<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">F</font>, <font color="Maroon" title="c3">f</font>, <font color="Maroon" title="c4">G</font>, <font color="Maroon" title="c5">g</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> , <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/>









<b>assume </b><a NAME="E1:15"/>
( <font color="Maroon" title="c2">F</font> <a href="integra7.html#R1" title="INTEGRA7:pred.1">is_integral_of</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c4">G</font> <a href="integra7.html#R1" title="INTEGRA7:pred.1">is_integral_of</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c1">X</font> )
 ;<br/>

<a NAME="E2:15"/><b>then </b><i><font color="Green" title="E13">A1</font></i>: 
( <font color="Maroon" title="c2">F</font> <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c2">F</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c1">X</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c4">G</font> <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c4">G</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c1">X</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c5">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font> )
 
<b>by </b><i><a class="txt" href="integra7.html#E13"><i><font color="Green" title="E11">Lm1</font></i></a></i>;<br/>
<a NAME="E3:15"/><b>then </b><i><font color="Green" title="E14">A2</font></i>: 
( <font color="Maroon" title="c2">F</font> <a href="seq_1.html#K6" title="SEQ_1:func.6">+</a> <font color="Maroon" title="c4">G</font> <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c2">F</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <font color="Maroon" title="c4">G</font> <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a> <font color="Maroon" title="c1">X</font> )
 
<b>by </b><i><a class="ref" href="integra7.html#T6" target="_self" title="INTEGRA7:th.6">Th6</a></i>;<br/>
<a NAME="E4:15"/>
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> )
 
<b>by </b><i><a class="txt" href="integra7.html#E2:15"><i><font color="Green" title="E13">A1</font></i></a>, <a class="ref" href="fdiff_1.html#D8" title="FDIFF_1:def.8">FDIFF_1:def 8</a></i>;<br/>
<a NAME="E5:15"/><b>then </b>
( <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> &amp; <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> )
 
<b>by </b><i><a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>;<br/>
<a NAME="E6:15"/><b>then </b>
( <font color="Maroon" title="c1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font> &amp; <font color="Maroon" title="c1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">g</font> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T18" title="XBOOLE_1:th.18">XBOOLE_1:18</a></i>;<br/>
<a NAME="E7:15"/><b>then </b>
<font color="Maroon" title="c1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">g</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T19" title="XBOOLE_1:th.19">XBOOLE_1:19</a></i>;<br/>
<a NAME="E8:15"/><b>then </b><i><font color="Green" title="E15">A3</font></i>: 
( <font color="Maroon" title="c1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="seq_1.html#K6" title="SEQ_1:func.6">+</a> <font color="Maroon" title="c5">g</font></span>)</span> &amp; <font color="Maroon" title="c1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <font color="Maroon" title="c5">g</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="seq_1.html#D3" title="SEQ_1:def.3">SEQ_1:def 3</a>, <a class="ref" href="seq_1.html#D4" title="SEQ_1:def.4">SEQ_1:def 4</a></i>;<br/>
<a NAME="E9:15"/><b>then </b>
( <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="seq_1.html#K6" title="SEQ_1:func.6">+</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font></span>)</span> &amp; <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a></i>;<br/>
<a NAME="E10:15"/><b>then </b><i><font color="Green" title="E16">A4</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="seq_1.html#K6" title="SEQ_1:func.6">+</a> <font color="Maroon" title="c4">G</font></span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="seq_1.html#K6" title="SEQ_1:func.6">+</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font></span>)</span> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <font color="Maroon" title="c4">G</font></span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="integra7.html#E3:15"><i><font color="Green" title="E14">A2</font></i></a>, <a class="ref" href="fdiff_1.html#D8" title="FDIFF_1:def.8">FDIFF_1:def 8</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/><b>assume </b><a NAME="E1:15_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="seq_1.html#K6" title="SEQ_1:func.6">+</a> <font color="Maroon" title="c4">G</font></span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c1">X</font></span>)</span>
 ;<br/><a NAME="E2:15_1"/><b>then </b><i><font color="Green" title="E17">A5</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="integra7.html#E3:15"><i><font color="Green" title="E14">A2</font></i></a>, <a class="ref" href="fdiff_1.html#D8" title="FDIFF_1:def.8">FDIFF_1:def 8</a></i>;<br/><a NAME="E3:15_1"/><b>then </b><i><font color="Green" title="E18">A6</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="seq_1.html#K6" title="SEQ_1:func.6">+</a> <font color="Maroon" title="c4">G</font></span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="seq_1.html#K6" title="SEQ_1:func.6">+</a> <font color="Maroon" title="c4">G</font></span>)</span>,<font color="Maroon" title="c6">x</font>
 <b>by </b><i><a class="txt" href="integra7.html#E3:15"><i><font color="Green" title="E14">A2</font></i></a>, <a class="ref" href="fdiff_1.html#D8" title="FDIFF_1:def.8">FDIFF_1:def 8</a></i>;<br/><a NAME="E4:15_1"/>
( <font color="Maroon" title="c2">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font> <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c6">x</font> &amp; <font color="Maroon" title="c4">G</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font> <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c6">x</font> )
 <b>by </b><i><a class="txt" href="integra7.html#E2:15_1"><i><font color="Green" title="E17">A5</font></i></a>, <a class="txt" href="integra7.html#E2:15"><i><font color="Green" title="E13">A1</font></i></a>, <a class="ref" href="fdiff_1.html#D7" title="FDIFF_1:def.7">FDIFF_1:def 7</a></i>;<br/><a NAME="E5:15_1"/><b>then </b>
( <font color="Maroon" title="c2">F</font> <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c6">x</font> &amp; <font color="Maroon" title="c4">G</font> <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c6">x</font> )
 <b>by </b><i><a class="txt" href="integra7.html#E2:15_1"><i><font color="Green" title="E17">A5</font></i></a>, <a class="ref" href="integra7.html#T4" target="_self" title="INTEGRA7:th.4">Th4</a></i>;<br/><a NAME="E6:15_1"/><b>then </b><i><font color="Green" title="E19">A7</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="seq_1.html#K6" title="SEQ_1:func.6">+</a> <font color="Maroon" title="c4">G</font></span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c6">x</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <font color="Maroon" title="c4">G</font>,<font color="Maroon" title="c6">x</font></span>)</span>
 <b>by </b><i><a class="txt" href="integra7.html#E3:15_1"><i><font color="Green" title="E18">A6</font></i></a>, <a class="ref" href="fdiff_1.html#T21" title="FDIFF_1:th.21">FDIFF_1:21</a></i>;<br/><a NAME="E7:15_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c6">x</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">G</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <font color="Maroon" title="c4">G</font>,<font color="Maroon" title="c6">x</font> )
 <b>by </b><i><a class="txt" href="integra7.html#E2:15"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="integra7.html#E2:15_1"><i><font color="Green" title="E17">A5</font></i></a>, <a class="ref" href="fdiff_1.html#D8" title="FDIFF_1:def.8">FDIFF_1:def 8</a></i>;<br/><a NAME="E8:15_1"/><b>then </b>
( <font color="Maroon" title="c3">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c6">x</font> &amp; <font color="Maroon" title="c5">g</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <font color="Maroon" title="c4">G</font>,<font color="Maroon" title="c6">x</font> )
 <b>by </b><i><a class="txt" href="integra7.html#E2:15"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="integra7.html#E2:15_1"><i><font color="Green" title="E17">A5</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>;<br/><a NAME="E9:15_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="seq_1.html#K6" title="SEQ_1:func.6">+</a> <font color="Maroon" title="c4">G</font></span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="seq_1.html#K6" title="SEQ_1:func.6">+</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">x</font>
 <b>by </b><i><a class="txt" href="integra7.html#E6:15_1"><i><font color="Green" title="E19">A7</font></i></a>, <a class="txt" href="integra7.html#E2:15_1"><i><font color="Green" title="E17">A5</font></i></a>, <a class="txt" href="integra7.html#E8:15"><i><font color="Green" title="E15">A3</font></i></a>, <a class="ref" href="seq_1.html#D3" title="SEQ_1:def.3">SEQ_1:def 3</a></i>;<br/><b>hence </b><a NAME="E10:15_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="seq_1.html#K6" title="SEQ_1:func.6">+</a> <font color="Maroon" title="c4">G</font></span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="seq_1.html#K6" title="SEQ_1:func.6">+</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">x</font>
 <b>by </b><i><a class="txt" href="integra7.html#E2:15_1"><i><font color="Green" title="E17">A5</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>;<br/></div><b>end;</b></div>
<a NAME="E12:15"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="seq_1.html#K6" title="SEQ_1:func.6">+</a> <font color="Maroon" title="c4">G</font></span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c1">X</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="seq_1.html#K6" title="SEQ_1:func.6">+</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="txt" href="integra7.html#E10:15"><i><font color="Green" title="E16">A4</font></i></a>, <a class="ref" href="partfun1.html#T34" title="PARTFUN1:th.34">PARTFUN1:34</a></i>;<br/>
<b>hence </b><a NAME="E13:15"/>
<font color="Maroon" title="c2">F</font> <a href="seq_1.html#K6" title="SEQ_1:func.6">+</a> <font color="Maroon" title="c4">G</font> <a href="integra7.html#R1" title="INTEGRA7:pred.1">is_integral_of</a> <font color="Maroon" title="c3">f</font> <a href="seq_1.html#K6" title="SEQ_1:func.6">+</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="integra7.html#E3:15"><i><font color="Green" title="E14">A2</font></i></a>, <a class="txt" href="integra7.html#E13"><i><font color="Green" title="E11">Lm1</font></i></a></i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/><b>assume </b><a NAME="E1:15_2"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <font color="Maroon" title="c4">G</font></span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c1">X</font></span>)</span>
 ;<br/><a NAME="E2:15_2"/><b>then </b><i><font color="Green" title="E17">A8</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="integra7.html#E3:15"><i><font color="Green" title="E14">A2</font></i></a>, <a class="ref" href="fdiff_1.html#D8" title="FDIFF_1:def.8">FDIFF_1:def 8</a></i>;<br/><a NAME="E3:15_2"/><b>then </b><i><font color="Green" title="E18">A9</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <font color="Maroon" title="c4">G</font></span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <font color="Maroon" title="c4">G</font></span>)</span>,<font color="Maroon" title="c6">x</font>
 <b>by </b><i><a class="txt" href="integra7.html#E3:15"><i><font color="Green" title="E14">A2</font></i></a>, <a class="ref" href="fdiff_1.html#D8" title="FDIFF_1:def.8">FDIFF_1:def 8</a></i>;<br/><a NAME="E4:15_2"/>
( <font color="Maroon" title="c2">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font> <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c6">x</font> &amp; <font color="Maroon" title="c4">G</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font> <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c6">x</font> )
 <b>by </b><i><a class="txt" href="integra7.html#E2:15_2"><i><font color="Green" title="E17">A8</font></i></a>, <a class="txt" href="integra7.html#E2:15"><i><font color="Green" title="E13">A1</font></i></a>, <a class="ref" href="fdiff_1.html#D7" title="FDIFF_1:def.7">FDIFF_1:def 7</a></i>;<br/><a NAME="E5:15_2"/><b>then </b>
( <font color="Maroon" title="c2">F</font> <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c6">x</font> &amp; <font color="Maroon" title="c4">G</font> <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c6">x</font> )
 <b>by </b><i><a class="txt" href="integra7.html#E2:15_2"><i><font color="Green" title="E17">A8</font></i></a>, <a class="ref" href="integra7.html#T4" target="_self" title="INTEGRA7:th.4">Th4</a></i>;<br/><a NAME="E6:15_2"/><b>then </b><i><font color="Green" title="E19">A10</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <font color="Maroon" title="c4">G</font></span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c6">x</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <font color="Maroon" title="c4">G</font>,<font color="Maroon" title="c6">x</font></span>)</span>
 <b>by </b><i><a class="txt" href="integra7.html#E3:15_2"><i><font color="Green" title="E18">A9</font></i></a>, <a class="ref" href="fdiff_1.html#T22" title="FDIFF_1:th.22">FDIFF_1:22</a></i>;<br/><a NAME="E7:15_2"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c6">x</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">G</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <font color="Maroon" title="c4">G</font>,<font color="Maroon" title="c6">x</font> )
 <b>by </b><i><a class="txt" href="integra7.html#E2:15"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="integra7.html#E2:15_2"><i><font color="Green" title="E17">A8</font></i></a>, <a class="ref" href="fdiff_1.html#D8" title="FDIFF_1:def.8">FDIFF_1:def 8</a></i>;<br/><a NAME="E8:15_2"/><b>then </b>
( <font color="Maroon" title="c3">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c6">x</font> &amp; <font color="Maroon" title="c5">g</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <font color="Maroon" title="c4">G</font>,<font color="Maroon" title="c6">x</font> )
 <b>by </b><i><a class="txt" href="integra7.html#E2:15"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="integra7.html#E2:15_2"><i><font color="Green" title="E17">A8</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>;<br/><a NAME="E9:15_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <font color="Maroon" title="c4">G</font></span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">x</font>
 <b>by </b><i><a class="txt" href="integra7.html#E6:15_2"><i><font color="Green" title="E19">A10</font></i></a>, <a class="txt" href="integra7.html#E2:15_2"><i><font color="Green" title="E17">A8</font></i></a>, <a class="txt" href="integra7.html#E8:15"><i><font color="Green" title="E15">A3</font></i></a>, <a class="ref" href="seq_1.html#D4" title="SEQ_1:def.4">SEQ_1:def 4</a></i>;<br/><b>hence </b><a NAME="E10:15_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <font color="Maroon" title="c4">G</font></span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">x</font>
 <b>by </b><i><a class="txt" href="integra7.html#E2:15_2"><i><font color="Green" title="E17">A8</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>;<br/></div><b>end;</b></div>
<a NAME="E15:15"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <font color="Maroon" title="c4">G</font></span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c1">X</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="txt" href="integra7.html#E10:15"><i><font color="Green" title="E16">A4</font></i></a>, <a class="ref" href="partfun1.html#T34" title="PARTFUN1:th.34">PARTFUN1:34</a></i>;<br/>
<b>hence </b><a NAME="E16:15"/>
<font color="Maroon" title="c2">F</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <font color="Maroon" title="c4">G</font> <a href="integra7.html#R1" title="INTEGRA7:pred.1">is_integral_of</a> <font color="Maroon" title="c3">f</font> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="integra7.html#E3:15"><i><font color="Green" title="E14">A2</font></i></a>, <a class="txt" href="integra7.html#E13"><i><font color="Green" title="E11">Lm1</font></i></a></i>;<br/>


</div>
