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<span class="kw">let </span><font color="Maroon" title="c2">F</font> be   non  <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a>  <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">Z</font> being   <a href="rcomp_1.html#V3" title="RCOMP_1:attr.3">open</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">f</font> being   <a href="ndiff_3.html#V3" title="NDIFF_3:attr.3">differentiable</a>  <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="c2">F</font>  st <font color="Olive" title="b1">Z</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">f</font> holds <br/><font color="Olive" title="b2">f</font> <a href="ndiff_3.html#R2" title="NDIFF_3:pred.2">is_differentiable_on</a> <font color="Olive" title="b1">Z</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">Z</font> be   <a href="rcomp_1.html#V3" title="RCOMP_1:attr.3">open</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">f</font> being   <a href="ndiff_3.html#V3" title="NDIFF_3:attr.3">differentiable</a>  <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="c2">F</font>  st <font color="Maroon" title="c3">Z</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">f</font> holds <br/><font color="Olive" title="b1">f</font> <a href="ndiff_3.html#R2" title="NDIFF_3:pred.2">is_differentiable_on</a> <font color="Maroon" title="c3">Z</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">f</font> be   <a href="ndiff_3.html#V3" title="NDIFF_3:attr.3">differentiable</a>  <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="c2">F</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">Z</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">f</font> implies <font color="Maroon" title="c4">f</font> <a href="ndiff_3.html#R2" title="NDIFF_3:pred.2">is_differentiable_on</a> <font color="Maroon" title="c3">Z</font> )</span><br/>





<span class="kw">assume </span><a NAME="E1:40"/><span class="lab"><font color="Green" title="E27">A1</font></span>: 
<font color="Maroon" title="c3">Z</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">f</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="c4">f</font> <a href="ndiff_3.html#R2" title="NDIFF_3:pred.2">is_differentiable_on</a> <font color="Maroon" title="c3">Z</font></span><br/>

<a NAME="E2:40"/>
<font color="Maroon" title="c4">f</font> <a href="ndiff_3.html#R2" title="NDIFF_3:pred.2">is_differentiable_on</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ndiff_3.html#D7" target="_self" title="NDIFF_3:def.7">Def7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:40"/>
<font color="Maroon" title="c4">f</font> <a href="ndiff_3.html#R2" title="NDIFF_3:pred.2">is_differentiable_on</a> <font color="Maroon" title="c3">Z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="ref" href="ndiff_3.html#T24" target="_self" title="NDIFF_3:th.24">Th24</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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