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<span class="kw">let </span><font color="Maroon" title="c4">m</font>, <font color="Maroon" title="c5">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>, <font color="Olive" title="b2">j</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><br/>  for <font color="Olive" title="b3">f</font> being   <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">m</font></span>)</span>,<span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c5">n</font></span>)</span><br/>  for <font color="Olive" title="b4">h</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">m</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c5">n</font></span>)</span><br/>  for <font color="Olive" title="b5">x</font> being   <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">m</font></span>)</span><br/>  for <font color="Olive" title="b6">z</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">m</font>  st <font color="Olive" title="b3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">h</font> &amp; <font color="Olive" title="b5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">z</font> holds <br/>( <font color="Olive" title="b3">f</font> <a href="pdiff_1.html#R5" title="PDIFF_1:pred.5">is_partial_differentiable_in</a> <font color="Olive" title="b5">x</font>,<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font> iff <font color="Olive" title="b4">h</font> <a href="pdiff_1.html#R6" title="PDIFF_1:pred.6">is_partial_differentiable_in</a> <font color="Olive" title="b6">z</font>,<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c6">i</font>, <font color="Maroon" title="c7">j</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">f</font> being   <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">m</font></span>)</span>,<span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c5">n</font></span>)</span><br/>  for <font color="Olive" title="b2">h</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">m</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c5">n</font></span>)</span><br/>  for <font color="Olive" title="b3">x</font> being   <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">m</font></span>)</span><br/>  for <font color="Olive" title="b4">z</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">m</font>  st <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">h</font> &amp; <font color="Olive" title="b3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">z</font> holds <br/>( <font color="Olive" title="b1">f</font> <a href="pdiff_1.html#R5" title="PDIFF_1:pred.5">is_partial_differentiable_in</a> <font color="Olive" title="b3">x</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> iff <font color="Olive" title="b2">h</font> <a href="pdiff_1.html#R6" title="PDIFF_1:pred.6">is_partial_differentiable_in</a> <font color="Olive" title="b4">z</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c8">f</font> be   <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">m</font></span>)</span>,<span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c5">n</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">  for <font color="Olive" title="b1">h</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">m</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c5">n</font></span>)</span><br/>  for <font color="Olive" title="b2">x</font> being   <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">m</font></span>)</span><br/>  for <font color="Olive" title="b3">z</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">m</font>  st <font color="Maroon" title="c8">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">h</font> &amp; <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">z</font> holds <br/>( <font color="Maroon" title="c8">f</font> <a href="pdiff_1.html#R5" title="PDIFF_1:pred.5">is_partial_differentiable_in</a> <font color="Olive" title="b2">x</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> iff <font color="Olive" title="b1">h</font> <a href="pdiff_1.html#R6" title="PDIFF_1:pred.6">is_partial_differentiable_in</a> <font color="Olive" title="b3">z</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c9">h</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">m</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c5">n</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">  for <font color="Olive" title="b1">x</font> being   <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">m</font></span>)</span><br/>  for <font color="Olive" title="b2">z</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">m</font>  st <font color="Maroon" title="c8">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">h</font> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">z</font> holds <br/>( <font color="Maroon" title="c8">f</font> <a href="pdiff_1.html#R5" title="PDIFF_1:pred.5">is_partial_differentiable_in</a> <font color="Olive" title="b1">x</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> iff <font color="Maroon" title="c9">h</font> <a href="pdiff_1.html#R6" title="PDIFF_1:pred.6">is_partial_differentiable_in</a> <font color="Olive" title="b2">z</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c10">x</font> be   <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">m</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">  for <font color="Olive" title="b1">z</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">m</font>  st <font color="Maroon" title="c8">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">h</font> &amp; <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">z</font> holds <br/>( <font color="Maroon" title="c8">f</font> <a href="pdiff_1.html#R5" title="PDIFF_1:pred.5">is_partial_differentiable_in</a> <font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> iff <font color="Maroon" title="c9">h</font> <a href="pdiff_1.html#R6" title="PDIFF_1:pred.6">is_partial_differentiable_in</a> <font color="Olive" title="b1">z</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c11">z</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">m</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="c8">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">h</font> &amp; <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">z</font> implies ( <font color="Maroon" title="c8">f</font> <a href="pdiff_1.html#R5" title="PDIFF_1:pred.5">is_partial_differentiable_in</a> <font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> iff <font color="Maroon" title="c9">h</font> <a href="pdiff_1.html#R6" title="PDIFF_1:pred.6">is_partial_differentiable_in</a> <font color="Maroon" title="c11">z</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> ) )</span><br/>















<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:42"/><span class="lab"><font color="Green" title="E24">A1</font></span>: 
<font color="Maroon" title="c8">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">h</font>
 <span class="kw">and </span><br/><a NAME="E2:42"/><span class="lab"><font color="Green" title="E25">A2</font></span>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">z</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="c8">f</font> <a href="pdiff_1.html#R5" title="PDIFF_1:pred.5">is_partial_differentiable_in</a> <font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> iff <font color="Maroon" title="c9">h</font> <a href="pdiff_1.html#R6" title="PDIFF_1:pred.6">is_partial_differentiable_in</a> <font color="Maroon" title="c11">z</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> )</span><br/>

<a NAME="E3:42"/><span class="lab"><font color="Green" title="E26">A3</font></span>: 
<span class="p1">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> (<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">m</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><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> (<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">m</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">z</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:42"><span class="lab"><font color="Green" title="E25">A2</font></span></a>, <a class="ref" href="pdiff_1.html#D4" target="_self" onmouseover="rs('pdiff_1/D4')" onmouseout="rh()">Def4</a></span>;<br/>
<a NAME="E4:42"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> (<font color="Maroon" title="c7">j</font>,<font color="Maroon" title="c5">n</font>)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K7" title="PDIFF_1:func.7">reproj</a> (<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c10">x</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pdiff_1.html#K2" title="PDIFF_1:func.2">&lt;&gt;*</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> (<font color="Maroon" title="c7">j</font>,<font color="Maroon" title="c5">n</font>)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c9">h</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="pdiff_1.html#K6" title="PDIFF_1:func.6">reproj</a> (<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c11">z</font>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42"><span class="lab"><font color="Green" title="E24">A1</font></span></a>, <a class="txt" href="#E2:42"><span class="lab"><font color="Green" title="E25">A2</font></span></a>, <a class="ref" href="pdiff_1.html#T22" target="_self" onmouseover="rs('pdiff_1/T22')" onmouseout="rh()">Th22</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:42"/>
( <font color="Maroon" title="c8">f</font> <a href="pdiff_1.html#R5" title="PDIFF_1:pred.5">is_partial_differentiable_in</a> <font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> iff <font color="Maroon" title="c9">h</font> <a href="pdiff_1.html#R6" title="PDIFF_1:pred.6">is_partial_differentiable_in</a> <font color="Maroon" title="c11">z</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:42"><span class="lab"><font color="Green" title="E26">A3</font></span></a>, <a class="ref" href="pdiff_1.html#T7" target="_self" onmouseover="rs('pdiff_1/T7')" onmouseout="rh()">Th7</a>, <a class="ref" href="pdiff_1.html#T8" target="_self" onmouseover="rs('pdiff_1/T8')" onmouseout="rh()">Th8</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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