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<span class="kw">let </span><font color="Maroon" title="c1">X</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</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">F</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>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> holds <br/> ( <font color="Olive" title="b2">F</font> <a href="integra7.html#R1" title="INTEGRA7:pred.1">is_integral_of</a> <font color="Olive" title="b1">f</font>,<font color="Maroon" title="c1">X</font> iff ( <font color="Olive" title="b2">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="Olive" title="b2">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="Olive" title="b1">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">f</font>, <font color="Maroon" title="c3">F</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>; <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="c3">F</font> <a href="integra7.html#R1" title="INTEGRA7:pred.1">is_integral_of</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">X</font> iff ( <font color="Maroon" title="c3">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="c3">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="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font> ) )</span><br/>





<div><span class="kw">hereby </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"> ( <font color="Maroon" title="c3">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="c3">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="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font> implies <font color="Maroon" title="c3">F</font> <a href="integra7.html#R1" title="INTEGRA7:pred.1">is_integral_of</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">X</font> )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:12_1"/>
<font color="Maroon" title="c3">F</font> <a href="integra7.html#R1" title="INTEGRA7:pred.1">is_integral_of</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">X</font>
 ; <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="c3">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="c3">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="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font> )</span><br/><a NAME="E2:12_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">F</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="integra7.html#K1" title="INTEGRA7:func.1">IntegralFuncs</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">X</font>)
 ;<br/><a NAME="E3:12_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">F9</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>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> st <br/>( <font color="Maroon" title="c3">F</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b1">F9</font> &amp; <font color="Olive" title="b1">F9</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="Olive" title="b1">F9</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="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="integra7.html#D1" target="_self" onmouseover="rs('integra7/D1')" onmouseout="rh()">Def1</a></span>;<br/><span class="kw">hence </span><a NAME="E4:12_1"/>
( <font color="Maroon" title="c3">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="c3">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="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font> )
 ; <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/></div>
<span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E2:12"/>
( <font color="Maroon" title="c3">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="c3">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="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font> )
 ; <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="c3">F</font> <a href="integra7.html#R1" title="INTEGRA7:pred.1">is_integral_of</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">X</font></span><br/>

<a NAME="E3:12"/><span class="kw">then </span>
<font color="Maroon" title="c3">F</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="integra7.html#K1" title="INTEGRA7:func.1">IntegralFuncs</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">X</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra7.html#D1" target="_self" onmouseover="rs('integra7/D1')" onmouseout="rh()">Def1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:12"/>
<font color="Maroon" title="c3">F</font> <a href="integra7.html#R1" title="INTEGRA7:pred.1">is_integral_of</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">X</font>
 ; <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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