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<span class="kw">let </span><font color="Maroon" title="c1">A</font> be   <a href="integra1.html#V1" title="INTEGRA1:attr.1">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="Red">::  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="Red">::  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="Red">::  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="E2:77"/><span class="lab"><font color="Green" title="E57">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="E3:77"/><span class="lab"><font color="Green" title="E58">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#T49" target="_self" title="INTEGRA1:th.49">Th49</a></span>;<br/>
<span class="kw">assume </span><a NAME="E4:77"/><span class="lab"><font color="Green" title="E59">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="Red">::  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="E5:77"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">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#T30" target="_self" title="INTEGRA1:th.30">Th30</a></span>;<br/>
<a NAME="E6:77"/>
 <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="#E4:77"><span class="lab"><font color="Green" title="E59">A3</font></span></a>, <a class="txt" href="#E3:77"><span class="lab"><font color="Green" title="E58">A2</font></span></a>, <a class="ref" href="integra1.html#T47" target="_self" title="INTEGRA1:th.47">Th47</a></span>;<br/>
<a NAME="E7:77"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">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="#E5:77"><span class="lab"><font color="Green" title="E60">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="E8:77"/>
 <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="#E4:77"><span class="lab"><font color="Green" title="E59">A3</font></span></a>, <a class="txt" href="#E2:77"><span class="lab"><font color="Green" title="E57">A1</font></span></a>, <a class="ref" href="integra1.html#T48" target="_self" title="INTEGRA1:th.48">Th48</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:77"/>
 <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="#E7:77"><span class="lab"><font color="Green" title="E61">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="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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