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<span class="kw">let </span><font color="Maroon" title="c1">X</font> be   <a href="normsp_1.html#NM1" title="NORMSP_1:NM.1">RealNormSpace</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">A</font> being   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><br/>  for <font color="Olive" title="b2">f1</font>, <font color="Olive" title="b3">f2</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b2">f1</font> <a href="integr18.html#R1" title="INTEGR18:pred.1">is_integrable_on</a> <font color="Olive" title="b1">A</font> &amp; <font color="Olive" title="b3">f2</font> <a href="integr18.html#R1" title="INTEGR18:pred.1">is_integrable_on</a> <font color="Olive" title="b1">A</font> &amp; <font color="Olive" title="b1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b2">f1</font> &amp; <font color="Olive" title="b1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b3">f2</font> holds <br/>( <font color="Olive" title="b2">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Olive" title="b3">f2</font> <a href="integr18.html#R1" title="INTEGR18:pred.1">is_integrable_on</a> <font color="Olive" title="b1">A</font> &amp;  <a href="integr18.html#K5" title="INTEGR18:func.5">integral</a> (<span class="p1">(<span class="default"><font color="Olive" title="b2">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Olive" title="b3">f2</font></span>)</span>,<font color="Olive" title="b1">A</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integr18.html#K5" title="INTEGR18:func.5">integral</a> (<font color="Olive" title="b2">f1</font>,<font color="Olive" title="b1">A</font>)</span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><a href="integr18.html#K5" title="INTEGR18:func.5">integral</a> (<font color="Olive" title="b3">f2</font>,<font color="Olive" title="b1">A</font>)</span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">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">f1</font>, <font color="Olive" title="b2">f2</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">f1</font> <a href="integr18.html#R1" title="INTEGR18:pred.1">is_integrable_on</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Olive" title="b2">f2</font> <a href="integr18.html#R1" title="INTEGR18:pred.1">is_integrable_on</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b1">f1</font> &amp; <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b2">f2</font> holds <br/>( <font color="Olive" title="b1">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Olive" title="b2">f2</font> <a href="integr18.html#R1" title="INTEGR18:pred.1">is_integrable_on</a> <font color="Maroon" title="c2">A</font> &amp;  <a href="integr18.html#K5" title="INTEGR18:func.5">integral</a> (<span class="p1">(<span class="default"><font color="Olive" title="b1">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Olive" title="b2">f2</font></span>)</span>,<font color="Maroon" title="c2">A</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integr18.html#K5" title="INTEGR18:func.5">integral</a> (<font color="Olive" title="b1">f1</font>,<font color="Maroon" title="c2">A</font>)</span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><a href="integr18.html#K5" title="INTEGR18:func.5">integral</a> (<font color="Olive" title="b2">f2</font>,<font color="Maroon" title="c2">A</font>)</span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">f1</font>, <font color="Maroon" title="c4">f2</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>; <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="c3">f1</font> <a href="integr18.html#R1" title="INTEGR18:pred.1">is_integrable_on</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c4">f2</font> <a href="integr18.html#R1" title="INTEGR18:pred.1">is_integrable_on</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">f1</font> &amp; <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">f2</font> implies ( <font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c4">f2</font> <a href="integr18.html#R1" title="INTEGR18:pred.1">is_integrable_on</a> <font color="Maroon" title="c2">A</font> &amp;  <a href="integr18.html#K5" title="INTEGR18:func.5">integral</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c4">f2</font></span>)</span>,<font color="Maroon" title="c2">A</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integr18.html#K5" title="INTEGR18:func.5">integral</a> (<font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c2">A</font>)</span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><a href="integr18.html#K5" title="INTEGR18:func.5">integral</a> (<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c2">A</font>)</span>)</span> ) )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:23"/><span class="lab"><font color="Green" title="E18">A1</font></span>: 
( <font color="Maroon" title="c3">f1</font> <a href="integr18.html#R1" title="INTEGR18:pred.1">is_integrable_on</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c4">f2</font> <a href="integr18.html#R1" title="INTEGR18:pred.1">is_integrable_on</a> <font color="Maroon" title="c2">A</font> )
 <span class="kw">and </span><br/><a NAME="E2:23"/><span class="lab"><font color="Green" title="E19">A2</font></span>: 
( <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">f1</font> &amp; <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">f2</font> )
 ; <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="c3">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c4">f2</font> <a href="integr18.html#R1" title="INTEGR18:pred.1">is_integrable_on</a> <font color="Maroon" title="c2">A</font> &amp;  <a href="integr18.html#K5" title="INTEGR18:func.5">integral</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c4">f2</font></span>)</span>,<font color="Maroon" title="c2">A</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integr18.html#K5" title="INTEGR18:func.5">integral</a> (<font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c2">A</font>)</span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><a href="integr18.html#K5" title="INTEGR18:func.5">integral</a> (<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c2">A</font>)</span>)</span> )</span><br/>

<a NAME="E3:23"/><span class="lab"><font color="Green" title="E20">A3</font></span>: 
<font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c4">f2</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K6" title="VFUNCT_1:func.6">+</a> <span class="p1">(<span class="default"><a href="vfunct_1.html#K5" title="VFUNCT_1:func.5">-</a> <font color="Maroon" title="c4">f2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vfunct_1.html#T25" title="VFUNCT_1:th.25">VFUNCT_1:25</a></span>;<br/>
<a NAME="E4:23"/><span class="lab"><font color="Green" title="E21">A4</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="vfunct_1.html#K5" title="VFUNCT_1:func.5">-</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">f2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vfunct_1.html#D5" title="VFUNCT_1:def.5">VFUNCT_1:def 5</a></span>;<br/>
<a NAME="E5:23"/><span class="lab"><font color="Green" title="E22">A5</font></span>: 
 <a href="vfunct_1.html#K5" title="VFUNCT_1:func.5">-</a> <font color="Maroon" title="c4">f2</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="vfunct_1.html#K4" title="VFUNCT_1:func.4">(#)</a> <font color="Maroon" title="c4">f2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vfunct_1.html#T23" title="VFUNCT_1:th.23">VFUNCT_1:23</a></span>;<br/>
<a NAME="E6:23"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A6</font></span>: 
 <a href="vfunct_1.html#K5" title="VFUNCT_1:func.5">-</a> <font color="Maroon" title="c4">f2</font> <a href="integr18.html#R1" title="INTEGR18:pred.1">is_integrable_on</a> <font color="Maroon" title="c2">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23"><span class="lab"><font color="Green" title="E18">A1</font></span></a>, <a class="txt" href="#E2:23"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="ref" href="integr18.html#T13" target="_self" title="INTEGR18:th.13">Th13</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:23"/>
<font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c4">f2</font> <a href="integr18.html#R1" title="INTEGR18:pred.1">is_integrable_on</a> <font color="Maroon" title="c2">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23"><span class="lab"><font color="Green" title="E18">A1</font></span></a>, <a class="txt" href="#E2:23"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="txt" href="#E3:23"><span class="lab"><font color="Green" title="E20">A3</font></span></a>, <a class="txt" href="#E4:23"><span class="lab"><font color="Green" title="E21">A4</font></span></a>, <a class="ref" href="integr18.html#T14" target="_self" title="INTEGR18:th.14">Th14</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="integr18.html#K5" title="INTEGR18:func.5">integral</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c4">f2</font></span>)</span>,<font color="Maroon" title="c2">A</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integr18.html#K5" title="INTEGR18:func.5">integral</a> (<font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c2">A</font>)</span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><a href="integr18.html#K5" title="INTEGR18:func.5">integral</a> (<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c2">A</font>)</span>)</span></span><br/>

<span class="kw">thus </span> <a href="integr18.html#K5" title="INTEGR18:func.5">integral</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c4">f2</font></span>)</span>,<font color="Maroon" title="c2">A</font>) = 
 <a href="integr18.html#K5" title="INTEGR18:func.5">integral</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K6" title="VFUNCT_1:func.6">+</a> <span class="p2">(<span class="default"><a href="vfunct_1.html#K5" title="VFUNCT_1:func.5">-</a> <font color="Maroon" title="c4">f2</font></span>)</span></span>)</span>,<font color="Maroon" title="c2">A</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="vfunct_1.html#T25" title="VFUNCT_1:th.25">VFUNCT_1:25</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="integr18.html#K5" title="INTEGR18:func.5">integral</a> (<font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c2">A</font>)</span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="integr18.html#K5" title="INTEGR18:func.5">integral</a> (<span class="p2">(<span class="default"><a href="vfunct_1.html#K5" title="VFUNCT_1:func.5">-</a> <font color="Maroon" title="c4">f2</font></span>)</span>,<font color="Maroon" title="c2">A</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23"><span class="lab"><font color="Green" title="E18">A1</font></span></a>, <a class="txt" href="#E2:23"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="txt" href="#E4:23"><span class="lab"><font color="Green" title="E21">A4</font></span></a>, <a class="txt" href="#E6:23"><span class="lab"><font color="Green" title="E23">A6</font></span></a>, <a class="ref" href="integr18.html#T14" target="_self" title="INTEGR18:th.14">Th14</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="integr18.html#K5" title="INTEGR18:func.5">integral</a> (<font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c2">A</font>)</span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><a href="integr18.html#K5" title="INTEGR18:func.5">integral</a> (<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c2">A</font>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23"><span class="lab"><font color="Green" title="E18">A1</font></span></a>, <a class="txt" href="#E2:23"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="txt" href="#E5:23"><span class="lab"><font color="Green" title="E22">A5</font></span></a>, <a class="ref" href="integr18.html#T13" target="_self" title="INTEGR18:th.13">Th13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="integr18.html#K5" title="INTEGR18:func.5">integral</a> (<font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c2">A</font>)</span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><a href="integr18.html#K5" title="INTEGR18:func.5">integral</a> (<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c2">A</font>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T16" title="RLVECT_1:th.16">RLVECT_1:16</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="integr18.html#K5" title="INTEGR18:func.5">integral</a> (<font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c2">A</font>)</span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><a href="integr18.html#K5" title="INTEGR18:func.5">integral</a> (<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c2">A</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D11" title="RLVECT_1:def.11">RLVECT_1:def 11</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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