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<span class="kw">let </span><font color="Maroon" title="c1">r</font> be   <a href="xreal_0.html#NM1" title="XREAL_0:NM.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">f1</font>, <font color="Olive" title="b2">f2</font>, <font color="Olive" title="b3">f3</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><br/>  for <font color="Olive" title="b4">t0</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st <font color="Olive" title="b1">f1</font> <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <font color="Olive" title="b4">t0</font> &amp; <font color="Olive" title="b2">f2</font> <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <font color="Olive" title="b4">t0</font> &amp; <font color="Olive" title="b3">f3</font> <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <font color="Olive" title="b4">t0</font> holds <br/> <a href="euclid_8.html#K8" title="EUCLID_8:func.8">VFuncdiff</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Olive" title="b1">f1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Olive" title="b2">f2</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Olive" title="b3">f3</font></span>)</span>,<font color="Olive" title="b4">t0</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">r</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p1">(<span class="default"><a href="euclid_8.html#K8" title="EUCLID_8:func.8">VFuncdiff</a> (<font color="Olive" title="b1">f1</font>,<font color="Olive" title="b2">f2</font>,<font color="Olive" title="b3">f3</font>,<font color="Olive" title="b4">t0</font>)</span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">f1</font>, <font color="Maroon" title="c3">f2</font>, <font color="Maroon" title="c4">f3</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">  for <font color="Olive" title="b1">t0</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st <font color="Maroon" title="c2">f1</font> <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <font color="Olive" title="b1">t0</font> &amp; <font color="Maroon" title="c3">f2</font> <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <font color="Olive" title="b1">t0</font> &amp; <font color="Maroon" title="c4">f3</font> <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <font color="Olive" title="b1">t0</font> holds <br/> <a href="euclid_8.html#K8" title="EUCLID_8:func.8">VFuncdiff</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c2">f1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c3">f2</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c4">f3</font></span>)</span>,<font color="Olive" title="b1">t0</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">r</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p1">(<span class="default"><a href="euclid_8.html#K8" title="EUCLID_8:func.8">VFuncdiff</a> (<font color="Maroon" title="c2">f1</font>,<font color="Maroon" title="c3">f2</font>,<font color="Maroon" title="c4">f3</font>,<font color="Olive" title="b1">t0</font>)</span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c5">t0</font> be   <a href="xreal_0.html#NM1" title="XREAL_0:NM.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">f1</font> <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c5">t0</font> &amp; <font color="Maroon" title="c3">f2</font> <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c5">t0</font> &amp; <font color="Maroon" title="c4">f3</font> <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c5">t0</font> implies  <a href="euclid_8.html#K8" title="EUCLID_8:func.8">VFuncdiff</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c2">f1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c3">f2</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c4">f3</font></span>)</span>,<font color="Maroon" title="c5">t0</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">r</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p1">(<span class="default"><a href="euclid_8.html#K8" title="EUCLID_8:func.8">VFuncdiff</a> (<font color="Maroon" title="c2">f1</font>,<font color="Maroon" title="c3">f2</font>,<font color="Maroon" title="c4">f3</font>,<font color="Maroon" title="c5">t0</font>)</span>)</span> )</span><br/>









<span class="kw">assume </span><a NAME="E1:103"/><span class="lab"><font color="Green" title="E41">A1</font></span>: 
( <font color="Maroon" title="c2">f1</font> <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c5">t0</font> &amp; <font color="Maroon" title="c3">f2</font> <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c5">t0</font> &amp; <font color="Maroon" title="c4">f3</font> <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c5">t0</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">  <a href="euclid_8.html#K8" title="EUCLID_8:func.8">VFuncdiff</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c2">f1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c3">f2</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c4">f3</font></span>)</span>,<font color="Maroon" title="c5">t0</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">r</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p1">(<span class="default"><a href="euclid_8.html#K8" title="EUCLID_8:func.8">VFuncdiff</a> (<font color="Maroon" title="c2">f1</font>,<font color="Maroon" title="c3">f2</font>,<font color="Maroon" title="c4">f3</font>,<font color="Maroon" title="c5">t0</font>)</span>)</span></span><br/>

<span class="kw">set </span><font color="Maroon" title="c6">p</font> = <span class="p1"><a href="euclid_8.html#K1" title="EUCLID_8:func.1">|[</a><span class="default"><span class="p2">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> (<font color="Maroon" title="c2">f1</font>,<font color="Maroon" title="c5">t0</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> (<font color="Maroon" title="c3">f2</font>,<font color="Maroon" title="c5">t0</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> (<font color="Maroon" title="c4">f3</font>,<font color="Maroon" title="c5">t0</font>)</span>)</span></span><a href="euclid_8.html#K1" title="EUCLID_8:func.1">]|</a></span>;<br/>
 <a href="euclid_8.html#K8" title="EUCLID_8:func.8">VFuncdiff</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c2">f1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c3">f2</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c4">f3</font></span>)</span>,<font color="Maroon" title="c5">t0</font>) = 
<span class="p1"><a href="euclid_8.html#K1" title="EUCLID_8:func.1">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> (<font color="Maroon" title="c2">f1</font>,<font color="Maroon" title="c5">t0</font>)</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c3">f2</font></span>)</span>,<font color="Maroon" title="c5">t0</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c4">f3</font></span>)</span>,<font color="Maroon" title="c5">t0</font>)</span>)</span></span><a href="euclid_8.html#K1" title="EUCLID_8:func.1">]|</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:103"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="ref" href="fdiff_1.html#T15" onmouseover="rs('fdiff_1/T15')" onmouseout="rh()">FDIFF_1:15</a></span>
<br/>.= 
<span class="p1"><a href="euclid_8.html#K1" title="EUCLID_8:func.1">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> (<font color="Maroon" title="c2">f1</font>,<font color="Maroon" title="c5">t0</font>)</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> (<font color="Maroon" title="c3">f2</font>,<font color="Maroon" title="c5">t0</font>)</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c4">f3</font></span>)</span>,<font color="Maroon" title="c5">t0</font>)</span>)</span></span><a href="euclid_8.html#K1" title="EUCLID_8:func.1">]|</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:103"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="ref" href="fdiff_1.html#T15" onmouseover="rs('fdiff_1/T15')" onmouseout="rh()">FDIFF_1:15</a></span>
<br/>.= 
<span class="p1"><a href="euclid_8.html#K1" title="EUCLID_8:func.1">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><span class="p4"><a href="euclid_8.html#K1" title="EUCLID_8:func.1">|[</a><span class="default"><span class="p5">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> (<font color="Maroon" title="c2">f1</font>,<font color="Maroon" title="c5">t0</font>)</span>)</span>,<span class="p5">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> (<font color="Maroon" title="c3">f2</font>,<font color="Maroon" title="c5">t0</font>)</span>)</span>,<span class="p5">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> (<font color="Maroon" title="c4">f3</font>,<font color="Maroon" title="c5">t0</font>)</span>)</span></span><a href="euclid_8.html#K1" title="EUCLID_8:func.1">]|</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><span class="p4"><a href="euclid_8.html#K1" title="EUCLID_8:func.1">|[</a><span class="default"><span class="p5">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> (<font color="Maroon" title="c2">f1</font>,<font color="Maroon" title="c5">t0</font>)</span>)</span>,<span class="p5">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> (<font color="Maroon" title="c3">f2</font>,<font color="Maroon" title="c5">t0</font>)</span>)</span>,<span class="p5">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> (<font color="Maroon" title="c4">f3</font>,<font color="Maroon" title="c5">t0</font>)</span>)</span></span><a href="euclid_8.html#K1" title="EUCLID_8:func.1">]|</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><span class="p4"><a href="euclid_8.html#K1" title="EUCLID_8:func.1">|[</a><span class="default"><span class="p5">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> (<font color="Maroon" title="c2">f1</font>,<font color="Maroon" title="c5">t0</font>)</span>)</span>,<span class="p5">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> (<font color="Maroon" title="c3">f2</font>,<font color="Maroon" title="c5">t0</font>)</span>)</span>,<span class="p5">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> (<font color="Maroon" title="c4">f3</font>,<font color="Maroon" title="c5">t0</font>)</span>)</span></span><a href="euclid_8.html#K1" title="EUCLID_8:func.1">]|</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3</span>)</span></span>)</span></span><a href="euclid_8.html#K1" title="EUCLID_8:func.1">]|</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:103"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="ref" href="fdiff_1.html#T15" onmouseover="rs('fdiff_1/T15')" onmouseout="rh()">FDIFF_1:15</a></span>
<br/>.= 
<font color="Maroon" title="c1">r</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p1">(<span class="default"><a href="euclid_8.html#K8" title="EUCLID_8:func.8">VFuncdiff</a> (<font color="Maroon" title="c2">f1</font>,<font color="Maroon" title="c3">f2</font>,<font color="Maroon" title="c4">f3</font>,<font color="Maroon" title="c5">t0</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="euclid_8.html#E2"><span class="lab"><font color="Green" title="E2">Lm1</font></span></a></span>
;<br/>
<span class="kw">hence </span><a NAME="E3:103"/>
 <a href="euclid_8.html#K8" title="EUCLID_8:func.8">VFuncdiff</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c2">f1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c3">f2</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c4">f3</font></span>)</span>,<font color="Maroon" title="c5">t0</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">r</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p1">(<span class="default"><a href="euclid_8.html#K8" title="EUCLID_8:func.8">VFuncdiff</a> (<font color="Maroon" title="c2">f1</font>,<font color="Maroon" title="c3">f2</font>,<font color="Maroon" title="c4">f3</font>,<font color="Maroon" title="c5">t0</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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