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

<b>let </b><font color="Maroon" title="c1">n</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c2">i</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c3">g1</font>, <font color="Maroon" title="c4">g2</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/><b>let </b><font color="Maroon" title="c5">y</font> be    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font>;<br/>









<b>assume </b><a NAME="E1:50"/><i><font color="Green" title="E36">A1</font></i>: 
( <font color="Maroon" title="c3">g1</font> <a href="pdiff_1.html#R3" title="PDIFF_1:pred.3">is_partial_differentiable_in</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c2">i</font> &amp; <font color="Maroon" title="c4">g2</font> <a href="pdiff_1.html#R3" title="PDIFF_1:pred.3">is_partial_differentiable_in</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c2">i</font> )
 ;<br/>

<a NAME="E2:50"/><i><font color="Green" title="E37">A2</font></i>: 
( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font> &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> 1 )
 
<b>by </b><i><a class="ref" href="real_ns1.html#D4" title="REAL_NS1:def.4">REAL_NS1:def 4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c6">f1</font> =  <a href="pdiff_1.html#K3" title="PDIFF_1:func.3">&lt;&gt;*</a> <font color="Maroon" title="c3">g1</font>, <font color="Maroon" title="c7">f2</font> =  <a href="pdiff_1.html#K3" title="PDIFF_1:func.3">&lt;&gt;*</a> <font color="Maroon" title="c4">g2</font> as   <a href="struct_0.html#NM10" title="STRUCT_0:NM.10">PartFunc</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> 1</span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon" title="c8">x</font> = <font color="Maroon" title="c5">y</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">n</font></span>)</span> <b>by </b><i><a class="ref" href="real_ns1.html#D4" title="REAL_NS1:def.4">REAL_NS1:def 4</a></i>;<br/>
<a NAME="E5:50"/>
(  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c3">g1</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"><span class="p2">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> 1,1</span>)</span> <a href="partfun2.html#K2" title="PARTFUN2:func.2">"</a> </span>)</span> &amp;  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c4">g2</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"><span class="p2">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> 1,1</span>)</span> <a href="partfun2.html#K2" title="PARTFUN2:func.2">"</a> </span>)</span> )
 
<b>by </b><i><a class="ref" href="pdiff_1.html#T2" target="_self" title="PDIFF_1:th.2">Th2</a></i>;<br/>
<a NAME="E6:50"/><b>then </b><i><font color="Green" title="E38">A3</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"><span class="p3">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> 1,1</span>)</span> <a href="partfun2.html#K2" title="PARTFUN2:func.2">"</a> </span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g1</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> <font color="Maroon" title="c3">g1</font> &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"><span class="p3">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> 1,1</span>)</span> <a href="partfun2.html#K2" title="PARTFUN2:func.2">"</a> </span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">g2</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> <font color="Maroon" title="c4">g2</font> )
 
<b>by </b><i><a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a></i>;<br/>
<a NAME="E7:50"/><b>then </b>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="vfunct_1.html#K1" title="VFUNCT_1:func.1">+</a> <font color="Maroon" title="c7">f2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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">g1</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="c4">g2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="vfunct_1.html#D1" title="VFUNCT_1:def.1">VFUNCT_1:def 1</a></i>;<br/>
<a NAME="E8:50"/><b>then </b><i><font color="Green" title="E39">A4</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="vfunct_1.html#K1" title="VFUNCT_1:func.1">+</a> <font color="Maroon" title="c7">f2</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"><font color="Maroon" title="c3">g1</font> <a href="seq_1.html#K6" title="SEQ_1:func.6">+</a> <font color="Maroon" title="c4">g2</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></i>;<br/>
<a NAME="E9:50"/><i><font color="Green" title="E40">A5</font></i>: 
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g1</font> <a href="seq_1.html#K6" title="SEQ_1:func.6">+</a> <font color="Maroon" title="c4">g2</font></span>)</span> <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"><span class="p2">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> 1,1</span>)</span> <a href="partfun2.html#K2" title="PARTFUN2:func.2">"</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="pdiff_1.html#T2" target="_self" title="PDIFF_1:th.2">Th2</a></i>;<br/>
<a NAME="E10:50"/><b>then </b><i><font color="Green" title="E41">A6</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"><span class="p3">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> 1,1</span>)</span> <a href="partfun2.html#K2" title="PARTFUN2:func.2">"</a> </span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g1</font> <a href="seq_1.html#K6" title="SEQ_1:func.6">+</a> <font color="Maroon" title="c4">g2</font></span>)</span></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"><font color="Maroon" title="c3">g1</font> <a href="seq_1.html#K6" title="SEQ_1:func.6">+</a> <font color="Maroon" title="c4">g2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a></i>;<br/>
<a NAME="E11:50"/><i><font color="Green" title="E42">A7</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="vfunct_1.html#K1" title="VFUNCT_1:func.1">+</a> <font color="Maroon" title="c7">f2</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"><a href="pdiff_1.html#K3" title="PDIFF_1:func.3">&lt;&gt;*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g1</font> <a href="seq_1.html#K6" title="SEQ_1:func.6">+</a> <font color="Maroon" title="c4">g2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="pdiff_1.html#E8:50"><i><font color="Green" title="E39">A4</font></i></a>, <a class="txt" href="pdiff_1.html#E9:50"><i><font color="Green" title="E40">A5</font></i></a>, <a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c9">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">n</font></span>)</span>;<br/><b>assume </b><a NAME="E1:50_1"/><i><font color="Green" title="E43">A8</font></i>: 
<font color="Maroon" title="c9">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"><font color="Maroon" title="c6">f1</font> <a href="vfunct_1.html#K1" title="VFUNCT_1:func.1">+</a> <font color="Maroon" title="c7">f2</font></span>)</span>
 ;<br/><a NAME="E2:50_1"/><b>then </b><i><font color="Green" title="E44">A9</font></i>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">f1</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="c7">f2</font></span>)</span>
 <b>by </b><i><a class="ref" href="vfunct_1.html#D1" title="VFUNCT_1:def.1">VFUNCT_1:def 1</a></i>;<br/><a NAME="E3:50_1"/><b>then </b><i><font color="Green" title="E45">A10</font></i>: 
( <font color="Maroon" title="c9">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> <font color="Maroon" title="c6">f1</font> &amp; <font color="Maroon" title="c9">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> <font color="Maroon" title="c7">f2</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E4:50_1"/><i><font color="Green" title="E46">A11</font></i>: 
( <font color="Maroon" title="c9">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> <font color="Maroon" title="c3">g1</font> &amp; <font color="Maroon" title="c9">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> <font color="Maroon" title="c4">g2</font> )
 <b>by </b><i><a class="txt" href="pdiff_1.html#E6:50"><i><font color="Green" title="E38">A3</font></i></a>, <a class="txt" href="pdiff_1.html#E2:50_1"><i><font color="Green" title="E44">A9</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E5:50_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="vfunct_1.html#K1" title="VFUNCT_1:func.1">+</a> <font color="Maroon" title="c7">f2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="vfunct_1.html#K1" title="VFUNCT_1:func.1">+</a> <font color="Maroon" title="c7">f2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">x</font>
 <b>by </b><i><a class="txt" href="pdiff_1.html#E1:50_1"><i><font color="Green" title="E43">A8</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><a NAME="E6:50_1"/><b>then </b><i><font color="Green" title="E47">A12</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="vfunct_1.html#K1" title="VFUNCT_1:func.1">+</a> <font color="Maroon" title="c7">f2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">x</font></span>)</span>
 <b>by </b><i><a class="txt" href="pdiff_1.html#E1:50_1"><i><font color="Green" title="E43">A8</font></i></a>, <a class="ref" href="vfunct_1.html#D1" title="VFUNCT_1:def.1">VFUNCT_1:def 1</a></i>;<br/><a NAME="E7:50_1"/>
<span class="p1">(<span class="default"><a href="pdiff_1.html#K3" title="PDIFF_1:func.3">&lt;&gt;*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g1</font> <a href="seq_1.html#K6" title="SEQ_1:func.6">+</a> <font color="Maroon" title="c4">g2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> 1,1</span>)</span> <a href="partfun2.html#K2" title="PARTFUN2:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">g1</font> <a href="seq_1.html#K6" title="SEQ_1:func.6">+</a> <font color="Maroon" title="c4">g2</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c9">x</font></span>)</span>
 <b>by </b><i><a class="txt" href="pdiff_1.html#E8:50"><i><font color="Green" title="E39">A4</font></i></a>, <a class="txt" href="pdiff_1.html#E10:50"><i><font color="Green" title="E41">A6</font></i></a>, <a class="txt" href="pdiff_1.html#E1:50_1"><i><font color="Green" title="E43">A8</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>;<br/><a NAME="E8:50_1"/><b>then </b><i><font color="Green" title="E48">A13</font></i>: 
<span class="p1">(<span class="default"><a href="pdiff_1.html#K3" title="PDIFF_1:func.3">&lt;&gt;*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g1</font> <a href="seq_1.html#K6" title="SEQ_1:func.6">+</a> <font color="Maroon" title="c4">g2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> 1,1</span>)</span> <a href="partfun2.html#K2" title="PARTFUN2:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">g1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c9">x</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">g2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c9">x</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="pdiff_1.html#E8:50"><i><font color="Green" title="E39">A4</font></i></a>, <a class="txt" href="pdiff_1.html#E1:50_1"><i><font color="Green" title="E43">A8</font></i></a>, <a class="ref" href="seq_1.html#D3" title="SEQ_1:def.3">SEQ_1:def 3</a></i>;<br/><a NAME="E9:50_1"/>
( <font color="Maroon" title="c6">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> 1,1</span>)</span> <a href="partfun2.html#K2" title="PARTFUN2:func.2">"</a> </span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> &amp; <font color="Maroon" title="c7">f2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> 1,1</span>)</span> <a href="partfun2.html#K2" title="PARTFUN2:func.2">"</a> </span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">g2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> )
 <b>by </b><i><a class="txt" href="pdiff_1.html#E3:50_1"><i><font color="Green" title="E45">A10</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><a NAME="E10:50_1"/><b>then </b>
( <font color="Maroon" title="c6">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> 1,1</span>)</span> <a href="partfun2.html#K2" title="PARTFUN2:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c9">x</font></span>)</span> &amp; <font color="Maroon" title="c7">f2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> 1,1</span>)</span> <a href="partfun2.html#K2" title="PARTFUN2:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c9">x</font></span>)</span> )
 <b>by </b><i><a class="txt" href="pdiff_1.html#E4:50_1"><i><font color="Green" title="E46">A11</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>;<br/><b>hence </b><a NAME="E11:50_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="vfunct_1.html#K1" title="VFUNCT_1:func.1">+</a> <font color="Maroon" title="c7">f2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K3" title="PDIFF_1:func.3">&lt;&gt;*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g1</font> <a href="seq_1.html#K6" title="SEQ_1:func.6">+</a> <font color="Maroon" title="c4">g2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font>
 <b>by </b><i><a class="txt" href="pdiff_1.html#E6:50_1"><i><font color="Green" title="E47">A12</font></i></a>, <a class="txt" href="pdiff_1.html#E8:50_1"><i><font color="Green" title="E48">A13</font></i></a>, <a class="ref" href="pdiff_1.html#T2" target="_self" title="PDIFF_1:th.2">Th2</a>, <a class="ref" href="pdiff_1.html#T3" target="_self" title="PDIFF_1:th.3">Th3</a></i>;<br/></div><b>end;</b></div>
<a NAME="E13:50"/><b>then </b><i><font color="Green" title="E43">A14</font></i>: 
<font color="Maroon" title="c6">f1</font> <a href="vfunct_1.html#K1" title="VFUNCT_1:func.1">+</a> <font color="Maroon" title="c7">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pdiff_1.html#K3" title="PDIFF_1:func.3">&lt;&gt;*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g1</font> <a href="seq_1.html#K6" title="SEQ_1:func.6">+</a> <font color="Maroon" title="c4">g2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="pdiff_1.html#E2:50"><i><font color="Green" title="E37">A2</font></i></a>, <a class="txt" href="pdiff_1.html#E11:50"><i><font color="Green" title="E42">A7</font></i></a>, <a class="ref" href="partfun1.html#T34" title="PARTFUN1:th.34">PARTFUN1:34</a></i>;<br/>
<a NAME="E14:50"/><i><font color="Green" title="E44">A15</font></i>: 
( <font color="Maroon" title="c6">f1</font> <a href="pdiff_1.html#R2" title="PDIFF_1:pred.2">is_partial_differentiable_in</a> <font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c2">i</font> &amp; <font color="Maroon" title="c7">f2</font> <a href="pdiff_1.html#R2" title="PDIFF_1:pred.2">is_partial_differentiable_in</a> <font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c2">i</font> )
 
<b>by </b><i><a class="txt" href="pdiff_1.html#E1:50"><i><font color="Green" title="E36">A1</font></i></a>, <a class="ref" href="pdiff_1.html#T14" target="_self" title="PDIFF_1:th.14">Th14</a></i>;<br/>
<a NAME="E15:50"/><b>then </b><i><font color="Green" title="E45">A16</font></i>: 
<font color="Maroon" title="c6">f1</font> <a href="vfunct_1.html#K1" title="VFUNCT_1:func.1">+</a> <font color="Maroon" title="c7">f2</font> <a href="pdiff_1.html#R2" title="PDIFF_1:pred.2">is_partial_differentiable_in</a> <font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c2">i</font>
 
<b>by </b><i><a class="ref" href="pdiff_1.html#T28" target="_self" title="PDIFF_1:th.28">Th28</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">One</font> = <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> as   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> 1</span>)</span> <b>by </b><i><a class="txt" href="pdiff_1.html#E2:50"><i><font color="Green" title="E37">A2</font></i></a>, <a class="ref" href="finseq_2.html#T118" title="FINSEQ_2:th.118">FINSEQ_2:118</a></i>;<br/>
<a NAME="E17:50"/><i><font color="Green" title="E46">A17</font></i>: 
( <span class="p1">(<span class="default"><a href="pdiff_1.html#K9" title="PDIFF_1:func.9">partdiff</a> <font color="Maroon" title="c6">f1</font>,<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c2">i</font></span>)</span> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Maroon" title="c9">One</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K10" title="PDIFF_1:func.10">partdiff</a> <font color="Maroon" title="c3">g1</font>,<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c2">i</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> &amp; <span class="p1">(<span class="default"><a href="pdiff_1.html#K9" title="PDIFF_1:func.9">partdiff</a> <font color="Maroon" title="c7">f2</font>,<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c2">i</font></span>)</span> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Maroon" title="c9">One</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K10" title="PDIFF_1:func.10">partdiff</a> <font color="Maroon" title="c4">g2</font>,<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c2">i</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> )
 
<b>by </b><i><a class="txt" href="pdiff_1.html#E14:50"><i><font color="Green" title="E44">A15</font></i></a>, <a class="ref" href="pdiff_1.html#T15" target="_self" title="PDIFF_1:th.15">Th15</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">Pd1</font> = <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K10" title="PDIFF_1:func.10">partdiff</a> <font color="Maroon" title="c3">g1</font>,<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c2">i</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> 1 <b>by </b><i><a class="ref" href="finseq_2.html#T118" title="FINSEQ_2:th.118">FINSEQ_2:118</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">Pd2</font> = <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K10" title="PDIFF_1:func.10">partdiff</a> <font color="Maroon" title="c4">g2</font>,<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c2">i</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> 1 <b>by </b><i><a class="ref" href="finseq_2.html#T118" title="FINSEQ_2:th.118">FINSEQ_2:118</a></i>;<br/>
<span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K10" title="PDIFF_1:func.10">partdiff</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">g1</font> <a href="seq_1.html#K6" title="SEQ_1:func.6">+</a> <font color="Maroon" title="c4">g2</font></span>)</span>,<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c2">i</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> = 
<span class="p1">(<span class="default"><a href="pdiff_1.html#K9" title="PDIFF_1:func.9">partdiff</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="vfunct_1.html#K1" title="VFUNCT_1:func.1">+</a> <font color="Maroon" title="c7">f2</font></span>)</span>,<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c2">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
<b>by </b><i><a class="txt" href="pdiff_1.html#E13:50"><i><font color="Green" title="E43">A14</font></i></a>, <a class="txt" href="pdiff_1.html#E15:50"><i><font color="Green" title="E45">A16</font></i></a>, <a class="ref" href="pdiff_1.html#T15" target="_self" title="PDIFF_1:th.15">Th15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K9" title="PDIFF_1:func.9">partdiff</a> <font color="Maroon" title="c6">f1</font>,<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c2">i</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><a href="pdiff_1.html#K9" title="PDIFF_1:func.9">partdiff</a> <font color="Maroon" title="c7">f2</font>,<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c2">i</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
<b>by </b><i><a class="txt" href="pdiff_1.html#E14:50"><i><font color="Green" title="E44">A15</font></i></a>, <a class="ref" href="pdiff_1.html#T28" target="_self" title="PDIFF_1:th.28">Th28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K9" title="PDIFF_1:func.9">partdiff</a> <font color="Maroon" title="c6">f1</font>,<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c2">i</font></span>)</span> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Maroon" title="c9">One</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="pdiff_1.html#K9" title="PDIFF_1:func.9">partdiff</a> <font color="Maroon" title="c7">f2</font>,<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c2">i</font></span>)</span> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Maroon" title="c9">One</font></span>)</span>
<b>by </b><i><a class="ref" href="lopban_1.html#T41" title="LOPBAN_1:th.41">LOPBAN_1:41</a></i>
<br/>.= 
<font color="Maroon" title="c10">Pd1</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c11">Pd2</font>
<b>by </b><i><a class="txt" href="pdiff_1.html#E17:50"><i><font color="Green" title="E46">A17</font></i></a>, <a class="ref" href="real_ns1.html#T2" title="REAL_NS1:th.2">REAL_NS1:2</a></i>
<br/>.= 
<span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K10" title="PDIFF_1:func.10">partdiff</a> <font color="Maroon" title="c3">g1</font>,<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c2">i</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><a href="pdiff_1.html#K10" title="PDIFF_1:func.10">partdiff</a> <font color="Maroon" title="c4">g2</font>,<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c2">i</font></span>)</span></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T29" title="RVSUM_1:th.29">RVSUM_1:29</a></i>
;<br/>
<a NAME="E21:50"/><b>then </b>
 <a href="pdiff_1.html#K10" title="PDIFF_1:func.10">partdiff</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g1</font> <a href="seq_1.html#K6" title="SEQ_1:func.6">+</a> <font color="Maroon" title="c4">g2</font></span>)</span>,<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K10" title="PDIFF_1:func.10">partdiff</a> <font color="Maroon" title="c3">g1</font>,<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c2">i</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><a href="pdiff_1.html#K10" title="PDIFF_1:func.10">partdiff</a> <font color="Maroon" title="c4">g2</font>,<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c2">i</font></span>)</span></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></i>;<br/>
<b>hence </b><a NAME="E22:50"/>
( <font color="Maroon" title="c3">g1</font> <a href="seq_1.html#K6" title="SEQ_1:func.6">+</a> <font color="Maroon" title="c4">g2</font> <a href="pdiff_1.html#R3" title="PDIFF_1:pred.3">is_partial_differentiable_in</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c2">i</font> &amp;  <a href="pdiff_1.html#K10" title="PDIFF_1:func.10">partdiff</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g1</font> <a href="seq_1.html#K6" title="SEQ_1:func.6">+</a> <font color="Maroon" title="c4">g2</font></span>)</span>,<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K10" title="PDIFF_1:func.10">partdiff</a> <font color="Maroon" title="c3">g1</font>,<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c2">i</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K10" title="PDIFF_1:func.10">partdiff</a> <font color="Maroon" title="c4">g2</font>,<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c2">i</font></span>)</span> )
 <b>by </b><i><a class="txt" href="pdiff_1.html#E13:50"><i><font color="Green" title="E43">A14</font></i></a>, <a class="txt" href="pdiff_1.html#E15:50"><i><font color="Green" title="E45">A16</font></i></a>, <a class="ref" href="pdiff_1.html#T14" target="_self" title="PDIFF_1:th.14">Th14</a>, <a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></i>;<br/>


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