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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#V2" title="MEASURE5:attr.2">closed_interval</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <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">  for <font color="Olive" title="b1">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">A</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <font color="Olive" title="b1">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">A</font> is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">bounded</a>  holds <br/> <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="integra1.html#K9" title="INTEGRA1:func.9">lower_sum_set</a> <font color="Olive" title="b1">f</font></span>)</span> is  <a href="xxreal_2.html#V4" title="XXREAL_2:attr.4">bounded_above</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">f</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">A</font>,<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">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">A</font> is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">bounded</a>  implies  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="integra1.html#K9" title="INTEGRA1:func.9">lower_sum_set</a> <font color="Maroon" title="c2">f</font></span>)</span> is  <a href="xxreal_2.html#V4" title="XXREAL_2:attr.4">bounded_above</a>  )</span><br/>



<span class="kw">set </span><font color="Maroon" title="c3">D1</font> =  the    <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Maroon" title="c1">A</font>;<br/>
<span class="kw">assume </span><a NAME="E1:38"/><span class="lab"><font color="Green" title="E25">A1</font></span>: 
<font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">A</font> is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">bounded</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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="integra1.html#K9" title="INTEGRA1:func.9">lower_sum_set</a> <font color="Maroon" title="c2">f</font></span>)</span> is  <a href="xxreal_2.html#V4" title="XXREAL_2:attr.4">bounded_above</a> </span><br/>

<span class="kw">take </span>
 <a href="integra1.html#K6" title="INTEGRA1:func.6">upper_sum</a> (<font color="Maroon" title="c2">f</font>, the    <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Maroon" title="c1">A</font>)
; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="xxreal_2.html#D10" onmouseover="rs('xxreal_2/D10')" onmouseout="rh()">XXREAL_2:def 10</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="integra1.html#K6" title="INTEGRA1:func.6">upper_sum</a> (<font color="Maroon" title="c2">f</font>, the    <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Maroon" title="c1">A</font>) is    <a href="xxreal_2.html#M1" title="XXREAL_2:mode.1">UpperBound</a> of  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="integra1.html#K9" title="INTEGRA1:func.9">lower_sum_set</a> <font color="Maroon" title="c2">f</font></span>)</span></span><br/>

<span class="kw">let </span><font color="Maroon" title="c4">a</font> be   <a href="xxreal_0.html#NM1" title="XXREAL_0:NM.1">ExtReal</a>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="xxreal_2.html#D1" onmouseover="rs('xxreal_2/D1')" onmouseout="rh()">XXREAL_2:def 1</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="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="integra1.html#K9" title="INTEGRA1:func.9">lower_sum_set</a> <font color="Maroon" title="c2">f</font></span>)</span> or <font color="Maroon" title="c4">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="integra1.html#K6" title="INTEGRA1:func.6">upper_sum</a> (<font color="Maroon" title="c2">f</font>, the    <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Maroon" title="c1">A</font>) )</span><br/>

<span class="kw">assume </span><a NAME="E2:38"/>
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="integra1.html#K9" title="INTEGRA1:func.9">lower_sum_set</a> <font color="Maroon" title="c2">f</font></span>)</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="c4">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="integra1.html#K6" title="INTEGRA1:func.6">upper_sum</a> (<font color="Maroon" title="c2">f</font>, the    <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Maroon" title="c1">A</font>)</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">D</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="integra1.html#K1" title="INTEGRA1:func.1">divs</a> <font color="Maroon" title="c1">A</font><span class="kw"> such that </span><br/><a NAME="E4:38"/><span class="lab"><font color="Green" title="E26">A2</font></span>: 
( <font color="Maroon" title="c5">D</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="integra1.html#K9" title="INTEGRA1:func.9">lower_sum_set</a> <font color="Maroon" title="c2">f</font></span>)</span> &amp; <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K9" title="INTEGRA1:func.9">lower_sum_set</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">D</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#T3" onmouseover="rs('partfun1/T3')" onmouseout="rh()">PARTFUN1:3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c6">D</font> = <font color="Maroon" title="c5">D</font> as    <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Maroon" title="c1">A</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#D3" onmouseover="rs('integra1/D3')" onmouseout="rh()">INTEGRA1:def 3</a></span>;<br/>
<a NAME="E6:38"/>
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">D</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:38"><span class="lab"><font color="Green" title="E26">A2</font></span></a>, <a class="ref" href="integra1.html#D11" onmouseover="rs('integra1/D11')" onmouseout="rh()">INTEGRA1:def 11</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:38"/>
<font color="Maroon" title="c4">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="integra1.html#K6" title="INTEGRA1:func.6">upper_sum</a> (<font color="Maroon" title="c2">f</font>, the    <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Maroon" title="c1">A</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:38"><span class="lab"><font color="Green" title="E25">A1</font></span></a>, <a class="ref" href="integra1.html#T48" onmouseover="rs('integra1/T48')" onmouseout="rh()">INTEGRA1:48</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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