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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  <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="Olive" title="b1">x</font> holds <br/> for <font color="Olive" title="b2">B</font> being   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>  st <font color="Olive" title="b2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">**</a> <font color="Maroon" title="c1">A</font> &amp; <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"><span class="p2">(<span class="default"><a href="supinf_2.html#K4" title="SUPINF_2:func.4">inf</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="supinf_2.html#K5" title="SUPINF_2:func.5">sup</a> <font color="Maroon" title="c1">A</font></span>)</span></span><a href="measure5.html#K1" title="MEASURE5:func.1">.[</a></span> holds <br/>( <font color="Olive" title="b2">B</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"><span class="p2">(<span class="default"><a href="supinf_2.html#K4" title="SUPINF_2:func.4">inf</a> <font color="Olive" title="b2">B</font></span>)</span>,<span class="p2">(<span class="default"><a href="supinf_2.html#K5" title="SUPINF_2:func.5">sup</a> <font color="Olive" title="b2">B</font></span>)</span></span><a href="measure5.html#K1" title="MEASURE5:func.1">.[</a></span> &amp; (  for <font color="Olive" title="b3">s</font>, <font color="Olive" title="b4">t</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st <font color="Olive" title="b3">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K4" title="SUPINF_2:func.4">inf</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Olive" title="b4">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K5" title="SUPINF_2:func.5">sup</a> <font color="Maroon" title="c1">A</font> holds <br/>(  <a href="supinf_2.html#K4" title="SUPINF_2:func.4">inf</a> <font color="Olive" title="b2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b3">s</font> &amp;  <a href="supinf_2.html#K5" title="SUPINF_2:func.5">sup</a> <font color="Olive" title="b2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b4">t</font> ) ) )</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"> (  <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> implies  for <font color="Olive" title="b1">B</font> being   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>  st <font color="Olive" title="b1">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> &amp; <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"><span class="p2">(<span class="default"><a href="supinf_2.html#K4" title="SUPINF_2:func.4">inf</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="supinf_2.html#K5" title="SUPINF_2:func.5">sup</a> <font color="Maroon" title="c1">A</font></span>)</span></span><a href="measure5.html#K1" title="MEASURE5:func.1">.[</a></span> holds <br/>( <font color="Olive" title="b1">B</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"><span class="p2">(<span class="default"><a href="supinf_2.html#K4" title="SUPINF_2:func.4">inf</a> <font color="Olive" title="b1">B</font></span>)</span>,<span class="p2">(<span class="default"><a href="supinf_2.html#K5" title="SUPINF_2:func.5">sup</a> <font color="Olive" title="b1">B</font></span>)</span></span><a href="measure5.html#K1" title="MEASURE5:func.1">.[</a></span> &amp; (  for <font color="Olive" title="b2">s</font>, <font color="Olive" title="b3">t</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st <font color="Olive" title="b2">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K4" title="SUPINF_2:func.4">inf</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Olive" title="b3">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K5" title="SUPINF_2:func.5">sup</a> <font color="Maroon" title="c1">A</font> holds <br/>(  <a href="supinf_2.html#K4" title="SUPINF_2:func.4">inf</a> <font color="Olive" title="b1">B</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="Olive" title="b2">s</font> &amp;  <a href="supinf_2.html#K5" title="SUPINF_2:func.5">sup</a> <font color="Olive" title="b1">B</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="Olive" title="b3">t</font> ) ) ) )</span><br/>



<span class="kw">assume </span><a NAME="E1:14"/><span class="lab"><font color="Green" title="E12">A1</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">  for <font color="Olive" title="b1">B</font> being   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>  st <font color="Olive" title="b1">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> &amp; <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"><span class="p2">(<span class="default"><a href="supinf_2.html#K4" title="SUPINF_2:func.4">inf</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="supinf_2.html#K5" title="SUPINF_2:func.5">sup</a> <font color="Maroon" title="c1">A</font></span>)</span></span><a href="measure5.html#K1" title="MEASURE5:func.1">.[</a></span> holds <br/>( <font color="Olive" title="b1">B</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"><span class="p2">(<span class="default"><a href="supinf_2.html#K4" title="SUPINF_2:func.4">inf</a> <font color="Olive" title="b1">B</font></span>)</span>,<span class="p2">(<span class="default"><a href="supinf_2.html#K5" title="SUPINF_2:func.5">sup</a> <font color="Olive" title="b1">B</font></span>)</span></span><a href="measure5.html#K1" title="MEASURE5:func.1">.[</a></span> &amp; (  for <font color="Olive" title="b2">s</font>, <font color="Olive" title="b3">t</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st <font color="Olive" title="b2">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K4" title="SUPINF_2:func.4">inf</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Olive" title="b3">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K5" title="SUPINF_2:func.5">sup</a> <font color="Maroon" title="c1">A</font> holds <br/>(  <a href="supinf_2.html#K4" title="SUPINF_2:func.4">inf</a> <font color="Olive" title="b1">B</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="Olive" title="b2">s</font> &amp;  <a href="supinf_2.html#K5" title="SUPINF_2:func.5">sup</a> <font color="Olive" title="b1">B</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="Olive" title="b3">t</font> ) ) )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c3">B</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"> ( <font color="Maroon" title="c3">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> &amp; <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"><span class="p2">(<span class="default"><a href="supinf_2.html#K4" title="SUPINF_2:func.4">inf</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="supinf_2.html#K5" title="SUPINF_2:func.5">sup</a> <font color="Maroon" title="c1">A</font></span>)</span></span><a href="measure5.html#K1" title="MEASURE5:func.1">.[</a></span> implies ( <font color="Maroon" title="c3">B</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"><span class="p2">(<span class="default"><a href="supinf_2.html#K4" title="SUPINF_2:func.4">inf</a> <font color="Maroon" title="c3">B</font></span>)</span>,<span class="p2">(<span class="default"><a href="supinf_2.html#K5" title="SUPINF_2:func.5">sup</a> <font color="Maroon" title="c3">B</font></span>)</span></span><a href="measure5.html#K1" title="MEASURE5:func.1">.[</a></span> &amp; (  for <font color="Olive" title="b1">s</font>, <font color="Olive" title="b2">t</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K4" title="SUPINF_2:func.4">inf</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Olive" title="b2">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K5" title="SUPINF_2:func.5">sup</a> <font color="Maroon" title="c1">A</font> holds <br/>(  <a href="supinf_2.html#K4" title="SUPINF_2:func.4">inf</a> <font color="Maroon" title="c3">B</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="Olive" title="b1">s</font> &amp;  <a href="supinf_2.html#K5" title="SUPINF_2:func.5">sup</a> <font color="Maroon" title="c3">B</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="Olive" title="b2">t</font> ) ) ) )</span><br/>

<span class="kw">assume </span><a NAME="E2:14"/><span class="lab"><font color="Green" title="E13">A2</font></span>: 
<font color="Maroon" title="c3">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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"> (  not <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"><span class="p2">(<span class="default"><a href="supinf_2.html#K4" title="SUPINF_2:func.4">inf</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="supinf_2.html#K5" title="SUPINF_2:func.5">sup</a> <font color="Maroon" title="c1">A</font></span>)</span></span><a href="measure5.html#K1" title="MEASURE5:func.1">.[</a></span> or ( <font color="Maroon" title="c3">B</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"><span class="p2">(<span class="default"><a href="supinf_2.html#K4" title="SUPINF_2:func.4">inf</a> <font color="Maroon" title="c3">B</font></span>)</span>,<span class="p2">(<span class="default"><a href="supinf_2.html#K5" title="SUPINF_2:func.5">sup</a> <font color="Maroon" title="c3">B</font></span>)</span></span><a href="measure5.html#K1" title="MEASURE5:func.1">.[</a></span> &amp; (  for <font color="Olive" title="b1">s</font>, <font color="Olive" title="b2">t</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K4" title="SUPINF_2:func.4">inf</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Olive" title="b2">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K5" title="SUPINF_2:func.5">sup</a> <font color="Maroon" title="c1">A</font> holds <br/>(  <a href="supinf_2.html#K4" title="SUPINF_2:func.4">inf</a> <font color="Maroon" title="c3">B</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="Olive" title="b1">s</font> &amp;  <a href="supinf_2.html#K5" title="SUPINF_2:func.5">sup</a> <font color="Maroon" title="c3">B</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="Olive" title="b2">t</font> ) ) ) )</span><br/>

<a NAME="E3:14"/>
( <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"><span class="p2">(<span class="default"><a href="supinf_2.html#K4" title="SUPINF_2:func.4">inf</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="supinf_2.html#K5" title="SUPINF_2:func.5">sup</a> <font color="Maroon" title="c1">A</font></span>)</span></span><a href="measure5.html#K1" title="MEASURE5:func.1">.[</a></span> implies ( <font color="Maroon" title="c3">B</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"><span class="p2">(<span class="default"><a href="supinf_2.html#K4" title="SUPINF_2:func.4">inf</a> <font color="Maroon" title="c3">B</font></span>)</span>,<span class="p2">(<span class="default"><a href="supinf_2.html#K5" title="SUPINF_2:func.5">sup</a> <font color="Maroon" title="c3">B</font></span>)</span></span><a href="measure5.html#K1" title="MEASURE5:func.1">.[</a></span> &amp; (  for <font color="Olive" title="b1">s</font>, <font color="Olive" title="b2">t</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K4" title="SUPINF_2:func.4">inf</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Olive" title="b2">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K5" title="SUPINF_2:func.5">sup</a> <font color="Maroon" title="c1">A</font> holds <br/>(  <a href="supinf_2.html#K4" title="SUPINF_2:func.4">inf</a> <font color="Maroon" title="c3">B</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="Olive" title="b1">s</font> &amp;  <a href="supinf_2.html#K5" title="SUPINF_2:func.5">sup</a> <font color="Maroon" title="c3">B</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="Olive" title="b2">t</font> ) ) ) )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="14_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:14_1"/><span class="lab"><font color="Green" title="E14">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"><span class="p2">(<span class="default"><a href="supinf_2.html#K4" title="SUPINF_2:func.4">inf</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="supinf_2.html#K5" title="SUPINF_2:func.5">sup</a> <font color="Maroon" title="c1">A</font></span>)</span></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="c3">B</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"><span class="p2">(<span class="default"><a href="supinf_2.html#K4" title="SUPINF_2:func.4">inf</a> <font color="Maroon" title="c3">B</font></span>)</span>,<span class="p2">(<span class="default"><a href="supinf_2.html#K5" title="SUPINF_2:func.5">sup</a> <font color="Maroon" title="c3">B</font></span>)</span></span><a href="measure5.html#K1" title="MEASURE5:func.1">.[</a></span> &amp; (  for <font color="Olive" title="b1">s</font>, <font color="Olive" title="b2">t</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K4" title="SUPINF_2:func.4">inf</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Olive" title="b2">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K5" title="SUPINF_2:func.5">sup</a> <font color="Maroon" title="c1">A</font> holds <br/>(  <a href="supinf_2.html#K4" title="SUPINF_2:func.4">inf</a> <font color="Maroon" title="c3">B</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="Olive" title="b1">s</font> &amp;  <a href="supinf_2.html#K5" title="SUPINF_2:func.5">sup</a> <font color="Maroon" title="c3">B</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="Olive" title="b2">t</font> ) ) )</span><br/>

<a NAME="E2:14_1"/><span class="lab"><font color="Green" title="E15">A4</font></span>: 
 for <font color="Olive" title="b1">s</font>, <font color="Olive" title="b2">t</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K4" title="SUPINF_2:func.4">inf</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Olive" title="b2">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K5" title="SUPINF_2:func.5">sup</a> <font color="Maroon" title="c1">A</font> holds <br/>(  <a href="supinf_2.html#K4" title="SUPINF_2:func.4">inf</a> <font color="Maroon" title="c3">B</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="Olive" title="b1">s</font> &amp;  <a href="supinf_2.html#K5" title="SUPINF_2:func.5">sup</a> <font color="Maroon" title="c3">B</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="Olive" title="b2">t</font> &amp; <font color="Maroon" title="c3">B</font> is  <a href="measure5.html#V1" title="MEASURE5:attr.1">open_interval</a>  )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="14_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">s</font>, <font color="Maroon" title="c5">t</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="c4">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K4" title="SUPINF_2:func.4">inf</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c5">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K5" title="SUPINF_2:func.5">sup</a> <font color="Maroon" title="c1">A</font> implies (  <a href="supinf_2.html#K4" title="SUPINF_2:func.4">inf</a> <font color="Maroon" title="c3">B</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="c4">s</font> &amp;  <a href="supinf_2.html#K5" title="SUPINF_2:func.5">sup</a> <font color="Maroon" title="c3">B</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">t</font> &amp; <font color="Maroon" title="c3">B</font> is  <a href="measure5.html#V1" title="MEASURE5:attr.1">open_interval</a>  ) )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:14_1_1"/><span class="lab"><font color="Green" title="E16">A5</font></span>: 
<font color="Maroon" title="c4">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K4" title="SUPINF_2:func.4">inf</a> <font color="Maroon" title="c1">A</font>
 <span class="kw">and </span><br/><a NAME="E2:14_1_1"/><span class="lab"><font color="Green" title="E17">A6</font></span>: 
<font color="Maroon" title="c5">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K5" title="SUPINF_2:func.5">sup</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"> (  <a href="supinf_2.html#K4" title="SUPINF_2:func.4">inf</a> <font color="Maroon" title="c3">B</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="c4">s</font> &amp;  <a href="supinf_2.html#K5" title="SUPINF_2:func.5">sup</a> <font color="Maroon" title="c3">B</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">t</font> &amp; <font color="Maroon" title="c3">B</font> is  <a href="measure5.html#V1" title="MEASURE5:attr.1">open_interval</a>  )</span><br/>

<a NAME="E3:14_1_1"/>
(  <a href="supinf_2.html#K4" title="SUPINF_2:func.4">inf</a> <font color="Maroon" title="c3">B</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="c4">s</font> &amp;  <a href="supinf_2.html#K5" title="SUPINF_2:func.5">sup</a> <font color="Maroon" title="c3">B</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">t</font> &amp; <font color="Maroon" title="c3">B</font> is  <a href="measure5.html#V1" title="MEASURE5:attr.1">open_interval</a>  )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="14_1_1_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:14_1_1_1"/>
<font color="Maroon" title="c4">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">t</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:14_1_1"><span class="lab"><font color="Green" title="E16">A5</font></span></a>, <a class="txt" href="#E2:14_1_1"><span class="lab"><font color="Green" title="E17">A6</font></span></a>, <a class="ref" href="xxreal_2.html#T40" onmouseover="rs('xxreal_2/T40')" onmouseout="rh()">XXREAL_2:40</a></span>;<br/>
<a NAME="E2:14_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E18">A7</font></span>: 
<font color="Maroon" title="c2">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&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">t</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:14"><span class="lab"><font color="Green" title="E12">A1</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="E3:14_1_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="c4">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="c6">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="E5:14_1_1_1"/><span class="lab"><font color="Green" title="E19">A8</font></span>: 
<font color="Maroon" title="c6">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="c4">s</font>
 ;<br/>
<a NAME="E6:14_1_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="c5">t</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">g</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="E8:14_1_1_1"/><span class="lab"><font color="Green" title="E20">A9</font></span>: 
<font color="Maroon" title="c7">g</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">t</font>
 ;<br/>
<a NAME="E9:14_1_1_1"/><span class="lab"><font color="Green" title="E21">A10</font></span>: 
<span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">].</a><span class="default"><font color="Maroon" title="c6">d</font>,<font color="Maroon" title="c7">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="14_1_1_1_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="c6">d</font>,<font color="Maroon" title="c7">g</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="E1:14_1_1_1_1"/><span class="lab"><font color="Green" title="E22">A11</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="c6">d</font>,<font color="Maroon" title="c7">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="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">set </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>;<br/>
<a NAME="E3:14_1_1_1_1"/>
<font color="Maroon" title="c9">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="c11">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="E5:14_1_1_1_1"/><span class="lab"><font color="Green" title="E23">A12</font></span>: 
<font color="Maroon" title="c11">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font>
 ;<br/>
<a NAME="E6:14_1_1_1_1"/><span class="lab"><font color="Green" title="E24">A13</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">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:14_1_1_1_1"><span class="lab"><font color="Green" title="E22">A11</font></span></a>, <a class="txt" href="#E5:14_1_1_1_1"><span class="lab"><font color="Green" title="E23">A12</font></span></a>, <a class="ref" href="measure5.html#D1" onmouseover="rs('measure5/D1')" onmouseout="rh()">MEASURE5:def 1</a></span>;<br/>
<a NAME="E7:14_1_1_1_1"/><span class="lab"><font color="Green" title="E25">A14</font></span>: 
<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> <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="14_1_1_1_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="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="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="E2:14_1_1_1_1_1"/><span class="lab"><font color="Green" title="E26">A16</font></span>: 
<font color="Maroon" title="c12">q3</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_2.html#K5" title="SUPINF_2:func.5">sup</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="14_1_1_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">consider </span><font color="Maroon" title="c13">p</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="E2:14_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E27">A17</font></span>: 
( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">q1</font> &amp; <font color="Maroon" title="c14">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">g</font> )
 <span class="kw">and </span><br/><a NAME="E3:14_1_1_1_1_1_1"/>
<font color="Maroon" title="c13">p</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="#E8:14_1_1_1"><span class="lab"><font color="Green" title="E20">A9</font></span></a>, <a class="txt" href="#E5:14_1_1_1_1"><span class="lab"><font color="Green" title="E23">A12</font></span></a>, <a class="txt" href="#E6:14_1_1_1_1"><span class="lab"><font color="Green" title="E24">A13</font></span></a></span>;<br/>
<a NAME="E4:14_1_1_1_1_1_1"/>
<font color="Maroon" title="c13">p</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> <font color="Maroon" title="c14">o</font> <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="#E1:14"><span class="lab"><font color="Green" title="E12">A1</font></span></a>, <a class="txt" href="#E6:14_1_1_1_1"><span class="lab"><font color="Green" title="E24">A13</font></span></a>, <a class="txt" href="#E2:14_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E27">A17</font></span></a>, <a class="ref" href="xreal_1.html#T74" onmouseover="rs('xreal_1/T74')" onmouseout="rh()">XREAL_1:74</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:14_1_1_1_1_1_1"/>
<font color="Maroon" title="c12">q3</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_2.html#K5" title="SUPINF_2:func.5">sup</a> <font color="Maroon" title="c1">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:14"><span class="lab"><font color="Green" title="E12">A1</font></span></a>, <a class="txt" href="#E2:14_1_1"><span class="lab"><font color="Green" title="E17">A6</font></span></a>, <a class="txt" href="#E8:14_1_1_1"><span class="lab"><font color="Green" title="E20">A9</font></span></a>, <a class="txt" href="#E5:14_1_1_1_1"><span class="lab"><font color="Green" title="E23">A12</font></span></a>, <a class="txt" href="#E2:14_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E27">A17</font></span></a>, <a class="ref" href="xcmplx_1.html#T89" onmouseover="rs('xcmplx_1/T89')" onmouseout="rh()">XCMPLX_1:89</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="E3:14_1_1_1_1_1"/>
( <font color="Maroon" title="c6">d</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">q1</font> &amp; <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> <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="#E1:14"><span class="lab"><font color="Green" title="E12">A1</font></span></a>, <a class="txt" href="#E1:14_1_1_1_1"><span class="lab"><font color="Green" title="E22">A11</font></span></a>, <a class="txt" href="#E5:14_1_1_1_1"><span class="lab"><font color="Green" title="E23">A12</font></span></a>, <a class="ref" href="measure5.html#D1" onmouseover="rs('measure5/D1')" onmouseout="rh()">MEASURE5:def 1</a>, <a class="ref" href="xcmplx_1.html#T87" onmouseover="rs('xcmplx_1/T87')" onmouseout="rh()">XCMPLX_1:87</a></span>;<br/>
<a NAME="E4:14_1_1_1_1_1"/><span class="kw">then </span>
 <a href="supinf_2.html#K4" title="SUPINF_2:func.4">inf</a> <font color="Maroon" title="c1">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="#E1:14"><span class="lab"><font color="Green" title="E12">A1</font></span></a>, <a class="txt" href="#E1:14_1_1"><span class="lab"><font color="Green" title="E16">A5</font></span></a>, <a class="txt" href="#E5:14_1_1_1"><span class="lab"><font color="Green" title="E19">A8</font></span></a>, <a class="txt" href="#E5:14_1_1_1_1"><span class="lab"><font color="Green" title="E23">A12</font></span></a>, <a class="ref" href="xreal_1.html#T64" onmouseover="rs('xreal_1/T64')" onmouseout="rh()">XREAL_1:64</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:14_1_1_1_1_1"/>
<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> <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="#E1:14_1"><span class="lab"><font color="Green" title="E14">A3</font></span></a>, <a class="txt" href="#E2:14_1_1_1_1_1"><span class="lab"><font color="Green" title="E26">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="E8:14_1_1_1_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="#E1:14"><span class="lab"><font color="Green" title="E12">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="E9:14_1_1_1_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="#E7:14_1_1_1_1"><span class="lab"><font color="Green" title="E25">A14</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="E10:14_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> <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="c6">d</font>,<font color="Maroon" title="c7">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="14_1_1_1_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="c6">d</font>,<font color="Maroon" title="c7">g</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.[</a></span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:14_1_1_1_2"/><span class="lab"><font color="Green" title="E22">A18</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="c6">d</font>,<font color="Maroon" title="c7">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="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">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="E4:14_1_1_1_2"/><span class="lab"><font color="Green" title="E23">A19</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="E5:14_1_1_1_2"/><span class="lab"><font color="Green" title="E24">A20</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="#E1:14_1_1_1_2"><span class="lab"><font color="Green" title="E22">A18</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> 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="E7:14_1_1_1_2"/>
 <a href="supinf_2.html#K4" title="SUPINF_2:func.4">inf</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">z2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:14_1"><span class="lab"><font color="Green" title="E14">A3</font></span></a>, <a class="txt" href="#E4:14_1_1_1_2"><span class="lab"><font color="Green" title="E23">A19</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="c12">1o</font>, <font color="Maroon" title="c13">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="E9:14_1_1_1_2"/><span class="lab"><font color="Green" title="E25">A21</font></span>: 
( <font color="Maroon" title="c12">1o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K4" title="SUPINF_2:func.4">inf</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c13">1ra</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="E10:14_1_1_1_2"/>
<font color="Maroon" title="c12">1o</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">1ra</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:14_1_1"><span class="lab"><font color="Green" title="E16">A5</font></span></a></span>;<br/>
<a NAME="E11:14_1_1_1_2"/>
<font color="Maroon" title="c12">1o</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">1ra</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:14_1"><span class="lab"><font color="Green" title="E14">A3</font></span></a>, <a class="txt" href="#E4:14_1_1_1_2"><span class="lab"><font color="Green" title="E23">A19</font></span></a>, <a class="txt" href="#E9:14_1_1_1_2"><span class="lab"><font color="Green" title="E25">A21</font></span></a>, <a class="ref" href="measure5.html#D1" onmouseover="rs('measure5/D1')" onmouseout="rh()">MEASURE5:def 1</a></span>;<br/>
<a NAME="E12:14_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A22</font></span>: 
<font color="Maroon" title="c2">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">1o</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="c13">1ra</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:14"><span class="lab"><font color="Green" title="E12">A1</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="E13:14_1_1_1_2"/>
<font color="Maroon" title="c11">z2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K5" title="SUPINF_2:func.5">sup</a> <font color="Maroon" title="c1">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:14_1"><span class="lab"><font color="Green" title="E14">A3</font></span></a>, <a class="txt" href="#E4:14_1_1_1_2"><span class="lab"><font color="Green" title="E23">A19</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="c14">2o</font>, <font color="Maroon" title="c15">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="E15:14_1_1_1_2"/><span class="lab"><font color="Green" title="E27">A23</font></span>: 
( <font color="Maroon" title="c14">2o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">z2</font> &amp; <font color="Maroon" title="c15">2r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K5" title="SUPINF_2:func.5">sup</a> <font color="Maroon" title="c1">A</font> )
 <span class="kw">and </span><br/><a NAME="E16:14_1_1_1_2"/>
<font color="Maroon" title="c14">2o</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">2r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:14_1_1"><span class="lab"><font color="Green" title="E17">A6</font></span></a></span>;<br/>
<a NAME="E17:14_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="c14">2o</font> is   <a href="supinf_1.html#NM1" title="SUPINF_1:NM.1">R_eal</a> &amp; <font color="Maroon" title="c2">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c15">2r</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="c16">2o1</font>, <font color="Maroon" title="c17">2r1</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="E19:14_1_1_1_2"/><span class="lab"><font color="Green" title="E28">A24</font></span>: 
( <font color="Maroon" title="c16">2o1</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="c14">2o</font> &amp; <font color="Maroon" title="c17">2r1</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="c15">2r</font> )
 ;<br/>
<a NAME="E20:14_1_1_1_2"/>
<font color="Maroon" title="c14">2o</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">2r</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:14_1"><span class="lab"><font color="Green" title="E14">A3</font></span></a>, <a class="txt" href="#E4:14_1_1_1_2"><span class="lab"><font color="Green" title="E23">A19</font></span></a>, <a class="txt" href="#E15:14_1_1_1_2"><span class="lab"><font color="Green" title="E27">A23</font></span></a>, <a class="ref" href="measure5.html#D1" onmouseover="rs('measure5/D1')" onmouseout="rh()">MEASURE5:def 1</a></span>;<br/>
<a NAME="E21:14_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c16">2o1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c17">2r1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:14"><span class="lab"><font color="Green" title="E12">A1</font></span></a>, <a class="txt" href="#E19:14_1_1_1_2"><span class="lab"><font color="Green" title="E28">A24</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="E22:14_1_1_1_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="c6">d</font>,<font color="Maroon" title="c7">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="#E1:14_1_1"><span class="lab"><font color="Green" title="E16">A5</font></span></a>, <a class="txt" href="#E2:14_1_1"><span class="lab"><font color="Green" title="E17">A6</font></span></a>, <a class="txt" href="#E5:14_1_1_1"><span class="lab"><font color="Green" title="E19">A8</font></span></a>, <a class="txt" href="#E8:14_1_1_1"><span class="lab"><font color="Green" title="E20">A9</font></span></a>, <a class="txt" href="#E5:14_1_1_1_2"><span class="lab"><font color="Green" title="E24">A20</font></span></a>, <a class="txt" href="#E9:14_1_1_1_2"><span class="lab"><font color="Green" title="E25">A21</font></span></a>, <a class="txt" href="#E15:14_1_1_1_2"><span class="lab"><font color="Green" title="E27">A23</font></span></a>, <a class="txt" href="#E12:14_1_1_1_2"><span class="lab"><font color="Green" title="E26">A22</font></span></a>, <a class="txt" href="#E19:14_1_1_1_2"><span class="lab"><font color="Green" title="E28">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="E11:14_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> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">].</a><span class="default"><font color="Maroon" title="c6">d</font>,<font color="Maroon" title="c7">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="#E9:14_1_1_1"><span class="lab"><font color="Green" title="E21">A10</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:14_1_1_1"/>
(  <a href="supinf_2.html#K4" title="SUPINF_2:func.4">inf</a> <font color="Maroon" title="c3">B</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="c4">s</font> &amp;  <a href="supinf_2.html#K5" title="SUPINF_2:func.5">sup</a> <font color="Maroon" title="c3">B</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">t</font> &amp; <font color="Maroon" title="c3">B</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="#E2:14"><span class="lab"><font color="Green" title="E13">A2</font></span></a>, <a class="txt" href="#E5:14_1_1_1"><span class="lab"><font color="Green" title="E19">A8</font></span></a>, <a class="txt" href="#E8:14_1_1_1"><span class="lab"><font color="Green" title="E20">A9</font></span></a>, <a class="txt" href="#E2:14_1_1_1"><span class="lab"><font color="Green" title="E18">A7</font></span></a>, <a class="ref" href="measure5.html#D2" onmouseover="rs('measure5/D2')" onmouseout="rh()">MEASURE5:def 2</a>, <a class="ref" href="measure6.html#T8" onmouseover="rs('measure6/T8')" onmouseout="rh()">MEASURE6:8</a>, <a class="ref" href="measure6.html#T12" onmouseover="rs('measure6/T12')" onmouseout="rh()">MEASURE6:12</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="E4:14_1_1"/>
(  <a href="supinf_2.html#K4" title="SUPINF_2:func.4">inf</a> <font color="Maroon" title="c3">B</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="c4">s</font> &amp;  <a href="supinf_2.html#K5" title="SUPINF_2:func.5">sup</a> <font color="Maroon" title="c3">B</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">t</font> &amp; <font color="Maroon" title="c3">B</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>
<a NAME="E3:14_1"/>
<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="#E1:14_1"><span class="lab"><font color="Green" title="E14">A3</font></span></a>, <a class="ref" href="measure5.html#D2" onmouseover="rs('measure5/D2')" onmouseout="rh()">MEASURE5:def 2</a></span>;<br/>
<a NAME="E4:14_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> 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="#E1:14"><span class="lab"><font color="Green" title="E12">A1</font></span></a>, <a class="ref" href="urysohn2.html#T7" target="_self" onmouseover="rs('urysohn2/T7')" onmouseout="rh()">Th7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:14_1"/>
( <font color="Maroon" title="c3">B</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"><span class="p2">(<span class="default"><a href="supinf_2.html#K4" title="SUPINF_2:func.4">inf</a> <font color="Maroon" title="c3">B</font></span>)</span>,<span class="p2">(<span class="default"><a href="supinf_2.html#K5" title="SUPINF_2:func.5">sup</a> <font color="Maroon" title="c3">B</font></span>)</span></span><a href="measure5.html#K1" title="MEASURE5:func.1">.[</a></span> &amp; (  for <font color="Olive" title="b1">s</font>, <font color="Olive" title="b2">t</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K4" title="SUPINF_2:func.4">inf</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Olive" title="b2">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K5" title="SUPINF_2:func.5">sup</a> <font color="Maroon" title="c1">A</font> holds <br/>(  <a href="supinf_2.html#K4" title="SUPINF_2:func.4">inf</a> <font color="Maroon" title="c3">B</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="Olive" title="b1">s</font> &amp;  <a href="supinf_2.html#K5" title="SUPINF_2:func.5">sup</a> <font color="Maroon" title="c3">B</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="Olive" title="b2">t</font> ) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:14"><span class="lab"><font color="Green" title="E13">A2</font></span></a>, <a class="txt" href="#E2:14_1"><span class="lab"><font color="Green" title="E15">A4</font></span></a>, <a class="ref" href="measure6.html#T16" onmouseover="rs('measure6/T16')" onmouseout="rh()">MEASURE6:16</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="E4:14"/>
(  not <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"><span class="p2">(<span class="default"><a href="supinf_2.html#K4" title="SUPINF_2:func.4">inf</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="supinf_2.html#K5" title="SUPINF_2:func.5">sup</a> <font color="Maroon" title="c1">A</font></span>)</span></span><a href="measure5.html#K1" title="MEASURE5:func.1">.[</a></span> or ( <font color="Maroon" title="c3">B</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"><span class="p2">(<span class="default"><a href="supinf_2.html#K4" title="SUPINF_2:func.4">inf</a> <font color="Maroon" title="c3">B</font></span>)</span>,<span class="p2">(<span class="default"><a href="supinf_2.html#K5" title="SUPINF_2:func.5">sup</a> <font color="Maroon" title="c3">B</font></span>)</span></span><a href="measure5.html#K1" title="MEASURE5:func.1">.[</a></span> &amp; (  for <font color="Olive" title="b1">s</font>, <font color="Olive" title="b2">t</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K4" title="SUPINF_2:func.4">inf</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Olive" title="b2">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K5" title="SUPINF_2:func.5">sup</a> <font color="Maroon" title="c1">A</font> holds <br/>(  <a href="supinf_2.html#K4" title="SUPINF_2:func.4">inf</a> <font color="Maroon" title="c3">B</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="Olive" title="b1">s</font> &amp;  <a href="supinf_2.html#K5" title="SUPINF_2:func.5">sup</a> <font color="Maroon" title="c3">B</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="Olive" title="b2">t</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"> verum</span><br/>


</div>
