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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">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="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="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> <br/>  for <font color="Olive" title="b5">M</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><br/>  for <font color="Olive" title="b6">L</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lopban_1.html#K16" title="LOPBAN_1:func.16">R_NormSpace_of_BoundedLinearOperators</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="prvect_2.html#K11" title="PRVECT_2:func.11">.</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="Olive" title="b4">i</font>)</span>)</span></span>)</span>,<font color="Olive" title="b1">F</font>)</span>)</span><br/>  for <font color="Olive" title="b7">p</font>, <font color="Olive" title="b8">q</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="prvect_2.html#K11" title="PRVECT_2:func.11">.</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="Olive" title="b4">i</font>)</span>)</span></span>)</span>  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; (  for <font color="Olive" title="b9">h</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="prvect_2.html#K11" title="PRVECT_2:func.11">.</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="Olive" title="b4">i</font>)</span>)</span></span>)</span>  st <font color="Olive" title="b9">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="ndiff_5.html#K2" title="NDIFF_5:func.2">].</a><span class="default"><font color="Olive" title="b7">p</font>,<font color="Olive" title="b8">q</font></span><a href="ndiff_5.html#K2" title="NDIFF_5:func.2">.[</a></span> holds <br/><span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ndiff_5.html#K6" title="NDIFF_5:func.6">partdiff</a> (<font color="Olive" title="b2">f</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p0">(<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="Olive" title="b4">i</font>)</span>)</span>,<font color="Olive" title="b3">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b9">h</font></span>)</span>,<font color="Olive" title="b4">i</font>)</span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Olive" title="b6">L</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b5">M</font> ) &amp; (  for <font color="Olive" title="b9">h</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="prvect_2.html#K11" title="PRVECT_2:func.11">.</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="Olive" title="b4">i</font>)</span>)</span></span>)</span>  st <font color="Olive" title="b9">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="ndiff_5.html#NK3" title="NDIFF_5:NK.3">[.</a><span class="default"><font color="Olive" title="b7">p</font>,<font color="Olive" title="b8">q</font></span><a href="ndiff_5.html#NK3" title="NDIFF_5:NK.3">.]</a></span> holds <br/><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="Olive" title="b4">i</font>)</span>)</span>,<font color="Olive" title="b3">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b9">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b2">f</font> ) &amp; (  for <font color="Olive" title="b9">h</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="prvect_2.html#K11" title="PRVECT_2:func.11">.</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="Olive" title="b4">i</font>)</span>)</span></span>)</span>  st <font color="Olive" title="b9">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="ndiff_5.html#NK3" title="NDIFF_5:NK.3">[.</a><span class="default"><font color="Olive" title="b7">p</font>,<font color="Olive" title="b8">q</font></span><a href="ndiff_5.html#NK3" title="NDIFF_5:NK.3">.]</a></span> holds <br/><font color="Olive" title="b2">f</font> <a href="ndiff_5.html#R1" title="NDIFF_5:pred.1">is_partial_differentiable_in</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="Olive" title="b4">i</font>)</span>)</span>,<font color="Olive" title="b3">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b9">h</font>,<font color="Olive" title="b4">i</font> ) holds <br/><span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Olive" title="b2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p1">(<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="Olive" title="b4">i</font>)</span>)</span>,<font color="Olive" title="b3">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b8">q</font></span>)</span></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p4">(<span class="default"><font color="Olive" title="b2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p1">(<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="Olive" title="b4">i</font>)</span>)</span>,<font color="Olive" title="b3">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b7">p</font></span>)</span></span>)</span></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p3">(<span class="default"><font color="Olive" title="b6">L</font> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <span class="p4">(<span class="default"><font color="Olive" title="b8">q</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Olive" title="b7">p</font></span>)</span></span>)</span></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b5">M</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b8">q</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Olive" title="b7">p</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></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">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="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="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="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="b3">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/>  for <font color="Olive" title="b4">M</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><br/>  for <font color="Olive" title="b5">L</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lopban_1.html#K16" title="LOPBAN_1:func.16">R_NormSpace_of_BoundedLinearOperators</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="prvect_2.html#K11" title="PRVECT_2:func.11">.</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="Olive" title="b3">i</font>)</span>)</span></span>)</span>,<font color="Maroon" title="c2">F</font>)</span>)</span><br/>  for <font color="Olive" title="b6">p</font>, <font color="Olive" title="b7">q</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="prvect_2.html#K11" title="PRVECT_2:func.11">.</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="Olive" title="b3">i</font>)</span>)</span></span>)</span>  st <font color="Olive" title="b3">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; (  for <font color="Olive" title="b8">h</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="prvect_2.html#K11" title="PRVECT_2:func.11">.</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="Olive" title="b3">i</font>)</span>)</span></span>)</span>  st <font color="Olive" title="b8">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="ndiff_5.html#K2" title="NDIFF_5:func.2">].</a><span class="default"><font color="Olive" title="b6">p</font>,<font color="Olive" title="b7">q</font></span><a href="ndiff_5.html#K2" title="NDIFF_5:func.2">.[</a></span> holds <br/><span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ndiff_5.html#K6" title="NDIFF_5:func.6">partdiff</a> (<font color="Olive" title="b1">f</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p0">(<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="Olive" title="b3">i</font>)</span>)</span>,<font color="Olive" title="b2">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b8">h</font></span>)</span>,<font color="Olive" title="b3">i</font>)</span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Olive" title="b5">L</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b4">M</font> ) &amp; (  for <font color="Olive" title="b8">h</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="prvect_2.html#K11" title="PRVECT_2:func.11">.</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="Olive" title="b3">i</font>)</span>)</span></span>)</span>  st <font color="Olive" title="b8">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="ndiff_5.html#NK3" title="NDIFF_5:NK.3">[.</a><span class="default"><font color="Olive" title="b6">p</font>,<font color="Olive" title="b7">q</font></span><a href="ndiff_5.html#NK3" title="NDIFF_5:NK.3">.]</a></span> holds <br/><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="Olive" title="b3">i</font>)</span>)</span>,<font color="Olive" title="b2">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b8">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b1">f</font> ) &amp; (  for <font color="Olive" title="b8">h</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="prvect_2.html#K11" title="PRVECT_2:func.11">.</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="Olive" title="b3">i</font>)</span>)</span></span>)</span>  st <font color="Olive" title="b8">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="ndiff_5.html#NK3" title="NDIFF_5:NK.3">[.</a><span class="default"><font color="Olive" title="b6">p</font>,<font color="Olive" title="b7">q</font></span><a href="ndiff_5.html#NK3" title="NDIFF_5:NK.3">.]</a></span> holds <br/><font color="Olive" title="b1">f</font> <a href="ndiff_5.html#R1" title="NDIFF_5:pred.1">is_partial_differentiable_in</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="Olive" title="b3">i</font>)</span>)</span>,<font color="Olive" title="b2">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b8">h</font>,<font color="Olive" title="b3">i</font> ) holds <br/><span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Olive" title="b1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p1">(<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="Olive" title="b3">i</font>)</span>)</span>,<font color="Olive" title="b2">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b7">q</font></span>)</span></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p4">(<span class="default"><font color="Olive" title="b1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p1">(<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="Olive" title="b3">i</font>)</span>)</span>,<font color="Olive" title="b2">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b6">p</font></span>)</span></span>)</span></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p3">(<span class="default"><font color="Olive" title="b5">L</font> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <span class="p4">(<span class="default"><font color="Olive" title="b7">q</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Olive" title="b6">p</font></span>)</span></span>)</span></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b4">M</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b7">q</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Olive" title="b6">p</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">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="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> <br/>  for <font color="Olive" title="b3">M</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><br/>  for <font color="Olive" title="b4">L</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lopban_1.html#K16" title="LOPBAN_1:func.16">R_NormSpace_of_BoundedLinearOperators</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="prvect_2.html#K11" title="PRVECT_2:func.11">.</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="Olive" title="b2">i</font>)</span>)</span></span>)</span>,<font color="Maroon" title="c2">F</font>)</span>)</span><br/>  for <font color="Olive" title="b5">p</font>, <font color="Olive" title="b6">q</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="prvect_2.html#K11" title="PRVECT_2:func.11">.</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="Olive" title="b2">i</font>)</span>)</span></span>)</span>  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; (  for <font color="Olive" title="b7">h</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="prvect_2.html#K11" title="PRVECT_2:func.11">.</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="Olive" title="b2">i</font>)</span>)</span></span>)</span>  st <font color="Olive" title="b7">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="ndiff_5.html#K2" title="NDIFF_5:func.2">].</a><span class="default"><font color="Olive" title="b5">p</font>,<font color="Olive" title="b6">q</font></span><a href="ndiff_5.html#K2" title="NDIFF_5:func.2">.[</a></span> holds <br/><span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ndiff_5.html#K6" title="NDIFF_5:func.6">partdiff</a> (<font color="Maroon" title="c3">f</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p0">(<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="Olive" title="b2">i</font>)</span>)</span>,<font color="Olive" title="b1">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b7">h</font></span>)</span>,<font color="Olive" title="b2">i</font>)</span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Olive" title="b4">L</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">M</font> ) &amp; (  for <font color="Olive" title="b7">h</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="prvect_2.html#K11" title="PRVECT_2:func.11">.</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="Olive" title="b2">i</font>)</span>)</span></span>)</span>  st <font color="Olive" title="b7">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="ndiff_5.html#NK3" title="NDIFF_5:NK.3">[.</a><span class="default"><font color="Olive" title="b5">p</font>,<font color="Olive" title="b6">q</font></span><a href="ndiff_5.html#NK3" title="NDIFF_5:NK.3">.]</a></span> holds <br/><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="Olive" title="b2">i</font>)</span>)</span>,<font color="Olive" title="b1">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b7">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">f</font> ) &amp; (  for <font color="Olive" title="b7">h</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="prvect_2.html#K11" title="PRVECT_2:func.11">.</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="Olive" title="b2">i</font>)</span>)</span></span>)</span>  st <font color="Olive" title="b7">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="ndiff_5.html#NK3" title="NDIFF_5:NK.3">[.</a><span class="default"><font color="Olive" title="b5">p</font>,<font color="Olive" title="b6">q</font></span><a href="ndiff_5.html#NK3" title="NDIFF_5:NK.3">.]</a></span> holds <br/><font color="Maroon" title="c3">f</font> <a href="ndiff_5.html#R1" title="NDIFF_5:pred.1">is_partial_differentiable_in</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="Olive" title="b2">i</font>)</span>)</span>,<font color="Olive" title="b1">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b7">h</font>,<font color="Olive" title="b2">i</font> ) holds <br/><span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p1">(<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="Olive" title="b2">i</font>)</span>)</span>,<font color="Olive" title="b1">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b6">q</font></span>)</span></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p1">(<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="Olive" title="b2">i</font>)</span>)</span>,<font color="Olive" title="b1">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b5">p</font></span>)</span></span>)</span></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p3">(<span class="default"><font color="Olive" title="b4">L</font> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <span class="p4">(<span class="default"><font color="Olive" title="b6">q</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Olive" title="b5">p</font></span>)</span></span>)</span></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">M</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b6">q</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Olive" title="b5">p</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c4">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> <br/>  for <font color="Olive" title="b2">M</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><br/>  for <font color="Olive" title="b3">L</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lopban_1.html#K16" title="LOPBAN_1:func.16">R_NormSpace_of_BoundedLinearOperators</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="prvect_2.html#K11" title="PRVECT_2:func.11">.</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="Olive" title="b1">i</font>)</span>)</span></span>)</span>,<font color="Maroon" title="c2">F</font>)</span>)</span><br/>  for <font color="Olive" title="b4">p</font>, <font color="Olive" title="b5">q</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="prvect_2.html#K11" title="PRVECT_2:func.11">.</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="Olive" title="b1">i</font>)</span>)</span></span>)</span>  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; (  for <font color="Olive" title="b6">h</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="prvect_2.html#K11" title="PRVECT_2:func.11">.</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="Olive" title="b1">i</font>)</span>)</span></span>)</span>  st <font color="Olive" title="b6">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="ndiff_5.html#K2" title="NDIFF_5:func.2">].</a><span class="default"><font color="Olive" title="b4">p</font>,<font color="Olive" title="b5">q</font></span><a href="ndiff_5.html#K2" title="NDIFF_5:func.2">.[</a></span> holds <br/><span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ndiff_5.html#K6" title="NDIFF_5:func.6">partdiff</a> (<font color="Maroon" title="c3">f</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p0">(<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="Olive" title="b1">i</font>)</span>)</span>,<font color="Maroon" title="c4">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b6">h</font></span>)</span>,<font color="Olive" title="b1">i</font>)</span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Olive" title="b3">L</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">M</font> ) &amp; (  for <font color="Olive" title="b6">h</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="prvect_2.html#K11" title="PRVECT_2:func.11">.</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="Olive" title="b1">i</font>)</span>)</span></span>)</span>  st <font color="Olive" title="b6">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="ndiff_5.html#NK3" title="NDIFF_5:NK.3">[.</a><span class="default"><font color="Olive" title="b4">p</font>,<font color="Olive" title="b5">q</font></span><a href="ndiff_5.html#NK3" title="NDIFF_5:NK.3">.]</a></span> holds <br/><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="Olive" title="b1">i</font>)</span>)</span>,<font color="Maroon" title="c4">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b6">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">f</font> ) &amp; (  for <font color="Olive" title="b6">h</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="prvect_2.html#K11" title="PRVECT_2:func.11">.</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="Olive" title="b1">i</font>)</span>)</span></span>)</span>  st <font color="Olive" title="b6">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="ndiff_5.html#NK3" title="NDIFF_5:NK.3">[.</a><span class="default"><font color="Olive" title="b4">p</font>,<font color="Olive" title="b5">q</font></span><a href="ndiff_5.html#NK3" title="NDIFF_5:NK.3">.]</a></span> holds <br/><font color="Maroon" title="c3">f</font> <a href="ndiff_5.html#R1" title="NDIFF_5:pred.1">is_partial_differentiable_in</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="Olive" title="b1">i</font>)</span>)</span>,<font color="Maroon" title="c4">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b6">h</font>,<font color="Olive" title="b1">i</font> ) holds <br/><span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p1">(<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="Olive" title="b1">i</font>)</span>)</span>,<font color="Maroon" title="c4">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b5">q</font></span>)</span></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p1">(<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="Olive" title="b1">i</font>)</span>)</span>,<font color="Maroon" title="c4">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">p</font></span>)</span></span>)</span></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p3">(<span class="default"><font color="Olive" title="b3">L</font> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <span class="p4">(<span class="default"><font color="Olive" title="b5">q</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Olive" title="b4">p</font></span>)</span></span>)</span></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">M</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b5">q</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Olive" title="b4">p</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c5">i</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">  for <font color="Olive" title="b1">M</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><br/>  for <font color="Olive" title="b2">L</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lopban_1.html#K16" title="LOPBAN_1:func.16">R_NormSpace_of_BoundedLinearOperators</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="prvect_2.html#K11" title="PRVECT_2:func.11">.</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="c5">i</font>)</span>)</span></span>)</span>,<font color="Maroon" title="c2">F</font>)</span>)</span><br/>  for <font color="Olive" title="b3">p</font>, <font color="Olive" title="b4">q</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="prvect_2.html#K11" title="PRVECT_2:func.11">.</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="c5">i</font>)</span>)</span></span>)</span>  st <font color="Maroon" title="c5">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; (  for <font color="Olive" title="b5">h</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="prvect_2.html#K11" title="PRVECT_2:func.11">.</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="c5">i</font>)</span>)</span></span>)</span>  st <font color="Olive" title="b5">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="ndiff_5.html#K2" title="NDIFF_5:func.2">].</a><span class="default"><font color="Olive" title="b3">p</font>,<font color="Olive" title="b4">q</font></span><a href="ndiff_5.html#K2" title="NDIFF_5:func.2">.[</a></span> holds <br/><span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ndiff_5.html#K6" title="NDIFF_5:func.6">partdiff</a> (<font color="Maroon" title="c3">f</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p0">(<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="c5">i</font>)</span>)</span>,<font color="Maroon" title="c4">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b5">h</font></span>)</span>,<font color="Maroon" title="c5">i</font>)</span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Olive" title="b2">L</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">M</font> ) &amp; (  for <font color="Olive" title="b5">h</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="prvect_2.html#K11" title="PRVECT_2:func.11">.</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="c5">i</font>)</span>)</span></span>)</span>  st <font color="Olive" title="b5">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="ndiff_5.html#NK3" title="NDIFF_5:NK.3">[.</a><span class="default"><font color="Olive" title="b3">p</font>,<font color="Olive" title="b4">q</font></span><a href="ndiff_5.html#NK3" title="NDIFF_5:NK.3">.]</a></span> holds <br/><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="c5">i</font>)</span>)</span>,<font color="Maroon" title="c4">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b5">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">f</font> ) &amp; (  for <font color="Olive" title="b5">h</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="prvect_2.html#K11" title="PRVECT_2:func.11">.</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="c5">i</font>)</span>)</span></span>)</span>  st <font color="Olive" title="b5">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="ndiff_5.html#NK3" title="NDIFF_5:NK.3">[.</a><span class="default"><font color="Olive" title="b3">p</font>,<font color="Olive" title="b4">q</font></span><a href="ndiff_5.html#NK3" title="NDIFF_5:NK.3">.]</a></span> holds <br/><font color="Maroon" title="c3">f</font> <a href="ndiff_5.html#R1" title="NDIFF_5:pred.1">is_partial_differentiable_in</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="c5">i</font>)</span>)</span>,<font color="Maroon" title="c4">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b5">h</font>,<font color="Maroon" title="c5">i</font> ) holds <br/><span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p1">(<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="c5">i</font>)</span>)</span>,<font color="Maroon" title="c4">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">q</font></span>)</span></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p1">(<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="c5">i</font>)</span>)</span>,<font color="Maroon" title="c4">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">p</font></span>)</span></span>)</span></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p3">(<span class="default"><font color="Olive" title="b2">L</font> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <span class="p4">(<span class="default"><font color="Olive" title="b4">q</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Olive" title="b3">p</font></span>)</span></span>)</span></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">M</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b4">q</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Olive" title="b3">p</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c6">M</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</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">L</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lopban_1.html#K16" title="LOPBAN_1:func.16">R_NormSpace_of_BoundedLinearOperators</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="prvect_2.html#K11" title="PRVECT_2:func.11">.</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="c5">i</font>)</span>)</span></span>)</span>,<font color="Maroon" title="c2">F</font>)</span>)</span><br/>  for <font color="Olive" title="b2">p</font>, <font color="Olive" title="b3">q</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="prvect_2.html#K11" title="PRVECT_2:func.11">.</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="c5">i</font>)</span>)</span></span>)</span>  st <font color="Maroon" title="c5">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; (  for <font color="Olive" title="b4">h</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="prvect_2.html#K11" title="PRVECT_2:func.11">.</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="c5">i</font>)</span>)</span></span>)</span>  st <font color="Olive" title="b4">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="ndiff_5.html#K2" title="NDIFF_5:func.2">].</a><span class="default"><font color="Olive" title="b2">p</font>,<font color="Olive" title="b3">q</font></span><a href="ndiff_5.html#K2" title="NDIFF_5:func.2">.[</a></span> holds <br/><span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ndiff_5.html#K6" title="NDIFF_5:func.6">partdiff</a> (<font color="Maroon" title="c3">f</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p0">(<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="c5">i</font>)</span>)</span>,<font color="Maroon" title="c4">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">h</font></span>)</span>,<font color="Maroon" title="c5">i</font>)</span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Olive" title="b1">L</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">M</font> ) &amp; (  for <font color="Olive" title="b4">h</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="prvect_2.html#K11" title="PRVECT_2:func.11">.</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="c5">i</font>)</span>)</span></span>)</span>  st <font color="Olive" title="b4">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="ndiff_5.html#NK3" title="NDIFF_5:NK.3">[.</a><span class="default"><font color="Olive" title="b2">p</font>,<font color="Olive" title="b3">q</font></span><a href="ndiff_5.html#NK3" title="NDIFF_5:NK.3">.]</a></span> holds <br/><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="c5">i</font>)</span>)</span>,<font color="Maroon" title="c4">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">f</font> ) &amp; (  for <font color="Olive" title="b4">h</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="prvect_2.html#K11" title="PRVECT_2:func.11">.</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="c5">i</font>)</span>)</span></span>)</span>  st <font color="Olive" title="b4">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="ndiff_5.html#NK3" title="NDIFF_5:NK.3">[.</a><span class="default"><font color="Olive" title="b2">p</font>,<font color="Olive" title="b3">q</font></span><a href="ndiff_5.html#NK3" title="NDIFF_5:NK.3">.]</a></span> holds <br/><font color="Maroon" title="c3">f</font> <a href="ndiff_5.html#R1" title="NDIFF_5:pred.1">is_partial_differentiable_in</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="c5">i</font>)</span>)</span>,<font color="Maroon" title="c4">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">h</font>,<font color="Maroon" title="c5">i</font> ) holds <br/><span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p1">(<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="c5">i</font>)</span>)</span>,<font color="Maroon" title="c4">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">q</font></span>)</span></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p1">(<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="c5">i</font>)</span>)</span>,<font color="Maroon" title="c4">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">p</font></span>)</span></span>)</span></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">L</font> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <span class="p4">(<span class="default"><font color="Olive" title="b3">q</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Olive" title="b2">p</font></span>)</span></span>)</span></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">M</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b3">q</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Olive" title="b2">p</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c7">L</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lopban_1.html#K16" title="LOPBAN_1:func.16">R_NormSpace_of_BoundedLinearOperators</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="prvect_2.html#K11" title="PRVECT_2:func.11">.</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="c5">i</font>)</span>)</span></span>)</span>,<font color="Maroon" title="c2">F</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">p</font>, <font color="Olive" title="b2">q</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="prvect_2.html#K11" title="PRVECT_2:func.11">.</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="c5">i</font>)</span>)</span></span>)</span>  st <font color="Maroon" title="c5">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; (  for <font color="Olive" title="b3">h</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="prvect_2.html#K11" title="PRVECT_2:func.11">.</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="c5">i</font>)</span>)</span></span>)</span>  st <font color="Olive" title="b3">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="ndiff_5.html#K2" title="NDIFF_5:func.2">].</a><span class="default"><font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font></span><a href="ndiff_5.html#K2" title="NDIFF_5:func.2">.[</a></span> holds <br/><span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ndiff_5.html#K6" title="NDIFF_5:func.6">partdiff</a> (<font color="Maroon" title="c3">f</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p0">(<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="c5">i</font>)</span>)</span>,<font color="Maroon" title="c4">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">h</font></span>)</span>,<font color="Maroon" title="c5">i</font>)</span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c7">L</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">M</font> ) &amp; (  for <font color="Olive" title="b3">h</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="prvect_2.html#K11" title="PRVECT_2:func.11">.</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="c5">i</font>)</span>)</span></span>)</span>  st <font color="Olive" title="b3">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="ndiff_5.html#NK3" title="NDIFF_5:NK.3">[.</a><span class="default"><font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font></span><a href="ndiff_5.html#NK3" title="NDIFF_5:NK.3">.]</a></span> holds <br/><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="c5">i</font>)</span>)</span>,<font color="Maroon" title="c4">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">f</font> ) &amp; (  for <font color="Olive" title="b3">h</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="prvect_2.html#K11" title="PRVECT_2:func.11">.</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="c5">i</font>)</span>)</span></span>)</span>  st <font color="Olive" title="b3">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="ndiff_5.html#NK3" title="NDIFF_5:NK.3">[.</a><span class="default"><font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font></span><a href="ndiff_5.html#NK3" title="NDIFF_5:NK.3">.]</a></span> holds <br/><font color="Maroon" title="c3">f</font> <a href="ndiff_5.html#R1" title="NDIFF_5:pred.1">is_partial_differentiable_in</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="c5">i</font>)</span>)</span>,<font color="Maroon" title="c4">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">h</font>,<font color="Maroon" title="c5">i</font> ) holds <br/><span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p1">(<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="c5">i</font>)</span>)</span>,<font color="Maroon" title="c4">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">q</font></span>)</span></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p1">(<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="c5">i</font>)</span>)</span>,<font color="Maroon" title="c4">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font></span>)</span></span>)</span></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">L</font> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <span class="p4">(<span class="default"><font color="Olive" title="b2">q</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Olive" title="b1">p</font></span>)</span></span>)</span></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">M</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b2">q</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Olive" title="b1">p</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c8">p</font>, <font color="Maroon" title="c9">q</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="prvect_2.html#K11" title="PRVECT_2:func.11">.</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="c5">i</font>)</span>)</span></span>)</span>; <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="c5">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; (  for <font color="Olive" title="b1">h</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="prvect_2.html#K11" title="PRVECT_2:func.11">.</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="c5">i</font>)</span>)</span></span>)</span>  st <font color="Olive" title="b1">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="ndiff_5.html#K2" title="NDIFF_5:func.2">].</a><span class="default"><font color="Maroon" title="c8">p</font>,<font color="Maroon" title="c9">q</font></span><a href="ndiff_5.html#K2" title="NDIFF_5:func.2">.[</a></span> holds <br/><span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ndiff_5.html#K6" title="NDIFF_5:func.6">partdiff</a> (<font color="Maroon" title="c3">f</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p0">(<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="c5">i</font>)</span>)</span>,<font color="Maroon" title="c4">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">h</font></span>)</span>,<font color="Maroon" title="c5">i</font>)</span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c7">L</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">M</font> ) &amp; (  for <font color="Olive" title="b1">h</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="prvect_2.html#K11" title="PRVECT_2:func.11">.</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="c5">i</font>)</span>)</span></span>)</span>  st <font color="Olive" title="b1">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="ndiff_5.html#NK3" title="NDIFF_5:NK.3">[.</a><span class="default"><font color="Maroon" title="c8">p</font>,<font color="Maroon" title="c9">q</font></span><a href="ndiff_5.html#NK3" title="NDIFF_5:NK.3">.]</a></span> holds <br/><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="c5">i</font>)</span>)</span>,<font color="Maroon" title="c4">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">f</font> ) &amp; (  for <font color="Olive" title="b1">h</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="prvect_2.html#K11" title="PRVECT_2:func.11">.</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="c5">i</font>)</span>)</span></span>)</span>  st <font color="Olive" title="b1">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="ndiff_5.html#NK3" title="NDIFF_5:NK.3">[.</a><span class="default"><font color="Maroon" title="c8">p</font>,<font color="Maroon" title="c9">q</font></span><a href="ndiff_5.html#NK3" title="NDIFF_5:NK.3">.]</a></span> holds <br/><font color="Maroon" title="c3">f</font> <a href="ndiff_5.html#R1" title="NDIFF_5:pred.1">is_partial_differentiable_in</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="c5">i</font>)</span>)</span>,<font color="Maroon" title="c4">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">h</font>,<font color="Maroon" title="c5">i</font> ) implies <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p1">(<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="c5">i</font>)</span>)</span>,<font color="Maroon" title="c4">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">q</font></span>)</span></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p1">(<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="c5">i</font>)</span>)</span>,<font color="Maroon" title="c4">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">p</font></span>)</span></span>)</span></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">L</font> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">q</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c8">p</font></span>)</span></span>)</span></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">M</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">q</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c8">p</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> )</span><br/>















<span class="kw">assume </span><a NAME="E1:71"/><span class="lab"><font color="Green" title="E58">A1</font></span>: 
( <font color="Maroon" title="c5">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; (  for <font color="Olive" title="b1">h</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="prvect_2.html#K11" title="PRVECT_2:func.11">.</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="c5">i</font>)</span>)</span></span>)</span>  st <font color="Olive" title="b1">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="ndiff_5.html#K2" title="NDIFF_5:func.2">].</a><span class="default"><font color="Maroon" title="c8">p</font>,<font color="Maroon" title="c9">q</font></span><a href="ndiff_5.html#K2" title="NDIFF_5:func.2">.[</a></span> holds <br/><span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ndiff_5.html#K6" title="NDIFF_5:func.6">partdiff</a> (<font color="Maroon" title="c3">f</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p0">(<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="c5">i</font>)</span>)</span>,<font color="Maroon" title="c4">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">h</font></span>)</span>,<font color="Maroon" title="c5">i</font>)</span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c7">L</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">M</font> ) &amp; (  for <font color="Olive" title="b1">h</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="prvect_2.html#K11" title="PRVECT_2:func.11">.</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="c5">i</font>)</span>)</span></span>)</span>  st <font color="Olive" title="b1">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="ndiff_5.html#NK3" title="NDIFF_5:NK.3">[.</a><span class="default"><font color="Maroon" title="c8">p</font>,<font color="Maroon" title="c9">q</font></span><a href="ndiff_5.html#NK3" title="NDIFF_5:NK.3">.]</a></span> holds <br/><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="c5">i</font>)</span>)</span>,<font color="Maroon" title="c4">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">f</font> ) &amp; (  for <font color="Olive" title="b1">h</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="prvect_2.html#K11" title="PRVECT_2:func.11">.</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="c5">i</font>)</span>)</span></span>)</span>  st <font color="Olive" title="b1">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="ndiff_5.html#NK3" title="NDIFF_5:NK.3">[.</a><span class="default"><font color="Maroon" title="c8">p</font>,<font color="Maroon" title="c9">q</font></span><a href="ndiff_5.html#NK3" title="NDIFF_5:NK.3">.]</a></span> holds <br/><font color="Maroon" title="c3">f</font> <a href="ndiff_5.html#R1" title="NDIFF_5:pred.1">is_partial_differentiable_in</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="c5">i</font>)</span>)</span>,<font color="Maroon" title="c4">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">h</font>,<font color="Maroon" title="c5">i</font> ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p1">(<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="c5">i</font>)</span>)</span>,<font color="Maroon" title="c4">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">q</font></span>)</span></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p1">(<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="c5">i</font>)</span>)</span>,<font color="Maroon" title="c4">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">p</font></span>)</span></span>)</span></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">L</font> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">q</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c8">p</font></span>)</span></span>)</span></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">M</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">q</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c8">p</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span></span><br/>

<span class="kw">set </span><font color="Maroon" title="c10">m</font> =  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">G</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c11">S</font> = <font color="Maroon" title="c1">G</font> <a href="prvect_2.html#K11" title="PRVECT_2:func.11">.</a> <span class="p1">(<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="c5">i</font>)</span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c12">g</font> = <font color="Maroon" title="c3">f</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="c5">i</font>)</span>)</span>,<font color="Maroon" title="c4">x</font>)</span>)</span>;<br/>
<div><span class="lab"><font color="Green" title="E59">A6</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c13">h</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="c13">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="ndiff_5.html#NK3" title="NDIFF_5:NK.3">[.</a><span class="default"><font color="Maroon" title="c8">p</font>,<font color="Maroon" title="c9">q</font></span><a href="ndiff_5.html#NK3" title="NDIFF_5:NK.3">.]</a></span> implies <font color="Maroon" title="c13">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</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="c5">i</font>)</span>)</span>,<font color="Maroon" title="c4">x</font>)</span>)</span></span>)</span> )</span><br/><span class="kw">assume </span><a NAME="E1:71_1"/><span class="lab"><font color="Green" title="E60">A2</font></span>: 
<font color="Maroon" title="c13">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="ndiff_5.html#NK3" title="NDIFF_5:NK.3">[.</a><span class="default"><font color="Maroon" title="c8">p</font>,<font color="Maroon" title="c9">q</font></span><a href="ndiff_5.html#NK3" title="NDIFF_5:NK.3">.]</a></span>
 ; <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="c13">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</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="c5">i</font>)</span>)</span>,<font color="Maroon" title="c4">x</font>)</span>)</span></span>)</span></span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c14">h1</font> = <font color="Maroon" title="c13">h</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="prvect_2.html#K11" title="PRVECT_2:func.11">.</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="c5">i</font>)</span>)</span></span>)</span> ;<br/><a NAME="E3:71_1"/><span class="lab"><font color="Green" title="E61">A3</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</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="c5">i</font>)</span>)</span>,<font color="Maroon" title="c4">x</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="prvect_2.html#K11" title="PRVECT_2:func.11">.</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="c5">i</font>)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/><a NAME="E4:71_1"/>
<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="c5">i</font>)</span>)</span>,<font color="Maroon" title="c4">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">h1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:71"><span class="lab"><font color="Green" title="E58">A1</font></span></a>, <a class="txt" href="#E1:71_1"><span class="lab"><font color="Green" title="E60">A2</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E5:71_1"/>
<font color="Maroon" title="c13">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</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="c5">i</font>)</span>)</span>,<font color="Maroon" title="c4">x</font>)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:71_1"><span class="lab"><font color="Green" title="E61">A3</font></span></a>, <a class="ref" href="funct_1.html#T11" title="FUNCT_1:th.11">FUNCT_1:11</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/></div><span class="kw">end;</span></div>
<a NAME="E3:71"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A4</font></span>: 
<span class="p1"><a href="ndiff_5.html#NK3" title="NDIFF_5:NK.3">[.</a><span class="default"><font color="Maroon" title="c8">p</font>,<font color="Maroon" title="c9">q</font></span><a href="ndiff_5.html#NK3" title="NDIFF_5:NK.3">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</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="c5">i</font>)</span>)</span>,<font color="Maroon" title="c4">x</font>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E61">A5</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="71_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c13">x0</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="prvect_2.html#K11" title="PRVECT_2:func.11">.</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="c5">i</font>)</span>)</span></span>)</span>; <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="c13">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="ndiff_5.html#NK3" title="NDIFF_5:NK.3">[.</a><span class="default"><font color="Maroon" title="c8">p</font>,<font color="Maroon" title="c9">q</font></span><a href="ndiff_5.html#NK3" title="NDIFF_5:NK.3">.]</a></span> implies <font color="Maroon" title="c3">f</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="c5">i</font>)</span>)</span>,<font color="Maroon" title="c4">x</font>)</span>)</span> <a href="ndiff_1.html#R1" title="NDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c13">x0</font> )</span><br/><span class="kw">assume </span><a NAME="E1:71_2"/><span class="lab"><font color="Green" title="E62">A6</font></span>: 
<font color="Maroon" title="c13">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="ndiff_5.html#NK3" title="NDIFF_5:NK.3">[.</a><span class="default"><font color="Maroon" title="c8">p</font>,<font color="Maroon" title="c9">q</font></span><a href="ndiff_5.html#NK3" title="NDIFF_5:NK.3">.]</a></span>
 ; <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">f</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="c5">i</font>)</span>)</span>,<font color="Maroon" title="c4">x</font>)</span>)</span> <a href="ndiff_1.html#R1" title="NDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c13">x0</font></span><br/><span class="kw">set </span><font color="Maroon" title="c14">y</font> = <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="c5">i</font>)</span>)</span>,<font color="Maroon" title="c4">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">x0</font>;<br/><a NAME="E2:71_2"/><span class="lab"><font color="Green" title="E63">A7</font></span>: 
<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="c5">i</font>)</span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><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="c5">i</font>)</span>)</span>,<font color="Maroon" title="c4">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">x0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">x0</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ndiff_5.html#T46" target="_self" title="NDIFF_5:th.46">XTh39</a></span>;<br/><a NAME="E3:71_2"/>
<font color="Maroon" title="c3">f</font> <a href="ndiff_5.html#R1" title="NDIFF_5:pred.1">is_partial_differentiable_in</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="c5">i</font>)</span>)</span>,<font color="Maroon" title="c4">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">x0</font>,<font color="Maroon" title="c5">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:71"><span class="lab"><font color="Green" title="E58">A1</font></span></a>, <a class="txt" href="#E1:71_2"><span class="lab"><font color="Green" title="E62">A6</font></span></a></span>;<br/><a NAME="E4:71_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">f</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="c5">i</font>)</span>)</span>,<span class="p2">(<span class="default"><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="c5">i</font>)</span>)</span>,<font color="Maroon" title="c4">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">x0</font></span>)</span>)</span>)</span> <a href="ndiff_1.html#R1" title="NDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c13">x0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:71_2"><span class="lab"><font color="Green" title="E63">A7</font></span></a>, <a class="ref" href="ndiff_5.html#D6" target="_self" title="NDIFF_5:def.6">Def9</a></span>;<br/><span class="kw">hence </span><a NAME="E5:71_2"/>
<font color="Maroon" title="c3">f</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="c5">i</font>)</span>)</span>,<font color="Maroon" title="c4">x</font>)</span>)</span> <a href="ndiff_1.html#R1" title="NDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c13">x0</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ndiff_5.html#T48" target="_self" title="NDIFF_5:th.48">XTh40</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/></div><span class="kw">end;</span></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_3"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c13">z</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="c13">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="ndiff_5.html#K2" title="NDIFF_5:func.2">].</a><span class="default"><font color="Maroon" title="c8">p</font>,<font color="Maroon" title="c9">q</font></span><a href="ndiff_5.html#K2" title="NDIFF_5:func.2">.[</a></span> implies <font color="Maroon" title="c13">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="ndiff_5.html#NK3" title="NDIFF_5:NK.3">[.</a><span class="default"><font color="Maroon" title="c8">p</font>,<font color="Maroon" title="c9">q</font></span><a href="ndiff_5.html#NK3" title="NDIFF_5:NK.3">.]</a></span> )</span><br/><span class="kw">assume </span><a NAME="E1:71_3"/>
<font color="Maroon" title="c13">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="ndiff_5.html#K2" title="NDIFF_5:func.2">].</a><span class="default"><font color="Maroon" title="c8">p</font>,<font color="Maroon" title="c9">q</font></span><a href="ndiff_5.html#K2" title="NDIFF_5:func.2">.[</a></span>
 ; <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="c13">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="ndiff_5.html#NK3" title="NDIFF_5:NK.3">[.</a><span class="default"><font color="Maroon" title="c8">p</font>,<font color="Maroon" title="c9">q</font></span><a href="ndiff_5.html#NK3" title="NDIFF_5:NK.3">.]</a></span></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">z1</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E3:71_3"/><span class="lab"><font color="Green" title="E62">AB</font></span>: 
( <font color="Maroon" title="c13">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">p</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">z1</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">q</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c8">p</font></span>)</span></span>)</span> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">z1</font> &amp; <font color="Maroon" title="c14">z1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 )
 ;<br/><a NAME="E4:71_3"/>
<font color="Maroon" title="c13">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c14">z1</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c8">p</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="c14">z1</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c9">q</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:71_3"><span class="lab"><font color="Green" title="E62">AB</font></span></a>, <a class="txt" href="ndiff_5.html#E19"><span class="lab"><font color="Green" title="E18">LmX</font></span></a></span>;<br/><a NAME="E5:71_3"/><span class="kw">then </span>
<font color="Maroon" title="c13">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Olive" title="b1">r1</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c8">p</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">r1</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c9">q</font></span>)</span></span>)</span> where <font color="Olive" title="b1">r1</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">r1</font> &amp; <font color="Olive" title="b1">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:71_3"><span class="lab"><font color="Green" title="E62">AB</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E6:71_3"/>
<font color="Maroon" title="c13">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="ndiff_5.html#NK3" title="NDIFF_5:NK.3">[.</a><span class="default"><font color="Maroon" title="c8">p</font>,<font color="Maroon" title="c9">q</font></span><a href="ndiff_5.html#NK3" title="NDIFF_5:NK.3">.]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rltopsp1.html#D2" title="RLTOPSP1:def.2">RLTOPSP1:def 2</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/></div><span class="kw">end;</span></div>
<a NAME="E6:71"/><span class="kw">then </span>
<span class="p1"><a href="ndiff_5.html#K2" title="NDIFF_5:func.2">].</a><span class="default"><font color="Maroon" title="c8">p</font>,<font color="Maroon" title="c9">q</font></span><a href="ndiff_5.html#K2" title="NDIFF_5:func.2">.[</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="ndiff_5.html#NK3" title="NDIFF_5:NK.3">[.</a><span class="default"><font color="Maroon" title="c8">p</font>,<font color="Maroon" title="c9">q</font></span><a href="ndiff_5.html#NK3" title="NDIFF_5:NK.3">.]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/>
<a NAME="E7:71"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A10</font></span>: 
 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"><font color="Maroon" title="c1">G</font> <a href="prvect_2.html#K11" title="PRVECT_2:func.11">.</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="c5">i</font>)</span>)</span></span>)</span>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="ndiff_5.html#K2" title="NDIFF_5:func.2">].</a><span class="default"><font color="Maroon" title="c8">p</font>,<font color="Maroon" title="c9">q</font></span><a href="ndiff_5.html#K2" title="NDIFF_5:func.2">.[</a></span> holds <br/><font color="Maroon" title="c3">f</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="c5">i</font>)</span>)</span>,<font color="Maroon" title="c4">x</font>)</span>)</span> <a href="ndiff_1.html#R1" title="NDIFF_1:pred.1">is_differentiable_in</a> <font color="Olive" title="b1">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:71"><span class="lab"><font color="Green" title="E61">A5</font></span></a></span>;<br/>
<a NAME="E8:71"/><span class="lab"><font color="Green" title="E63">A11</font></span>: 
 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"><font color="Maroon" title="c1">G</font> <a href="prvect_2.html#K11" title="PRVECT_2:func.11">.</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="c5">i</font>)</span>)</span></span>)</span>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="ndiff_5.html#NK3" title="NDIFF_5:NK.3">[.</a><span class="default"><font color="Maroon" title="c8">p</font>,<font color="Maroon" title="c9">q</font></span><a href="ndiff_5.html#NK3" title="NDIFF_5:NK.3">.]</a></span> holds <br/><font color="Maroon" title="c3">f</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="c5">i</font>)</span>)</span>,<font color="Maroon" title="c4">x</font>)</span>)</span> <a href="nfcont_1.html#R1" title="NFCONT_1:pred.1">is_continuous_in</a> <font color="Olive" title="b1">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:71"><span class="lab"><font color="Green" title="E61">A5</font></span></a>, <a class="ref" href="ndiff_1.html#T44" title="NDIFF_1:th.44">NDIFF_1:44</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E64">U0</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="71_4"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c13">h</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="prvect_2.html#K11" title="PRVECT_2:func.11">.</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="c5">i</font>)</span>)</span></span>)</span>; <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="c13">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="ndiff_5.html#K2" title="NDIFF_5:func.2">].</a><span class="default"><font color="Maroon" title="c8">p</font>,<font color="Maroon" title="c9">q</font></span><a href="ndiff_5.html#K2" title="NDIFF_5:func.2">.[</a></span> implies <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> (<span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p5">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p0">(<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="c5">i</font>)</span>)</span>,<font color="Maroon" title="c4">x</font>)</span>)</span></span>)</span>,<font color="Maroon" title="c13">h</font>)</span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c7">L</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">M</font> )</span><br/><span class="kw">set </span><font color="Maroon" title="c14">y</font> = <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="c5">i</font>)</span>)</span>,<font color="Maroon" title="c4">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">h</font>;<br/><span class="kw">assume </span><a NAME="E1:71_4"/>
<font color="Maroon" title="c13">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="ndiff_5.html#K2" title="NDIFF_5:func.2">].</a><span class="default"><font color="Maroon" title="c8">p</font>,<font color="Maroon" title="c9">q</font></span><a href="ndiff_5.html#K2" title="NDIFF_5:func.2">.[</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> (<span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p5">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p0">(<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="c5">i</font>)</span>)</span>,<font color="Maroon" title="c4">x</font>)</span>)</span></span>)</span>,<font color="Maroon" title="c13">h</font>)</span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c7">L</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">M</font></span><br/><a NAME="E2:71_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">X1</font></span>: 
<span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ndiff_5.html#K6" title="NDIFF_5:func.6">partdiff</a> (<font color="Maroon" title="c3">f</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p0">(<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="c5">i</font>)</span>)</span>,<font color="Maroon" title="c4">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">h</font></span>)</span>,<font color="Maroon" title="c5">i</font>)</span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c7">L</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">M</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:71"><span class="lab"><font color="Green" title="E58">A1</font></span></a></span>;<br/><a NAME="E3:71_4"/>
<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="c5">i</font>)</span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><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="c5">i</font>)</span>)</span>,<font color="Maroon" title="c4">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">h</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">h</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ndiff_5.html#T46" target="_self" title="NDIFF_5:th.46">XTh39</a></span>;<br/><span class="kw">hence </span><a NAME="E4:71_4"/>
<span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> (<span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p5">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p0">(<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="c5">i</font>)</span>)</span>,<font color="Maroon" title="c4">x</font>)</span>)</span></span>)</span>,<font color="Maroon" title="c13">h</font>)</span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c7">L</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">M</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:71_4"><span class="lab"><font color="Green" title="E65">X1</font></span></a>, <a class="ref" href="ndiff_5.html#T48" target="_self" title="NDIFF_5:th.48">XTh40</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/></div><span class="kw">end;</span></div>
<a NAME="E10:71"/><span class="lab"><font color="Green" title="E65">U1</font></span>: 
( <font color="Maroon" title="c8">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</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="c5">i</font>)</span>)</span>,<font color="Maroon" title="c4">x</font>)</span>)</span></span>)</span> &amp; <font color="Maroon" title="c9">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</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="c5">i</font>)</span>)</span>,<font color="Maroon" title="c4">x</font>)</span>)</span></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:71"><span class="lab"><font color="Green" title="E59">A6</font></span></a>, <a class="ref" href="rltopsp1.html#T68" title="RLTOPSP1:th.68">RLTOPSP1:68</a></span>;<br/>
<a NAME="E11:71"/>
( <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><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="c5">i</font>)</span>)</span>,<font color="Maroon" title="c4">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><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="c5">i</font>)</span>)</span>,<font color="Maroon" title="c4">x</font>)</span>)</span> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c8">p</font></span>)</span> &amp; <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><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="c5">i</font>)</span>)</span>,<font color="Maroon" title="c4">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><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="c5">i</font>)</span>)</span>,<font color="Maroon" title="c4">x</font>)</span>)</span> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c9">q</font></span>)</span> )
 
;<br/>
<a NAME="E12:71"/><span class="kw">then </span>
( <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><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="c5">i</font>)</span>)</span>,<font color="Maroon" title="c4">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</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="c5">i</font>)</span>)</span>,<font color="Maroon" title="c4">x</font>)</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c8">p</font> &amp; <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><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="c5">i</font>)</span>)</span>,<font color="Maroon" title="c4">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</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="c5">i</font>)</span>)</span>,<font color="Maroon" title="c4">x</font>)</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">q</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:71"><span class="lab"><font color="Green" title="E65">U1</font></span></a>, <a class="ref" href="partfun2.html#T3" title="PARTFUN2:th.3">PARTFUN2:3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:71"/>
<span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p1">(<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="c5">i</font>)</span>)</span>,<font color="Maroon" title="c4">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">q</font></span>)</span></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p1">(<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="c5">i</font>)</span>)</span>,<font color="Maroon" title="c4">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">p</font></span>)</span></span>)</span></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">L</font> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">q</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c8">p</font></span>)</span></span>)</span></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">M</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">q</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c8">p</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:71"><span class="lab"><font color="Green" title="E64">U0</font></span></a>, <a class="ref" href="ndiff_5.html#T20" target="_self" title="NDIFF_5:th.20">FDAF20</a>, <a class="txt" href="#E3:71"><span class="lab"><font color="Green" title="E60">A4</font></span></a>, <a class="txt" href="#E7:71"><span class="lab"><font color="Green" title="E62">A10</font></span></a>, <a class="txt" href="#E8:71"><span class="lab"><font color="Green" title="E63">A11</font></span></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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