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<span class="kw">let </span><font color="Maroon" title="c1">G</font> be   <a href="ndiff_5.html#V1" title="NDIFF_5:attr.1">non-trivial</a>  <a href="prvect_2.html#NM3" title="PRVECT_2:NM.3">RealNormSpace-Sequence</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   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><br/>  for <font color="Olive" title="b2">f1</font>, <font color="Olive" title="b3">f2</font> being   <a href="struct_0.html#NM12" title="STRUCT_0:NM.12">PartFunc</a> of <span class="p1">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">G</font></span>)</span>,<font color="Olive" title="b1">F</font><br/>  for <font color="Olive" title="b4">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="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">G</font></span>)</span><br/>  for <font color="Olive" title="b5">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">G</font> &amp; <font color="Olive" title="b2">f1</font> <a href="ndiff_5.html#R1" title="NDIFF_5:pred.1">is_partial_differentiable_in</a> <font color="Olive" title="b4">x</font>,<font color="Olive" title="b5">i</font> &amp; <font color="Olive" title="b3">f2</font> <a href="ndiff_5.html#R1" title="NDIFF_5:pred.1">is_partial_differentiable_in</a> <font color="Olive" title="b4">x</font>,<font color="Olive" title="b5">i</font> holds <br/>( <font color="Olive" title="b2">f1</font> <a href="vfunct_1.html#K6" title="VFUNCT_1:func.6">+</a> <font color="Olive" title="b3">f2</font> <a href="ndiff_5.html#R1" title="NDIFF_5:pred.1">is_partial_differentiable_in</a> <font color="Olive" title="b4">x</font>,<font color="Olive" title="b5">i</font> &amp;  <a href="ndiff_5.html#K6" title="NDIFF_5:func.6">partdiff</a> (<span class="p1">(<span class="default"><font color="Olive" title="b2">f1</font> <a href="vfunct_1.html#K6" title="VFUNCT_1:func.6">+</a> <font color="Olive" title="b3">f2</font></span>)</span>,<font color="Olive" title="b4">x</font>,<font color="Olive" title="b5">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ndiff_5.html#K6" title="NDIFF_5:func.6">partdiff</a> (<font color="Olive" title="b2">f1</font>,<font color="Olive" title="b4">x</font>,<font color="Olive" title="b5">i</font>)</span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="ndiff_5.html#K6" title="NDIFF_5:func.6">partdiff</a> (<font color="Olive" title="b3">f2</font>,<font color="Olive" title="b4">x</font>,<font color="Olive" title="b5">i</font>)</span>)</span> )</span><br/><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">f1</font>, <font color="Olive" title="b2">f2</font> being   <a href="struct_0.html#NM12" title="STRUCT_0:NM.12">PartFunc</a> of <span class="p1">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">G</font></span>)</span>,<font color="Maroon" title="c2">F</font><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="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">G</font></span>)</span><br/>  for <font color="Olive" title="b4">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">G</font> &amp; <font color="Olive" title="b1">f1</font> <a href="ndiff_5.html#R1" title="NDIFF_5:pred.1">is_partial_differentiable_in</a> <font color="Olive" title="b3">x</font>,<font color="Olive" title="b4">i</font> &amp; <font color="Olive" title="b2">f2</font> <a href="ndiff_5.html#R1" title="NDIFF_5:pred.1">is_partial_differentiable_in</a> <font color="Olive" title="b3">x</font>,<font color="Olive" title="b4">i</font> holds <br/>( <font color="Olive" title="b1">f1</font> <a href="vfunct_1.html#K6" title="VFUNCT_1:func.6">+</a> <font color="Olive" title="b2">f2</font> <a href="ndiff_5.html#R1" title="NDIFF_5:pred.1">is_partial_differentiable_in</a> <font color="Olive" title="b3">x</font>,<font color="Olive" title="b4">i</font> &amp;  <a href="ndiff_5.html#K6" title="NDIFF_5:func.6">partdiff</a> (<span class="p1">(<span class="default"><font color="Olive" title="b1">f1</font> <a href="vfunct_1.html#K6" title="VFUNCT_1:func.6">+</a> <font color="Olive" title="b2">f2</font></span>)</span>,<font color="Olive" title="b3">x</font>,<font color="Olive" title="b4">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ndiff_5.html#K6" title="NDIFF_5:func.6">partdiff</a> (<font color="Olive" title="b1">f1</font>,<font color="Olive" title="b3">x</font>,<font color="Olive" title="b4">i</font>)</span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="ndiff_5.html#K6" title="NDIFF_5:func.6">partdiff</a> (<font color="Olive" title="b2">f2</font>,<font color="Olive" title="b3">x</font>,<font color="Olive" title="b4">i</font>)</span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">f1</font>, <font color="Maroon" title="c4">f2</font> be   <a href="struct_0.html#NM12" title="STRUCT_0:NM.12">PartFunc</a> of <span class="p1">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">G</font></span>)</span>,<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">  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="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">G</font></span>)</span><br/>  for <font color="Olive" title="b2">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">G</font> &amp; <font color="Maroon" title="c3">f1</font> <a href="ndiff_5.html#R1" title="NDIFF_5:pred.1">is_partial_differentiable_in</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">i</font> &amp; <font color="Maroon" title="c4">f2</font> <a href="ndiff_5.html#R1" title="NDIFF_5:pred.1">is_partial_differentiable_in</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">i</font> holds <br/>( <font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K6" title="VFUNCT_1:func.6">+</a> <font color="Maroon" title="c4">f2</font> <a href="ndiff_5.html#R1" title="NDIFF_5:pred.1">is_partial_differentiable_in</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">i</font> &amp;  <a href="ndiff_5.html#K6" title="NDIFF_5:func.6">partdiff</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K6" title="VFUNCT_1:func.6">+</a> <font color="Maroon" title="c4">f2</font></span>)</span>,<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ndiff_5.html#K6" title="NDIFF_5:func.6">partdiff</a> (<font color="Maroon" title="c3">f1</font>,<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">i</font>)</span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="ndiff_5.html#K6" title="NDIFF_5:func.6">partdiff</a> (<font color="Maroon" title="c4">f2</font>,<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">i</font>)</span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c5">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="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">G</font></span>)</span>; <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">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">G</font> &amp; <font color="Maroon" title="c3">f1</font> <a href="ndiff_5.html#R1" title="NDIFF_5:pred.1">is_partial_differentiable_in</a> <font color="Maroon" title="c5">x</font>,<font color="Olive" title="b1">i</font> &amp; <font color="Maroon" title="c4">f2</font> <a href="ndiff_5.html#R1" title="NDIFF_5:pred.1">is_partial_differentiable_in</a> <font color="Maroon" title="c5">x</font>,<font color="Olive" title="b1">i</font> holds <br/>( <font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K6" title="VFUNCT_1:func.6">+</a> <font color="Maroon" title="c4">f2</font> <a href="ndiff_5.html#R1" title="NDIFF_5:pred.1">is_partial_differentiable_in</a> <font color="Maroon" title="c5">x</font>,<font color="Olive" title="b1">i</font> &amp;  <a href="ndiff_5.html#K6" title="NDIFF_5:func.6">partdiff</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K6" title="VFUNCT_1:func.6">+</a> <font color="Maroon" title="c4">f2</font></span>)</span>,<font color="Maroon" title="c5">x</font>,<font color="Olive" title="b1">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ndiff_5.html#K6" title="NDIFF_5:func.6">partdiff</a> (<font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c5">x</font>,<font color="Olive" title="b1">i</font>)</span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="ndiff_5.html#K6" title="NDIFF_5:func.6">partdiff</a> (<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c5">x</font>,<font color="Olive" title="b1">i</font>)</span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c6">i0</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <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="c6">i0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">G</font> &amp; <font color="Maroon" title="c3">f1</font> <a href="ndiff_5.html#R1" title="NDIFF_5:pred.1">is_partial_differentiable_in</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">i0</font> &amp; <font color="Maroon" title="c4">f2</font> <a href="ndiff_5.html#R1" title="NDIFF_5:pred.1">is_partial_differentiable_in</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">i0</font> implies ( <font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K6" title="VFUNCT_1:func.6">+</a> <font color="Maroon" title="c4">f2</font> <a href="ndiff_5.html#R1" title="NDIFF_5:pred.1">is_partial_differentiable_in</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">i0</font> &amp;  <a href="ndiff_5.html#K6" title="NDIFF_5:func.6">partdiff</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K6" title="VFUNCT_1:func.6">+</a> <font color="Maroon" title="c4">f2</font></span>)</span>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">i0</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ndiff_5.html#K6" title="NDIFF_5:func.6">partdiff</a> (<font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="ndiff_5.html#K6" title="NDIFF_5:func.6">partdiff</a> (<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span> ) )</span><br/>











<span class="kw">set </span><font color="Maroon" title="c7">i</font> =  <a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>);<br/>
<span class="kw">assume </span><a NAME="E1:47"/><span class="lab"><font color="Green" title="E39">AS</font></span>: 
<font color="Maroon" title="c6">i0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">G</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c3">f1</font> <a href="ndiff_5.html#R1" title="NDIFF_5:pred.1">is_partial_differentiable_in</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">i0</font> or  not <font color="Maroon" title="c4">f2</font> <a href="ndiff_5.html#R1" title="NDIFF_5:pred.1">is_partial_differentiable_in</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">i0</font> or ( <font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K6" title="VFUNCT_1:func.6">+</a> <font color="Maroon" title="c4">f2</font> <a href="ndiff_5.html#R1" title="NDIFF_5:pred.1">is_partial_differentiable_in</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">i0</font> &amp;  <a href="ndiff_5.html#K6" title="NDIFF_5:func.6">partdiff</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K6" title="VFUNCT_1:func.6">+</a> <font color="Maroon" title="c4">f2</font></span>)</span>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">i0</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ndiff_5.html#K6" title="NDIFF_5:func.6">partdiff</a> (<font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="ndiff_5.html#K6" title="NDIFF_5:func.6">partdiff</a> (<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span> ) )</span><br/>

<a NAME="E2:47"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A0</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K6" title="VFUNCT_1:func.6">+</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p2">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p3">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span></span>)</span> <a href="vfunct_1.html#K6" title="VFUNCT_1:func.6">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p3">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ndiff_5.html#T26" target="_self" title="NDIFF_5:th.26">Th26</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:47"/><span class="lab"><font color="Green" title="E41">A1</font></span>: 
( <font color="Maroon" title="c3">f1</font> <a href="ndiff_5.html#R1" title="NDIFF_5:pred.1">is_partial_differentiable_in</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">i0</font> &amp; <font color="Maroon" title="c4">f2</font> <a href="ndiff_5.html#R1" title="NDIFF_5:pred.1">is_partial_differentiable_in</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">i0</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">f1</font> <a href="vfunct_1.html#K6" title="VFUNCT_1:func.6">+</a> <font color="Maroon" title="c4">f2</font> <a href="ndiff_5.html#R1" title="NDIFF_5:pred.1">is_partial_differentiable_in</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">i0</font> &amp;  <a href="ndiff_5.html#K6" title="NDIFF_5:func.6">partdiff</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K6" title="VFUNCT_1:func.6">+</a> <font color="Maroon" title="c4">f2</font></span>)</span>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">i0</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ndiff_5.html#K6" title="NDIFF_5:func.6">partdiff</a> (<font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="ndiff_5.html#K6" title="NDIFF_5:func.6">partdiff</a> (<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span> )</span><br/>

<a NAME="E4:47"/><span class="lab"><font color="Green" title="E42">A2</font></span>: 
( <font color="Maroon" title="c3">f1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p2">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span> <a href="ndiff_1.html#R1" title="NDIFF_1:pred.1">is_differentiable_in</a> <span class="p1">(<span class="default"><a href="ndiff_5.html#K3" title="NDIFF_5:func.3">proj</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x</font> &amp; <font color="Maroon" title="c4">f2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p2">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span> <a href="ndiff_1.html#R1" title="NDIFF_1:pred.1">is_differentiable_in</a> <span class="p1">(<span class="default"><a href="ndiff_5.html#K3" title="NDIFF_5:func.3">proj</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:47"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="ref" href="ndiff_5.html#D6" target="_self" title="NDIFF_5:def.6">Def9</a></span>;<br/>
<a NAME="E5:47"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K6" title="VFUNCT_1:func.6">+</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p2">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span> <a href="ndiff_1.html#R1" title="NDIFF_1:pred.1">is_differentiable_in</a> <span class="p1">(<span class="default"><a href="ndiff_5.html#K3" title="NDIFF_5:func.3">proj</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:47"><span class="lab"><font color="Green" title="E40">A0</font></span></a>, <a class="ref" href="ndiff_1.html#T35" title="NDIFF_1:th.35">NDIFF_1:35</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:47"/>
<font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K6" title="VFUNCT_1:func.6">+</a> <font color="Maroon" title="c4">f2</font> <a href="ndiff_5.html#R1" title="NDIFF_5:pred.1">is_partial_differentiable_in</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">i0</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ndiff_5.html#D6" target="_self" title="NDIFF_5:def.6">Def9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="ndiff_5.html#K6" title="NDIFF_5:func.6">partdiff</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K6" title="VFUNCT_1:func.6">+</a> <font color="Maroon" title="c4">f2</font></span>)</span>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">i0</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ndiff_5.html#K6" title="NDIFF_5:func.6">partdiff</a> (<font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="ndiff_5.html#K6" title="NDIFF_5:func.6">partdiff</a> (<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span></span><br/>

<span class="kw">thus </span><span class="p1">(<span class="default"><a href="ndiff_5.html#K6" title="NDIFF_5:func.6">partdiff</a> (<font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="ndiff_5.html#K6" title="NDIFF_5:func.6">partdiff</a> (<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span> = 
 <a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p4">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span></span>)</span> <a href="vfunct_1.html#K6" title="VFUNCT_1:func.6">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p4">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ndiff_5.html#K3" title="NDIFF_5:func.3">proj</a> <span class="p3">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x</font></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:47"><span class="lab"><font color="Green" title="E42">A2</font></span></a>, <a class="ref" href="ndiff_1.html#T35" title="NDIFF_1:th.35">NDIFF_1:35</a></span>
<br/>.= 
 <a href="ndiff_5.html#K6" title="NDIFF_5:func.6">partdiff</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K6" title="VFUNCT_1:func.6">+</a> <font color="Maroon" title="c4">f2</font></span>)</span>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">i0</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47"><span class="lab"><font color="Green" title="E39">AS</font></span></a>, <a class="ref" href="ndiff_5.html#T26" target="_self" title="NDIFF_5:th.26">Th26</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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