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<span class="kw">let </span><font color="Maroon" title="c1">S</font>, <font color="Maroon" title="c2">T</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 <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font><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 <font color="Maroon" title="c1">S</font><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> (<font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font>)</span>)</span>  st <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> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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="b6">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">S</font>  st <font color="Olive" title="b6">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="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="Olive" title="b1">f</font> <a href="nfcont_1.html#R1" title="NFCONT_1:pred.1">is_continuous_in</a> <font color="Olive" title="b6">x</font> ) &amp; (  for <font color="Olive" title="b6">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">S</font>  st <font color="Olive" title="b6">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="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/><font color="Olive" title="b1">f</font> <a href="ndiff_1.html#R1" title="NDIFF_1:pred.1">is_differentiable_in</a> <font color="Olive" title="b6">x</font> ) &amp; (  for <font color="Olive" title="b6">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">S</font>  st <font color="Olive" title="b6">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="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_1.html#K3" title="NDIFF_1:func.3">diff</a> (<font color="Olive" title="b1">f</font>,<font color="Olive" title="b6">x</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> ) 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> <font color="Olive" title="b3">q</font></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> <font color="Olive" title="b2">p</font></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="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="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="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="c3">f</font> be   <a href="struct_0.html#NM12" title="STRUCT_0:NM.12">PartFunc</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</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">p</font>, <font color="Olive" title="b2">q</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">S</font><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> (<font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font>)</span>)</span>  st <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> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">S</font>  st <font color="Olive" title="b5">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="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="nfcont_1.html#R1" title="NFCONT_1:pred.1">is_continuous_in</a> <font color="Olive" title="b5">x</font> ) &amp; (  for <font color="Olive" title="b5">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">S</font>  st <font color="Olive" title="b5">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="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/><font color="Maroon" title="c3">f</font> <a href="ndiff_1.html#R1" title="NDIFF_1:pred.1">is_differentiable_in</a> <font color="Olive" title="b5">x</font> ) &amp; (  for <font color="Olive" title="b5">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">S</font>  st <font color="Olive" title="b5">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="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_1.html#K3" title="NDIFF_1:func.3">diff</a> (<font color="Maroon" title="c3">f</font>,<font color="Olive" title="b5">x</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> ) 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> <font color="Olive" title="b2">q</font></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> <font color="Olive" title="b1">p</font></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="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="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="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="c4">p</font>, <font color="Maroon" title="c5">q</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">S</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">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> (<font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font>)</span>)</span>  st <span class="p1"><a href="ndiff_5.html#NK3" title="NDIFF_5:NK.3">[.</a><span class="default"><font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">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> <font color="Maroon" title="c3">f</font> &amp; (  for <font color="Olive" title="b3">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">S</font>  st <font color="Olive" title="b3">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="c4">p</font>,<font color="Maroon" title="c5">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="nfcont_1.html#R1" title="NFCONT_1:pred.1">is_continuous_in</a> <font color="Olive" title="b3">x</font> ) &amp; (  for <font color="Olive" title="b3">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">S</font>  st <font color="Olive" title="b3">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="c4">p</font>,<font color="Maroon" title="c5">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="ndiff_1.html#R1" title="NDIFF_1:pred.1">is_differentiable_in</a> <font color="Olive" title="b3">x</font> ) &amp; (  for <font color="Olive" title="b3">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">S</font>  st <font color="Olive" title="b3">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="c4">p</font>,<font color="Maroon" title="c5">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_1.html#K3" title="NDIFF_1:func.3">diff</a> (<font color="Maroon" title="c3">f</font>,<font color="Olive" title="b3">x</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> ) 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> <font color="Maroon" title="c5">q</font></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> <font color="Maroon" title="c4">p</font></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="Maroon" title="c5">q</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c4">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="Maroon" title="c5">q</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c4">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> (<font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font>)</span>)</span>  st <span class="p1"><a href="ndiff_5.html#NK3" title="NDIFF_5:NK.3">[.</a><span class="default"><font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">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> <font color="Maroon" title="c3">f</font> &amp; (  for <font color="Olive" title="b2">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">S</font>  st <font color="Olive" title="b2">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="c4">p</font>,<font color="Maroon" title="c5">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="nfcont_1.html#R1" title="NFCONT_1:pred.1">is_continuous_in</a> <font color="Olive" title="b2">x</font> ) &amp; (  for <font color="Olive" title="b2">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">S</font>  st <font color="Olive" title="b2">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="c4">p</font>,<font color="Maroon" title="c5">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="ndiff_1.html#R1" title="NDIFF_1:pred.1">is_differentiable_in</a> <font color="Olive" title="b2">x</font> ) &amp; (  for <font color="Olive" title="b2">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">S</font>  st <font color="Olive" title="b2">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="c4">p</font>,<font color="Maroon" title="c5">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_1.html#K3" title="NDIFF_1:func.3">diff</a> (<font color="Maroon" title="c3">f</font>,<font color="Olive" title="b2">x</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> ) 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> <font color="Maroon" title="c5">q</font></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> <font color="Maroon" title="c4">p</font></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="Maroon" title="c5">q</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c4">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="c5">q</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c4">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> (<font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font>)</span>)</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="ndiff_5.html#NK3" title="NDIFF_5:NK.3">[.</a><span class="default"><font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">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> <font color="Maroon" title="c3">f</font> &amp; (  for <font color="Olive" title="b1">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">S</font>  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="c4">p</font>,<font color="Maroon" title="c5">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="nfcont_1.html#R1" title="NFCONT_1:pred.1">is_continuous_in</a> <font color="Olive" title="b1">x</font> ) &amp; (  for <font color="Olive" title="b1">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">S</font>  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="c4">p</font>,<font color="Maroon" title="c5">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="ndiff_1.html#R1" title="NDIFF_1:pred.1">is_differentiable_in</a> <font color="Olive" title="b1">x</font> ) &amp; (  for <font color="Olive" title="b1">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">S</font>  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="c4">p</font>,<font color="Maroon" title="c5">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_1.html#K3" title="NDIFF_1:func.3">diff</a> (<font color="Maroon" title="c3">f</font>,<font color="Olive" title="b1">x</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> ) 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> <font color="Maroon" title="c5">q</font></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> <font color="Maroon" title="c4">p</font></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="c5">q</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c4">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="c5">q</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c4">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><span class="kw">that </span><br/><a NAME="E1:32"/><span class="lab"><font color="Green" title="E26">A1</font></span>: 
<span class="p1"><a href="ndiff_5.html#NK3" title="NDIFF_5:NK.3">[.</a><span class="default"><font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">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> <font color="Maroon" title="c3">f</font>
 <span class="kw">and </span><br/><a NAME="E2:32"/><span class="lab"><font color="Green" title="E27">A2</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 <font color="Maroon" title="c1">S</font>  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="c4">p</font>,<font color="Maroon" title="c5">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="nfcont_1.html#R1" title="NFCONT_1:pred.1">is_continuous_in</a> <font color="Olive" title="b1">x</font> ) &amp; (  for <font color="Olive" title="b1">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">S</font>  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="c4">p</font>,<font color="Maroon" title="c5">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="ndiff_1.html#R1" title="NDIFF_1:pred.1">is_differentiable_in</a> <font color="Olive" title="b1">x</font> ) &amp; (  for <font color="Olive" title="b1">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">S</font>  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="c4">p</font>,<font color="Maroon" title="c5">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_1.html#K3" title="NDIFF_1:func.3">diff</a> (<font color="Maroon" title="c3">f</font>,<font color="Olive" title="b1">x</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> ) )
 ; <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> <font color="Maroon" title="c5">q</font></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> <font color="Maroon" title="c4">p</font></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="c5">q</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c4">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="c5">q</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c4">p</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c8">LP</font> = <font color="Maroon" title="c7">L</font> as   <a href="lopban_1.html#V2" title="LOPBAN_1:attr.2">bounded</a>  <a href="lopban_1.html#NM1" title="LOPBAN_1:NM.1">LinearOperator</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="lopban_1.html#D9" title="LOPBAN_1:def.9">LOPBAN_1:def 9</a></span>;<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">S</font>) <span class="kw">-&gt; </span>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">T</font> = <font color="Maroon" title="c7">L</font> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <span class="p1">(<span class="default">$1 <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c4">p</font></span>)</span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c9">L0</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">T</font><span class="kw"> such that </span><br/><a NAME="E5:32"/><span class="lab"><font color="Green" title="E28">P0</font></span>: 
 for <font color="Olive" title="b1">t</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> holds  <font color="Maroon" title="c9">L0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">t</font>)
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S4" title="FUNCT_2:sch.4">FUNCT_2:sch 4</a>();<br/></span>
<a NAME="E6:32"/><span class="lab"><font color="Green" title="E29">D1</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c9">L0</font> <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 <font color="Maroon" title="c1">S</font>
 
<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/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c10">x1</font>, <font color="Maroon" title="c11">x2</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c10">x1</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="c9">L0</font> &amp; <font color="Maroon" title="c11">x2</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="c9">L0</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"><font color="Maroon" title="c9">L0</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c10">x1</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">L0</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c11">x2</font></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> <span class="p1">(<span class="default"><span class="p2"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c7">L</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> 1</span>)</span> <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="c10">x1</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c11">x2</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="E1:32_1"/>
( <font color="Maroon" title="c10">x1</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="c9">L0</font> &amp; <font color="Maroon" title="c11">x2</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="c9">L0</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"><font color="Maroon" title="c9">L0</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c10">x1</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">L0</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c11">x2</font></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> <span class="p1">(<span class="default"><span class="p2"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c7">L</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> 1</span>)</span> <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="c10">x1</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c11">x2</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span></span><br/><a NAME="E2:32_1"/><span class="lab"><font color="Green" title="E30">A4</font></span>: 
 <a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1 is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></span>;<br/><a NAME="E3:32_1"/>
( <font color="Maroon" title="c9">L0</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c10">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">L</font> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">x1</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c4">p</font></span>)</span> &amp; <font color="Maroon" title="c9">L0</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c11">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">L</font> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">x2</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c4">p</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:32"><span class="lab"><font color="Green" title="E28">P0</font></span></a></span>;<br/><span class="kw">then </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"><font color="Maroon" title="c9">L0</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c10">x1</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">L0</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c11">x2</font></span>)</span></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></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"><font color="Maroon" title="c8">LP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c10">x1</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c4">p</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1</span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">LP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c11">x2</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c4">p</font></span>)</span></span>)</span></span>)</span></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="ref" href="rlvect_1.html#T16" title="RLVECT_1:th.16">RLVECT_1:16</a></span>
<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"><font color="Maroon" title="c8">LP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c10">x1</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c4">p</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">LP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1</span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c11">x2</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c4">p</font></span>)</span></span>)</span></span>)</span></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="#E2:32_1"><span class="lab"><font color="Green" title="E30">A4</font></span></a>, <a class="ref" href="lopban_1.html#D5" title="LOPBAN_1:def.5">LOPBAN_1:def 5</a></span>
<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"><font color="Maroon" title="c8">LP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">x1</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1</span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c11">x2</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c4">p</font></span>)</span></span>)</span></span>)</span></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="ref" href="grcat_1.html#D8" title="GRCAT_1:def.8">GRCAT_1:def 8</a></span>
<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"><font color="Maroon" title="c8">LP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">x1</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c11">x2</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c4">p</font></span>)</span></span>)</span></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="ref" href="rlvect_1.html#T16" title="RLVECT_1:th.16">RLVECT_1:16</a></span>
<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"><font color="Maroon" title="c8">LP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">x1</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c11">x2</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">p</font></span>)</span></span>)</span></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="ref" href="rlvect_1.html#T27" title="RLVECT_1:th.27">RLVECT_1:27</a></span>
<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"><font color="Maroon" title="c8">LP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">x1</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c11">x2</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c4">p</font></span>)</span></span>)</span></span>)</span></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="ref" href="rlvect_1.html#T29" title="RLVECT_1:th.29">RLVECT_1:29</a></span>
<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"><font color="Maroon" title="c8">LP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">x1</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c11">x2</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p5">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">S</font></span>)</span></span>)</span></span>)</span></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="ref" href="rlvect_1.html#T15" title="RLVECT_1:th.15">RLVECT_1:15</a></span>
<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"><font color="Maroon" title="c8">LP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">x1</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c11">x2</font></span>)</span></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="ref" href="rlvect_1.html#T13" title="RLVECT_1:th.13">RLVECT_1:13</a></span>
;<br/><a NAME="E5:32_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A5</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"><font color="Maroon" title="c9">L0</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c10">x1</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">L0</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c11">x2</font></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> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c7">L</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <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="c10">x1</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c11">x2</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="ref" href="lopban_1.html#T32" title="LOPBAN_1:th.32">LOPBAN_1:32</a></span>;<br/><a NAME="E6:32_1"/>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c7">L</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c7">L</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><a NAME="E7:32_1"/><span class="kw">then </span>
<span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c7">L</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <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="c10">x1</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c11">x2</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> <span class="p1">(<span class="default"><span class="p2"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c7">L</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> 1</span>)</span> <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="c10">x1</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c11">x2</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="ref" href="xreal_1.html#T64" title="XREAL_1:th.64">XREAL_1:64</a></span>;<br/><span class="kw">hence </span><a NAME="E8:32_1"/>
<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"><font color="Maroon" title="c9">L0</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c10">x1</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">L0</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c11">x2</font></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> <span class="p1">(<span class="default"><span class="p2"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c7">L</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> 1</span>)</span> <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="c10">x1</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c11">x2</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="#E5:32_1"><span class="lab"><font color="Green" title="E31">A5</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</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="E8:32"/><span class="kw">then </span>
<font color="Maroon" title="c9">L0</font> <a href="nfcont_1.html#R5" title="NFCONT_1:pred.5">is_Lipschitzian_on</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c9">L0</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nfcont_1.html#D9" title="NFCONT_1:def.9">NFCONT_1:def 9</a></span>;<br/>
<a NAME="E9:32"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">D3</font></span>: 
<font color="Maroon" title="c9">L0</font> <a href="nfcont_1.html#R3" title="NFCONT_1:pred.3">is_continuous_on</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c9">L0</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nfcont_1.html#T45" title="NFCONT_1:th.45">NFCONT_1:45</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">R</font> =  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">T</font></span>)</span> as   <a href="struct_0.html#NM12" title="STRUCT_0:NM.12">PartFunc</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font> ;<br/>
<a NAME="E11:32"/><span class="lab"><font color="Green" title="E31">A6</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c10">R</font> <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 <font color="Maroon" title="c1">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c11">h</font> be   <a href="ndiff_1.html#V1" title="NDIFF_1:attr.1">convergent_to_0</a>  <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c1">S</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">(<span class="default"><span class="p2"><a href="normsp_0.html#K4" title="NORMSP_0:func.4">||.</a><span class="default"><font color="Maroon" title="c11">h</font></span><a href="normsp_0.html#K4" title="NORMSP_0:func.4">.||</a></span> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="ndiff_1.html#K1" title="NDIFF_1:func.1">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">R</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c11">h</font></span>)</span> is  <a href="normsp_1.html#V3" title="NORMSP_1:attr.3">convergent</a>  &amp;  <a href="normsp_1.html#K6" title="NORMSP_1:func.6">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="normsp_0.html#K4" title="NORMSP_0:func.4">||.</a><span class="default"><font color="Maroon" title="c11">h</font></span><a href="normsp_0.html#K4" title="NORMSP_0:func.4">.||</a></span> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="ndiff_1.html#K1" title="NDIFF_1:func.1">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">R</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c11">h</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">T</font> )</span><br/><div><span class="lab"><font color="Green" title="E32">A7</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c12">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="normsp_0.html#K4" title="NORMSP_0:func.4">||.</a><span class="default"><font color="Maroon" title="c11">h</font></span><a href="normsp_0.html#K4" title="NORMSP_0:func.4">.||</a></span> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="ndiff_1.html#K1" title="NDIFF_1:func.1">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">R</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c11">h</font></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">T</font></span><br/><span class="lab"><font color="Green" title="E33">A8</font></span>: <font color="Maroon" title="c10">R</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">h</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">n</font></span>)</span> = 
<font color="Maroon" title="c10">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">h</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">n</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:32"><span class="lab"><font color="Green" title="E31">A6</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">T</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T7" title="FUNCOP_1:th.7">FUNCOP_1:7</a></span>
;<br/><a NAME="E2:32_2_1"/><span class="lab"><font color="Green" title="E34">A9</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c11">h</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c10">R</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:32"><span class="lab"><font color="Green" title="E31">A6</font></span></a></span>;<br/><a NAME="E3:32_2_1"/><span class="lab"><font color="Green" title="E35">A10</font></span>: 
<font color="Maroon" title="c12">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/><span class="kw">hence </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="normsp_0.html#K4" title="NORMSP_0:func.4">||.</a><span class="default"><font color="Maroon" title="c11">h</font></span><a href="normsp_0.html#K4" title="NORMSP_0:func.4">.||</a></span> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="ndiff_1.html#K1" title="NDIFF_1:func.1">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">R</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c11">h</font></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">n</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="normsp_0.html#K4" title="NORMSP_0:func.4">||.</a><span class="default"><font color="Maroon" title="c11">h</font></span><a href="normsp_0.html#K4" title="NORMSP_0:func.4">.||</a></span> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">n</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">R</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c11">h</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">n</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ndiff_1.html#D2" title="NDIFF_1:def.2">NDIFF_1:def 2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="normsp_0.html#K4" title="NORMSP_0:func.4">||.</a><span class="default"><font color="Maroon" title="c11">h</font></span><a href="normsp_0.html#K4" title="NORMSP_0:func.4">.||</a></span> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">n</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">R</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">h</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">n</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:32_2_1"><span class="lab"><font color="Green" title="E35">A10</font></span></a>, <a class="txt" href="#E2:32_2_1"><span class="lab"><font color="Green" title="E34">A9</font></span></a>, <a class="ref" href="funct_2.html#T109" title="FUNCT_2:th.109">FUNCT_2:109</a></span>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">T</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32_2_1"><span class="lab"><font color="Green" title="E33">A8</font></span></a>, <a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</a></span>
;<br/> <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="E2:32_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A11</font></span>: 
<span class="p1">(<span class="default"><span class="p2"><a href="normsp_0.html#K4" title="NORMSP_0:func.4">||.</a><span class="default"><font color="Maroon" title="c11">h</font></span><a href="normsp_0.html#K4" title="NORMSP_0:func.4">.||</a></span> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="ndiff_1.html#K1" title="NDIFF_1:func.1">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">R</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c11">h</font></span>)</span> is  <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="valued_0.html#D18" title="VALUED_0:def.18">VALUED_0:def 18</a></span>;<br/><span class="kw">hence </span><a NAME="E3:32_2"/>
<span class="p1">(<span class="default"><span class="p2"><a href="normsp_0.html#K4" title="NORMSP_0:func.4">||.</a><span class="default"><font color="Maroon" title="c11">h</font></span><a href="normsp_0.html#K4" title="NORMSP_0:func.4">.||</a></span> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="ndiff_1.html#K1" title="NDIFF_1:func.1">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">R</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c11">h</font></span>)</span> is  <a href="normsp_1.html#V3" title="NORMSP_1:attr.3">convergent</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ndiff_1.html#T18" title="NDIFF_1:th.18">NDIFF_1:18</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="normsp_1.html#K6" title="NORMSP_1:func.6">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="normsp_0.html#K4" title="NORMSP_0:func.4">||.</a><span class="default"><font color="Maroon" title="c11">h</font></span><a href="normsp_0.html#K4" title="NORMSP_0:func.4">.||</a></span> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="ndiff_1.html#K1" title="NDIFF_1:func.1">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">R</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c11">h</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">T</font></span><br/><a NAME="E4:32_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="normsp_0.html#K4" title="NORMSP_0:func.4">||.</a><span class="default"><font color="Maroon" title="c11">h</font></span><a href="normsp_0.html#K4" title="NORMSP_0:func.4">.||</a></span> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="ndiff_1.html#K1" title="NDIFF_1:func.1">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">R</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c11">h</font></span>)</span></span>)</span> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32_2"><span class="lab"><font color="Green" title="E32">A7</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E5:32_2"/>
 <a href="normsp_1.html#K6" title="NORMSP_1:func.6">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="normsp_0.html#K4" title="NORMSP_0:func.4">||.</a><span class="default"><font color="Maroon" title="c11">h</font></span><a href="normsp_0.html#K4" title="NORMSP_0:func.4">.||</a></span> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="ndiff_1.html#K1" title="NDIFF_1:func.1">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">R</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c11">h</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:32_2"><span class="lab"><font color="Green" title="E33">A11</font></span></a>, <a class="ref" href="ndiff_1.html#T18" title="NDIFF_1:th.18">NDIFF_1:18</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>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">R</font> = <font color="Maroon" title="c10">R</font> as   <a href="ndiff_1.html#NM1" title="NDIFF_1:NM.1">REST</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="ndiff_1.html#D5" title="NDIFF_1:def.5">NDIFF_1:def 5</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E32">A12</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c12">x0</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">L0</font> <a href="ndiff_1.html#R1" title="NDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c12">x0</font> &amp;  <a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> (<font color="Maroon" title="c9">L0</font>,<font color="Maroon" title="c12">x0</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">L</font> )</span><br/><span class="kw">set </span><font color="Maroon" title="c13">N</font> =  the    <a href="nfcont_1.html#M1" title="NFCONT_1:mode.1">Neighbourhood</a> of <font color="Maroon" title="c12">x0</font>;<br/><a NAME="E1:32_3"/><span class="lab"><font color="Green" title="E33">A15</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 <font color="Maroon" title="c1">S</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the    <a href="nfcont_1.html#M1" title="NFCONT_1:mode.1">Neighbourhood</a> of <font color="Maroon" title="c12">x0</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c9">L0</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">L0</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c12">x0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">L</font> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">x</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c12">x0</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">R</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">x</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c12">x0</font></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c14">x</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the    <a href="nfcont_1.html#M1" title="NFCONT_1:mode.1">Neighbourhood</a> of <font color="Maroon" title="c12">x0</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c9">L0</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c14">x</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">L0</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c12">x0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">L</font> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">x</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c12">x0</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">R</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">x</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c12">x0</font></span>)</span></span>)</span> )</span><br/>

<span class="lab"><font color="Green" title="E34">A16</font></span>: <font color="Maroon" title="c11">R</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">x</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c12">x0</font></span>)</span> = 
<font color="Maroon" title="c11">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">x</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c12">x0</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:32"><span class="lab"><font color="Green" title="E31">A6</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">T</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T7" title="FUNCOP_1:th.7">FUNCOP_1:7</a></span>
;<br/>
<a NAME="E2:32_3_1"/><span class="lab"><font color="Green" title="E35">A40</font></span>: 
 <a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1 is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:32_3_1"/>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the    <a href="nfcont_1.html#M1" title="NFCONT_1:mode.1">Neighbourhood</a> of <font color="Maroon" title="c12">x0</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">(<span class="default"><font color="Maroon" title="c9">L0</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c14">x</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">L0</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c12">x0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">L</font> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">x</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c12">x0</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">R</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">x</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c12">x0</font></span>)</span></span>)</span></span><br/>

<span class="kw">thus </span><span class="p1">(<span class="default"><font color="Maroon" title="c9">L0</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c14">x</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">L0</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c12">x0</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">L</font> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">x</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c4">p</font></span>)</span></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">L0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x0</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:32"><span class="lab"><font color="Green" title="E28">P0</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">L</font> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">x</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c4">p</font></span>)</span></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">L</font> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">x0</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c4">p</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:32"><span class="lab"><font color="Green" title="E28">P0</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">LP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">x</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c4">p</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1</span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">LP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">x0</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c4">p</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T16" title="RLVECT_1:th.16">RLVECT_1:16</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">LP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">x</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c4">p</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">LP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1</span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">x0</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c4">p</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:32_3_1"><span class="lab"><font color="Green" title="E35">A40</font></span></a>, <a class="ref" href="lopban_1.html#D5" title="LOPBAN_1:def.5">LOPBAN_1:def 5</a></span>
<br/>.= 
<font color="Maroon" title="c8">LP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">x</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1</span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">x0</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c4">p</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="grcat_1.html#D8" title="GRCAT_1:def.8">GRCAT_1:def 8</a></span>
<br/>.= 
<font color="Maroon" title="c8">LP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">x</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">x0</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c4">p</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T16" title="RLVECT_1:th.16">RLVECT_1:16</a></span>
<br/>.= 
<font color="Maroon" title="c8">LP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">x</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">x0</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">p</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T27" title="RLVECT_1:th.27">RLVECT_1:27</a></span>
<br/>.= 
<font color="Maroon" title="c8">LP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">x</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">x0</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c4">p</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T29" title="RLVECT_1:th.29">RLVECT_1:29</a></span>
<br/>.= 
<font color="Maroon" title="c8">LP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">x</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">x0</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">S</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T15" title="RLVECT_1:th.15">RLVECT_1:15</a></span>
<br/>.= 
<font color="Maroon" title="c8">LP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">x</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c12">x0</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T13" title="RLVECT_1:th.13">RLVECT_1:13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">L</font> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">x</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c12">x0</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">R</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">x</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c12">x0</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32_3_1"><span class="lab"><font color="Green" title="E34">A16</font></span></a>, <a class="ref" href="rlvect_1.html#T4" title="RLVECT_1:th.4">RLVECT_1:4</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><span class="kw">hence </span><a NAME="E2:32_3"/>
<font color="Maroon" title="c9">L0</font> <a href="ndiff_1.html#R1" title="NDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c12">x0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:32"><span class="lab"><font color="Green" title="E29">D1</font></span></a>, <a class="ref" href="ndiff_1.html#D6" title="NDIFF_1:def.6">NDIFF_1:def 6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> (<font color="Maroon" title="c9">L0</font>,<font color="Maroon" title="c12">x0</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">L</font></span><br/><span class="kw">hence </span><a NAME="E3:32_3"/>
 <a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> (<font color="Maroon" title="c9">L0</font>,<font color="Maroon" title="c12">x0</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">L</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:32"><span class="lab"><font color="Green" title="E29">D1</font></span></a>, <a class="txt" href="#E1:32_3"><span class="lab"><font color="Green" title="E33">A15</font></span></a>, <a class="ref" href="ndiff_1.html#D7" title="NDIFF_1:def.7">NDIFF_1:def 7</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>
<span class="kw">set </span><font color="Maroon" title="c12">g</font> = <font color="Maroon" title="c3">f</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c9">L0</font>;<br/>
<span class="lab"><font color="Green" title="E33">P1</font></span>:  <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="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c9">L0</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c9">L0</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vfunct_1.html#D2" title="VFUNCT_1:def.2">VFUNCT_1:def 2</a></span>
<br/>.= 
 <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="#E6:32"><span class="lab"><font color="Green" title="E29">D1</font></span></a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></span>
;<br/>
<a NAME="E16:32"/><span class="lab"><font color="Green" title="E34">F0</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 <font color="Maroon" title="c1">S</font>  st <font color="Olive" title="b1">x</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="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c9">L0</font></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c9">L0</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">x</font> <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#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">L</font> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">x</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c4">p</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c13">x</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c13">x</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="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c9">L0</font></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c9">L0</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c13">x</font> <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#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c13">x</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">L</font> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">x</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c4">p</font></span>)</span></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:32_5"/>
<font color="Maroon" title="c13">x</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="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c9">L0</font></span>)</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">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c9">L0</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c13">x</font> <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#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c13">x</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">L</font> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">x</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c4">p</font></span>)</span></span>)</span></span><br/>

<span class="kw">hence </span><span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c9">L0</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c13">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c13">x</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">L0</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c13">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vfunct_1.html#D2" title="VFUNCT_1:def.2">VFUNCT_1:def 2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c13">x</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">L</font> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">x</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c4">p</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:32"><span class="lab"><font color="Green" title="E28">P0</font></span></a></span>
;<br/> <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="E17:32"/><span class="lab"><font color="Green" title="E35">P2</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 <font color="Maroon" title="c1">S</font>  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="c4">p</font>,<font color="Maroon" title="c5">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="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c9">L0</font> <a href="nfcont_1.html#R1" title="NFCONT_1:pred.1">is_continuous_in</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c13">x</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c13">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="c4">p</font>,<font color="Maroon" title="c5">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="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c9">L0</font> <a href="nfcont_1.html#R1" title="NFCONT_1:pred.1">is_continuous_in</a> <font color="Maroon" title="c13">x</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:32_6"/>
<font color="Maroon" title="c13">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="c4">p</font>,<font color="Maroon" title="c5">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="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c9">L0</font> <a href="nfcont_1.html#R1" title="NFCONT_1:pred.1">is_continuous_in</a> <font color="Maroon" title="c13">x</font></span><br/>

<a NAME="E2:32_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">P20</font></span>: 
<font color="Maroon" title="c3">f</font> <a href="nfcont_1.html#R1" title="NFCONT_1:pred.1">is_continuous_in</a> <font color="Maroon" title="c13">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:32"><span class="lab"><font color="Green" title="E27">A2</font></span></a></span>;<br/>
<a NAME="E3:32_6"/>
<font color="Maroon" title="c9">L0</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c9">L0</font></span>)</span> <a href="nfcont_1.html#R1" title="NFCONT_1:pred.1">is_continuous_in</a> <font color="Maroon" title="c13">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:32"><span class="lab"><font color="Green" title="E29">D1</font></span></a>, <a class="txt" href="#E9:32"><span class="lab"><font color="Green" title="E30">D3</font></span></a>, <a class="ref" href="nfcont_1.html#D7" title="NFCONT_1:def.7">NFCONT_1:def 7</a></span>;<br/>
<a NAME="E4:32_6"/><span class="kw">then </span>
<font color="Maroon" title="c9">L0</font> <a href="nfcont_1.html#R1" title="NFCONT_1:pred.1">is_continuous_in</a> <font color="Maroon" title="c13">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T68" title="RELAT_1:th.68">RELAT_1:68</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:32_6"/>
<font color="Maroon" title="c3">f</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c9">L0</font> <a href="nfcont_1.html#R1" title="NFCONT_1:pred.1">is_continuous_in</a> <font color="Maroon" title="c13">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:32_6"><span class="lab"><font color="Green" title="E36">P20</font></span></a>, <a class="ref" href="nfcont_1.html#T15" title="NFCONT_1:th.15">NFCONT_1:15</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="E18:32"/><span class="lab"><font color="Green" title="E36">P3</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 <font color="Maroon" title="c1">S</font>  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="c4">p</font>,<font color="Maroon" title="c5">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="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c9">L0</font> <a href="ndiff_1.html#R1" title="NDIFF_1:pred.1">is_differentiable_in</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_7"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c13">x</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c13">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="c4">p</font>,<font color="Maroon" title="c5">q</font></span><a href="ndiff_5.html#K2" title="NDIFF_5:func.2">.[</a></span> implies <font color="Maroon" title="c3">f</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c9">L0</font> <a href="ndiff_1.html#R1" title="NDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c13">x</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:32_7"/>
<font color="Maroon" title="c13">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="c4">p</font>,<font color="Maroon" title="c5">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="c3">f</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c9">L0</font> <a href="ndiff_1.html#R1" title="NDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c13">x</font></span><br/>

<a NAME="E2:32_7"/><span class="kw">then </span>
( <font color="Maroon" title="c3">f</font> <a href="ndiff_1.html#R1" title="NDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c13">x</font> &amp; <font color="Maroon" title="c9">L0</font> <a href="ndiff_1.html#R1" title="NDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c13">x</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:32"><span class="lab"><font color="Green" title="E27">A2</font></span></a>, <a class="txt" href="#E14:32"><span class="lab"><font color="Green" title="E32">A12</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:32_7"/>
<font color="Maroon" title="c3">f</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c9">L0</font> <a href="ndiff_1.html#R1" title="NDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c13">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ndiff_1.html#T36" title="NDIFF_1:th.36">NDIFF_1:36</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="E19:32"/>
 for <font color="Olive" title="b1">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">S</font>  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="c4">p</font>,<font color="Maroon" title="c5">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"><a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c9">L0</font></span>)</span>,<font color="Olive" title="b1">x</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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_8"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c13">x</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c13">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="c4">p</font>,<font color="Maroon" title="c5">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"><a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c9">L0</font></span>)</span>,<font color="Maroon" title="c13">x</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">assume </span><a NAME="E1:32_8"/><span class="lab"><font color="Green" title="E37">P40</font></span>: 
<font color="Maroon" title="c13">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="c4">p</font>,<font color="Maroon" title="c5">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"><a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c9">L0</font></span>)</span>,<font color="Maroon" title="c13">x</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="E2:32_8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">Z1</font></span>: 
<font color="Maroon" title="c3">f</font> <a href="ndiff_1.html#R1" title="NDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c13">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:32"><span class="lab"><font color="Green" title="E27">A2</font></span></a></span>;<br/>
<a NAME="E3:32_8"/>
( <font color="Maroon" title="c9">L0</font> <a href="ndiff_1.html#R1" title="NDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c13">x</font> &amp;  <a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> (<font color="Maroon" title="c9">L0</font>,<font color="Maroon" title="c13">x</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">L</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:32"><span class="lab"><font color="Green" title="E32">A12</font></span></a></span>;<br/>
<a NAME="E4:32_8"/><span class="kw">then </span>
 <a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c9">L0</font></span>)</span>,<font color="Maroon" title="c13">x</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c13">x</font>)</span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c7">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:32_8"><span class="lab"><font color="Green" title="E38">Z1</font></span></a>, <a class="ref" href="ndiff_1.html#T36" title="NDIFF_1:th.36">NDIFF_1:36</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:32_8"/>
<span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c9">L0</font></span>)</span>,<font color="Maroon" title="c13">x</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="#E2:32"><span class="lab"><font color="Green" title="E27">A2</font></span></a>, <a class="txt" href="#E1:32_8"><span class="lab"><font color="Green" title="E37">P40</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/>


</div><span class="kw">end;</span></div>
<a NAME="E20:32"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">P5</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"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c9">L0</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">q</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c9">L0</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">p</font></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="c5">q</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c4">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="ref" href="ndiff_5.html#T19" target="_self" title="NDIFF_5:th.19">FDAF10</a>, <a class="txt" href="#E1:32"><span class="lab"><font color="Green" title="E26">A1</font></span></a>, <a class="txt" href="#E15:32"><span class="lab"><font color="Green" title="E33">P1</font></span></a>, <a class="txt" href="#E17:32"><span class="lab"><font color="Green" title="E35">P2</font></span></a>, <a class="txt" href="#E18:32"><span class="lab"><font color="Green" title="E36">P3</font></span></a></span>;<br/>
<a NAME="E21:32"/>
<font color="Maroon" title="c4">p</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="c4">p</font>,<font color="Maroon" title="c5">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#T68" title="RLTOPSP1:th.68">RLTOPSP1:68</a></span>;<br/>
<a NAME="E22:32"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c9">L0</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">p</font> <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#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">L</font> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c4">p</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32"><span class="lab"><font color="Green" title="E26">A1</font></span></a>, <a class="txt" href="#E15:32"><span class="lab"><font color="Green" title="E33">P1</font></span></a>, <a class="txt" href="#E16:32"><span class="lab"><font color="Green" title="E34">F0</font></span></a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E38">H1</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c9">L0</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">p</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">LP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">S</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T15" title="RLVECT_1:th.15">RLVECT_1:15</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">LP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c4">p</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">LP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lopban_1.html#D5" title="LOPBAN_1:def.5">LOPBAN_1:def 5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">T</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</a></span>
<br/>.= 
<font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">p</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T13" title="RLVECT_1:th.13">RLVECT_1:13</a></span>
;<br/>
<a NAME="E24:32"/>
<font color="Maroon" title="c5">q</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="c4">p</font>,<font color="Maroon" title="c5">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#T68" title="RLTOPSP1:th.68">RLTOPSP1:68</a></span>;<br/>
<a NAME="E25:32"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c9">L0</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">q</font> <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#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">q</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">L</font> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c4">p</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32"><span class="lab"><font color="Green" title="E26">A1</font></span></a>, <a class="txt" href="#E15:32"><span class="lab"><font color="Green" title="E33">P1</font></span></a>, <a class="txt" href="#E16:32"><span class="lab"><font color="Green" title="E34">F0</font></span></a></span>;<br/>
<a NAME="E26:32"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">q</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">L</font> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c4">p</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">p</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c9">L0</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">q</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c9">L0</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">p</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:32"><span class="lab"><font color="Green" title="E38">H1</font></span></a>, <a class="ref" href="rlvect_1.html#T27" title="RLVECT_1:th.27">RLVECT_1:27</a></span>;<br/>
<span class="kw">hence </span><a NAME="E27:32"/>
<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> <font color="Maroon" title="c5">q</font></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> <font color="Maroon" title="c4">p</font></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="c5">q</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c4">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="c5">q</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c4">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="#E20:32"><span class="lab"><font color="Green" title="E37">P5</font></span></a>, <a class="ref" href="rlvect_1.html#T27" title="RLVECT_1:th.27">RLVECT_1:27</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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