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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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">D1</font>, <font color="Olive" title="b2">D2</font> being    <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Maroon" title="c1">A</font><br/>  for <font color="Olive" title="b3">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="b3">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">A</font> is  <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a>  holds <br/> <a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> (<font color="Olive" title="b3">f</font>,<font color="Olive" title="b1">D1</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="Olive" title="b3">f</font>,<font color="Olive" title="b2">D2</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c2">D1</font>, <font color="Maroon" title="c3">D2</font> be    <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Maroon" title="c1">A</font>; <a class="txt" 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="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a>  holds <br/> <a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> (<font color="Olive" title="b1">f</font>,<font color="Maroon" title="c2">D1</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="Olive" title="b1">f</font>,<font color="Maroon" title="c3">D2</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c4">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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">A</font> is  <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a>  implies  <a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c2">D1</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="c4">f</font>,<font color="Maroon" title="c3">D2</font>) )</span><br/>







<span class="kw">consider </span><font color="Maroon" title="c5">D</font> being    <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Maroon" title="c1">A</font><span class="kw"> such that </span><br/><a NAME="E2:70"/><span class="lab"><font color="Green" title="E53">A1</font></span>: 
<font color="Maroon" title="c2">D1</font> <a href="integra1.html#R1" title="INTEGRA1:pred.1">&lt;=</a> <font color="Maroon" title="c5">D</font>
 <span class="kw">and </span><br/><a NAME="E3:70"/><span class="lab"><font color="Green" title="E54">A2</font></span>: 
<font color="Maroon" title="c3">D2</font> <a href="integra1.html#R1" title="INTEGRA1:pred.1">&lt;=</a> <font color="Maroon" title="c5">D</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#T47" target="_self" title="INTEGRA1:th.47">Th49</a></span>;<br/>
<span class="kw">assume </span><a NAME="E4:70"/><span class="lab"><font color="Green" title="E55">A3</font></span>: 
<font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">A</font> is  <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c2">D1</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="c4">f</font>,<font color="Maroon" title="c3">D2</font>)</span><br/>

<a NAME="E5:70"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A4</font></span>: 
 <a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">D</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="c4">f</font>,<font color="Maroon" title="c5">D</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#T28" target="_self" title="INTEGRA1:th.28">Th30</a></span>;<br/>
<a NAME="E6:70"/>
 <a href="integra1.html#K6" title="INTEGRA1:func.6">upper_sum</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">D</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="c4">f</font>,<font color="Maroon" title="c3">D2</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:70"><span class="lab"><font color="Green" title="E55">A3</font></span></a>, <a class="txt" href="#E3:70"><span class="lab"><font color="Green" title="E54">A2</font></span></a>, <a class="ref" href="integra1.html#T45" target="_self" title="INTEGRA1:th.45">Th47</a></span>;<br/>
<a NAME="E7:70"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A5</font></span>: 
 <a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">D</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="c4">f</font>,<font color="Maroon" title="c3">D2</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:70"><span class="lab"><font color="Green" title="E56">A4</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E8:70"/>
 <a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c2">D1</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">D</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:70"><span class="lab"><font color="Green" title="E55">A3</font></span></a>, <a class="txt" href="#E2:70"><span class="lab"><font color="Green" title="E53">A1</font></span></a>, <a class="ref" href="integra1.html#T46" target="_self" title="INTEGRA1:th.46">Th48</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:70"/>
 <a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c2">D1</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="c4">f</font>,<font color="Maroon" title="c3">D2</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:70"><span class="lab"><font color="Green" title="E57">A5</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>; <a class="txt" 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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