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<span class="kw">let </span><font color="Maroon" title="c4">n</font> be   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</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">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><br/>  for <font color="Olive" title="b2">g</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1">(<span class="default"><a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c4">n</font></span>)</span>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a><br/>  for <font color="Olive" title="b3">y</font> being    <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="c4">n</font><br/>  for <font color="Olive" title="b4">g1</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1">(<span class="default"><a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c4">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> 1</span>)</span>  st <font color="Olive" title="b4">g1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="pdiff_1.html#K3" title="PDIFF_1:func.3">&lt;&gt;*</a> <font color="Olive" title="b2">g</font> &amp; <font color="Olive" title="b4">g1</font> <a href="pdiff_1.html#R4" title="PDIFF_1:pred.4">is_partial_differentiable_in</a> <font color="Olive" title="b3">y</font>,<font color="Olive" title="b1">i</font> holds <br/> <a href="pdiff_1.html#K11" title="PDIFF_1:func.11">partdiff</a> (<font color="Olive" title="b4">g1</font>,<font color="Olive" title="b3">y</font>,<font color="Olive" title="b1">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"><a href="pdiff_1.html#K10" title="PDIFF_1:func.10">partdiff</a> (<font color="Olive" title="b2">g</font>,<font color="Olive" title="b3">y</font>,<font color="Olive" title="b1">i</font>)</span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c5">i</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</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">g</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1">(<span class="default"><a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c4">n</font></span>)</span>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a><br/>  for <font color="Olive" title="b2">y</font> being    <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="c4">n</font><br/>  for <font color="Olive" title="b3">g1</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1">(<span class="default"><a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c4">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> 1</span>)</span>  st <font color="Olive" title="b3">g1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="pdiff_1.html#K3" title="PDIFF_1:func.3">&lt;&gt;*</a> <font color="Olive" title="b1">g</font> &amp; <font color="Olive" title="b3">g1</font> <a href="pdiff_1.html#R4" title="PDIFF_1:pred.4">is_partial_differentiable_in</a> <font color="Olive" title="b2">y</font>,<font color="Maroon" title="c5">i</font> holds <br/> <a href="pdiff_1.html#K11" title="PDIFF_1:func.11">partdiff</a> (<font color="Olive" title="b3">g1</font>,<font color="Olive" title="b2">y</font>,<font color="Maroon" title="c5">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"><a href="pdiff_1.html#K10" title="PDIFF_1:func.10">partdiff</a> (<font color="Olive" title="b1">g</font>,<font color="Olive" title="b2">y</font>,<font color="Maroon" title="c5">i</font>)</span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c6">g</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1">(<span class="default"><a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c4">n</font></span>)</span>,<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">y</font> being    <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="c4">n</font><br/>  for <font color="Olive" title="b2">g1</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1">(<span class="default"><a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c4">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> 1</span>)</span>  st <font color="Olive" title="b2">g1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="pdiff_1.html#K3" title="PDIFF_1:func.3">&lt;&gt;*</a> <font color="Maroon" title="c6">g</font> &amp; <font color="Olive" title="b2">g1</font> <a href="pdiff_1.html#R4" title="PDIFF_1:pred.4">is_partial_differentiable_in</a> <font color="Olive" title="b1">y</font>,<font color="Maroon" title="c5">i</font> holds <br/> <a href="pdiff_1.html#K11" title="PDIFF_1:func.11">partdiff</a> (<font color="Olive" title="b2">g1</font>,<font color="Olive" title="b1">y</font>,<font color="Maroon" title="c5">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"><a href="pdiff_1.html#K10" title="PDIFF_1:func.10">partdiff</a> (<font color="Maroon" title="c6">g</font>,<font color="Olive" title="b1">y</font>,<font color="Maroon" title="c5">i</font>)</span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c7">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="c4">n</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">  for <font color="Olive" title="b1">g1</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1">(<span class="default"><a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c4">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> 1</span>)</span>  st <font color="Olive" title="b1">g1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="pdiff_1.html#K3" title="PDIFF_1:func.3">&lt;&gt;*</a> <font color="Maroon" title="c6">g</font> &amp; <font color="Olive" title="b1">g1</font> <a href="pdiff_1.html#R4" title="PDIFF_1:pred.4">is_partial_differentiable_in</a> <font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c5">i</font> holds <br/> <a href="pdiff_1.html#K11" title="PDIFF_1:func.11">partdiff</a> (<font color="Olive" title="b1">g1</font>,<font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c5">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"><a href="pdiff_1.html#K10" title="PDIFF_1:func.10">partdiff</a> (<font color="Maroon" title="c6">g</font>,<font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c5">i</font>)</span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c8">g1</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1">(<span class="default"><a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c4">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> 1</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"> ( <font color="Maroon" title="c8">g1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="pdiff_1.html#K3" title="PDIFF_1:func.3">&lt;&gt;*</a> <font color="Maroon" title="c6">g</font> &amp; <font color="Maroon" title="c8">g1</font> <a href="pdiff_1.html#R4" title="PDIFF_1:pred.4">is_partial_differentiable_in</a> <font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c5">i</font> implies  <a href="pdiff_1.html#K11" title="PDIFF_1:func.11">partdiff</a> (<font color="Maroon" title="c8">g1</font>,<font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c5">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"><a href="pdiff_1.html#K10" title="PDIFF_1:func.10">partdiff</a> (<font color="Maroon" title="c6">g</font>,<font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c5">i</font>)</span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> )</span><br/>









<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:35"/><span class="lab"><font color="Green" title="E23">A1</font></span>: 
<font color="Maroon" title="c8">g1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="pdiff_1.html#K3" title="PDIFF_1:func.3">&lt;&gt;*</a> <font color="Maroon" title="c6">g</font>
 <span class="kw">and </span><br/><a NAME="E2:35"/><span class="lab"><font color="Green" title="E24">A2</font></span>: 
<font color="Maroon" title="c8">g1</font> <a href="pdiff_1.html#R4" title="PDIFF_1:pred.4">is_partial_differentiable_in</a> <font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c5">i</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="pdiff_1.html#K11" title="PDIFF_1:func.11">partdiff</a> (<font color="Maroon" title="c8">g1</font>,<font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c5">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"><a href="pdiff_1.html#K10" title="PDIFF_1:func.10">partdiff</a> (<font color="Maroon" title="c6">g</font>,<font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c5">i</font>)</span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c9">y9</font> = <font color="Maroon" title="c7">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="c4">n</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="real_ns1.html#D4" onmouseover="rs('real_ns1/D4')" onmouseout="rh()">REAL_NS1:def 4</a></span>;<br/>
<a NAME="E4:35"/>
 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
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="real_ns1.html#D4" onmouseover="rs('real_ns1/D4')" onmouseout="rh()">REAL_NS1:def 4</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">h</font> = <font color="Maroon" title="c8">g1</font> as   <a href="struct_0.html#NM12" title="STRUCT_0:NM.12">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="c4">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> <span class="kw">by</span> <span class="lab"><a class="ref" href="real_ns1.html#D4" onmouseover="rs('real_ns1/D4')" onmouseout="rh()">REAL_NS1:def 4</a></span>;<br/>
<a NAME="E6:35"/><span class="lab"><font color="Green" title="E25">A3</font></span>: 
<font color="Maroon" title="c10">h</font> <a href="pdiff_1.html#R2" title="PDIFF_1:pred.2">is_partial_differentiable_in</a> <font color="Maroon" title="c9">y9</font>,<font color="Maroon" title="c5">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35"><span class="lab"><font color="Green" title="E24">A2</font></span></a></span>;<br/>
<a NAME="E7:35"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="pdiff_1.html#K9" title="PDIFF_1:func.9">partdiff</a> (<font color="Maroon" title="c10">h</font>,<font color="Maroon" title="c9">y9</font>,<font color="Maroon" title="c5">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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pdiff_1.html#K11" title="PDIFF_1:func.11">partdiff</a> (<font color="Maroon" title="c8">g1</font>,<font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c5">i</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pdiff_1.html#T17" target="_self" onmouseover="rs('pdiff_1/T17')" onmouseout="rh()">Th17</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:35"/>
 <a href="pdiff_1.html#K11" title="PDIFF_1:func.11">partdiff</a> (<font color="Maroon" title="c8">g1</font>,<font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c5">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"><a href="pdiff_1.html#K10" title="PDIFF_1:func.10">partdiff</a> (<font color="Maroon" title="c6">g</font>,<font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c5">i</font>)</span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35"><span class="lab"><font color="Green" title="E23">A1</font></span></a>, <a class="txt" href="#E6:35"><span class="lab"><font color="Green" title="E25">A3</font></span></a>, <a class="ref" href="pdiff_1.html#T15" target="_self" onmouseover="rs('pdiff_1/T15')" onmouseout="rh()">Th15</a></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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