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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>, <font color="Olive" title="b2">g</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>  &amp; <font color="Olive" title="b1">f</font> is  <a href="integra1.html#V3" title="INTEGRA1:attr.3">integrable</a>  &amp; <font color="Olive" title="b2">g</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>  &amp; <font color="Olive" title="b2">g</font> is  <a href="integra1.html#V3" title="INTEGRA1:attr.3">integrable</a>  holds <br/>( <font color="Olive" title="b1">f</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Olive" title="b2">g</font> is  <a href="integra1.html#V3" title="INTEGRA1:attr.3">integrable</a>  &amp;  <a href="integra1.html#K12" title="INTEGRA1:func.12">integral</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Olive" title="b2">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K12" title="INTEGRA1:func.12">integral</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K12" title="INTEGRA1:func.12">integral</a> <font color="Olive" title="b2">g</font></span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">f</font>, <font color="Maroon" title="c3">g</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>  &amp; <font color="Maroon" title="c2">f</font> is  <a href="integra1.html#V3" title="INTEGRA1:attr.3">integrable</a>  &amp; <font color="Maroon" title="c3">g</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>  &amp; <font color="Maroon" title="c3">g</font> is  <a href="integra1.html#V3" title="INTEGRA1:attr.3">integrable</a>  implies ( <font color="Maroon" title="c2">f</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c3">g</font> is  <a href="integra1.html#V3" title="INTEGRA1:attr.3">integrable</a>  &amp;  <a href="integra1.html#K12" title="INTEGRA1:func.12">integral</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K12" title="INTEGRA1:func.12">integral</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K12" title="INTEGRA1:func.12">integral</a> <font color="Maroon" title="c3">g</font></span>)</span> ) )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:35"/><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>  &amp; <font color="Maroon" title="c2">f</font> is  <a href="integra1.html#V3" title="INTEGRA1:attr.3">integrable</a>  )
 <span class="kw">and </span><br/><a NAME="E2:35"/><span class="lab"><font color="Green" title="E26">A2</font></span>: 
<font color="Maroon" title="c3">g</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> 
 <span class="kw">and </span><br/><a NAME="E3:35"/><span class="lab"><font color="Green" title="E27">A3</font></span>: 
<font color="Maroon" title="c3">g</font> is  <a href="integra1.html#V3" title="INTEGRA1:attr.3">integrable</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="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c3">g</font> is  <a href="integra1.html#V3" title="INTEGRA1:attr.3">integrable</a>  &amp;  <a href="integra1.html#K12" title="INTEGRA1:func.12">integral</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K12" title="INTEGRA1:func.12">integral</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K12" title="INTEGRA1:func.12">integral</a> <font color="Maroon" title="c3">g</font></span>)</span> )</span><br/>

<a NAME="E4:35"/><span class="lab"><font color="Green" title="E28">A4</font></span>: 
 <a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <font color="Maroon" title="c3">g</font> is  <a href="integra1.html#V3" title="INTEGRA1:attr.3">integrable</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35"><span class="lab"><font color="Green" title="E26">A2</font></span></a>, <a class="txt" href="#E3:35"><span class="lab"><font color="Green" title="E27">A3</font></span></a>, <a class="ref" href="integra2.html#T31" target="_self" onmouseover="rs('integra2/T31')" onmouseout="rh()">Th31</a></span>;<br/>
<a NAME="E5:35"/><span class="lab"><font color="Green" title="E29">A5</font></span>: 
<span class="p1">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <font color="Maroon" title="c3">g</font></span>)</span> <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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35"><span class="lab"><font color="Green" title="E26">A2</font></span></a>, <a class="ref" href="rfunct_1.html#T82" onmouseover="rs('rfunct_1/T82')" onmouseout="rh()">RFUNCT_1:82</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:35"/>
<font color="Maroon" title="c2">f</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c3">g</font> is  <a href="integra1.html#V3" title="INTEGRA1:attr.3">integrable</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35"><span class="lab"><font color="Green" title="E25">A1</font></span></a>, <a class="txt" href="#E4:35"><span class="lab"><font color="Green" title="E28">A4</font></span></a>, <a class="ref" href="integra1.html#T57" onmouseover="rs('integra1/T57')" onmouseout="rh()">INTEGRA1:57</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">  <a href="integra1.html#K12" title="INTEGRA1:func.12">integral</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K12" title="INTEGRA1:func.12">integral</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K12" title="INTEGRA1:func.12">integral</a> <font color="Maroon" title="c3">g</font></span>)</span></span><br/>

<a NAME="E7:35"/>
 <a href="integra1.html#K12" title="INTEGRA1:func.12">integral</a> <span class="p1">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K12" title="INTEGRA1:func.12">integral</a> <font color="Maroon" title="c3">g</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35"><span class="lab"><font color="Green" title="E26">A2</font></span></a>, <a class="txt" href="#E3:35"><span class="lab"><font color="Green" title="E27">A3</font></span></a>, <a class="ref" href="integra2.html#T31" target="_self" onmouseover="rs('integra2/T31')" onmouseout="rh()">Th31</a></span>;<br/>
<a NAME="E8:35"/><span class="kw">then </span>
 <a href="integra1.html#K12" title="INTEGRA1:func.12">integral</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K12" title="INTEGRA1:func.12">integral</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p2">(<span class="default"><a href="integra1.html#K12" title="INTEGRA1:func.12">integral</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35"><span class="lab"><font color="Green" title="E25">A1</font></span></a>, <a class="txt" href="#E5:35"><span class="lab"><font color="Green" title="E29">A5</font></span></a>, <a class="txt" href="#E4:35"><span class="lab"><font color="Green" title="E28">A4</font></span></a>, <a class="ref" href="integra1.html#T57" onmouseover="rs('integra1/T57')" onmouseout="rh()">INTEGRA1:57</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:35"/>
 <a href="integra1.html#K12" title="INTEGRA1:func.12">integral</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K12" title="INTEGRA1:func.12">integral</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K12" title="INTEGRA1:func.12">integral</a> <font color="Maroon" title="c3">g</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"> verum</span><br/>


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