6:6_1_1_3_1_1_5_2"><span class="lab"><font color="Green" title="E16">A76</font></span></a>, <a class="txt" href="#E5:6_1_1_3_1_1_5_2"><span class="lab"><font color="Green" title="E15">A75</font></span></a>, <a class="ref" href="xreal_1.html#T64" onmouseover="rs('xreal_1/T64')" onmouseout="rh()">XREAL_1:64</a>, <a class="ref" href="xxreal_0.html#T9" onmouseover="rs('xxreal_0/T9')" onmouseout="rh()">XXREAL_0:9</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:6_1_1_3_1_1_5_2_1"/>
<font color="Maroon" title="c10">q2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:6"><span class="lab"><font color="Green" title="E9">A3</font></span></a>, <a class="ref" href="measure5.html#D1" onmouseover="rs('measure5/D1')" onmouseout="rh()">MEASURE5:def 1</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/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E8:6_1_1_3_1_1_5_2"/>
<font color="Maroon" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:6_1_1_3_1_1_5_2"><span class="lab"><font color="Green" title="E15">A75</font></span></a>, <a class="ref" href="member_1.html#T193" onmouseover="rs('member_1/T193')" onmouseout="rh()">MEMBER_1:193</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/>


</div><span class="kw">end;</span></div><a NAME="E6:6_1_1_3_1_1_5"/><span class="kw">then </span>
<font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">].</a><span class="default"><font color="Maroon" title="c7">d</font>,<a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a></span><a href="measure5.html#K1" title="MEASURE5:func.1">.[</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:6_1_1_3_1_1_5"><span class="lab"><font color="Green" title="E13">A68</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E7:6_1_1_3_1_1_5"/>
<font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> is  <a href="measure5.html#V1" title="MEASURE5:attr.1">open_interval</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="measure5.html#D2" onmouseover="rs('measure5/D2')" onmouseout="rh()">MEASURE5:def 2</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/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_1_3_1_1_6"><span class="kw">case </span></a><a NAME="E1:6_1_1_3_1_1_6"/>
( <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  &amp; <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</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="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> is  <a href="measure5.html#V1" title="MEASURE5:attr.1">open_interval</a> </span><br/><div class="add"><a NAME="E2:6_1_1_3_1_1_6"/><span class="kw">then </span>
<span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">].</a><span class="default"><font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.[</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><a NAME="E3:6_1_1_3_1_1_6"/><span class="kw">then </span>
<font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:6"><span class="lab"><font color="Green" title="E9">A3</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:6_1_1_3_1_1_6"/>
<font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> is  <a href="measure5.html#V1" title="MEASURE5:attr.1">open_interval</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"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:6_1_1_3"/>
<font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> is  <a href="measure5.html#V1" title="MEASURE5:attr.1">open_interval</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"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E7:6"/>
<font color="Maroon" title="c2">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> is  <a href="measure5.html#V1" title="MEASURE5:attr.1">open_interval</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"> verum</span><br/>


</div>
