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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>  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="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="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="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="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="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="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"><a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> (<font color="Olive" title="b1">f</font>,<font color="Olive" title="b5">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="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"><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="p3">(<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></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>  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="b4">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="b4">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="b4">x</font> ) &amp; (  for <font color="Olive" title="b4">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="b4">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="b4">x</font> ) &amp; (  for <font color="Olive" title="b4">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="b4">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"><a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> (<font color="Maroon" title="c3">f</font>,<font color="Olive" title="b4">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="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"><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="p3">(<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></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>  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"><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></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"><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="p3">(<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></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"> ( <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"><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></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"><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="p3">(<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></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><a NAME="E1:31"/><span class="lab"><font color="Green" title="E25">AS1</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> &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"><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></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"><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="p3">(<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></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">deffunc </span><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ) <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="c1">S</font> = <span class="p1">(<span class="default">$1 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</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> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">p</font>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">pt0</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</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"> such that </span><br/><a NAME="E3:31"/><span class="lab"><font color="Green" title="E26">D0</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  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  holds  <font color="Maroon" title="c7">pt0</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>
<span class="kw">set </span><font color="Maroon" title="c8">pt</font> = <font color="Maroon" title="c7">pt0</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>;<br/>
<a NAME="E4:31"/><span class="lab"><font color="Green" title="E27">D1</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c7">pt0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E5:31"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">D3</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="c7">pt0</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T62" title="RELAT_1:th.62">RELAT_1:62</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c9">t</font> be   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> implies <font color="Maroon" title="c7">pt0</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c9">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">t</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</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> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">p</font> )</span><br/><span class="kw">assume </span><a NAME="E1:31_1"/>
<font color="Maroon" title="c9">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</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="c7">pt0</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c9">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">t</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</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> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">p</font></span><br/><a NAME="E2:31_1"/><span class="lab"><font color="Green" title="E29">D11</font></span>: 
<font color="Maroon" title="c9">t</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.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="E3:31_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">pt0</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c9">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">pt0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">t</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:31"><span class="lab"><font color="Green" title="E27">D1</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>;<br/><span class="kw">hence </span><a NAME="E4:31_1"/>
<font color="Maroon" title="c7">pt0</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c9">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">t</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</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> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:31"><span class="lab"><font color="Green" title="E26">D0</font></span></a>, <a class="txt" href="#E2:31_1"><span class="lab"><font color="Green" title="E29">D11</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="E7:31"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">D2</font></span>: 
<font color="Maroon" title="c7">pt0</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> is  <a href="nfcont_3.html#V1" title="NFCONT_3:attr.1">continuous</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nfcont_3.html#T33" title="NFCONT_3:th.33">NFCONT_3:33</a></span>;<br/>
<a NAME="E8:31"/><span class="lab"><font color="Green" title="E30">D4</font></span>: 
<span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_1.html#T25" title="XXREAL_1:th.25">XXREAL_1:25</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E31">D5</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c9">t</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"> ( <font color="Maroon" title="c9">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c7">pt0</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">t</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</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> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">p</font> )</span><br/><span class="kw">assume </span><a NAME="E1:31_2"/>
<font color="Maroon" title="c9">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="rcomp_1.html#K2" title="RCOMP_1: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">(<span class="default"><font color="Maroon" title="c7">pt0</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">t</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</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> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">p</font></span><br/><span class="kw">hence </span><span class="p1">(<span class="default"><font color="Maroon" title="c7">pt0</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">t</font> = 
<font color="Maroon" title="c7">pt0</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c9">t</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:31"><span class="lab"><font color="Green" title="E28">D3</font></span></a>, <a class="txt" href="#E8:31"><span class="lab"><font color="Green" title="E30">D4</font></span></a>, <a class="ref" href="partfun2.html#T15" title="PARTFUN2:th.15">PARTFUN2:15</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">t</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</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> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">p</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:31"><span class="lab"><font color="Green" title="E26">D0</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="E10:31"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">D6</font></span>: 
( <font color="Maroon" title="c7">pt0</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="ndiff_3.html#R2" title="NDIFF_3:pred.2">is_differentiable_on</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> &amp; (  for <font color="Olive" title="b1">t</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">pt0</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span> <a href="ndiff_3.html#K2" title="NDIFF_3:func.2">`|</a> <span class="p2"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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 class="kw">by</span> <span class="lab"><a class="txt" href="#E5:31"><span class="lab"><font color="Green" title="E28">D3</font></span></a>, <a class="txt" href="#E8:31"><span class="lab"><font color="Green" title="E30">D4</font></span></a>, <a class="ref" href="ndiff_3.html#T21" title="NDIFF_3:th.21">NDIFF_3:21</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">phi</font> = <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">pt0</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span> as   <a href="struct_0.html#NM11" title="STRUCT_0:NM.11">PartFunc</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>,<font color="Maroon" title="c2">T</font> ;<br/>
<a NAME="E12:31"/><span class="lab"><font color="Green" title="E33">D7</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">pt0</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="ndiff_5.html#NK3" title="NDIFF_5:NK.3">[.</a><span class="default"><font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font></span><a href="ndiff_5.html#NK3" title="NDIFF_5:NK.3">.]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">pt0</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span> or <font color="Maroon" title="c10">y</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><br/>

<span class="kw">assume </span><a NAME="E1:31_3"/>
<font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">pt0</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c10">y</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><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:31_3"/><span class="lab"><font color="Green" title="E34">X1</font></span>: 
( <font color="Maroon" title="c11">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="c7">pt0</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span> &amp; <font color="Maroon" title="c10">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">pt0</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
<a NAME="E4:31_3"/><span class="lab"><font color="Green" title="E35">X2</font></span>: 
<font color="Maroon" title="c10">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">pt0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:31_3"><span class="lab"><font color="Green" title="E34">X1</font></span></a>, <a class="ref" href="funct_1.html#T47" title="FUNCT_1:th.47">FUNCT_1:47</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c12">x</font> = <font color="Maroon" title="c11">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:31_3"><span class="lab"><font color="Green" title="E34">X1</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c13">r</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E7:31_3"/><span class="lab"><font color="Green" title="E36">X3</font></span>: 
( <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">r</font> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">r</font> &amp; <font color="Maroon" title="c13">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:31_3"><span class="lab"><font color="Green" title="E34">X1</font></span></a>, <a class="txt" href="#E5:31"><span class="lab"><font color="Green" title="E28">D3</font></span></a></span>;<br/>
<font color="Maroon" title="c10">y</font> = 
<font color="Maroon" title="c4">p</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">x</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</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="#E3:31"><span class="lab"><font color="Green" title="E26">D0</font></span></a>, <a class="txt" href="#E4:31_3"><span class="lab"><font color="Green" title="E35">X2</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c12">x</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">x</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">q</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="ndiff_5.html#E19"><span class="lab"><font color="Green" title="E18">LmX</font></span></a></span>
;<br/>
<a NAME="E9:31_3"/><span class="kw">then </span>
<font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Olive" title="b1">r1</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">r1</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">q</font></span>)</span></span>)</span> where <font color="Olive" title="b1">r1</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">r1</font> &amp; <font color="Olive" title="b1">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 ) </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:31_3"><span class="lab"><font color="Green" title="E36">X3</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:31_3"/>
<font color="Maroon" title="c10">y</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#D2" title="RLTOPSP1:def.2">RLTOPSP1:def 2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E13:31"/><span class="kw">then </span>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">pt0</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</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">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E25">AS1</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E14:31"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">P1A</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c9">phi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:31"><span class="lab"><font color="Green" title="E28">D3</font></span></a>, <a class="ref" href="relat_1.html#T27" title="RELAT_1:th.27">RELAT_1:27</a></span>;<br/>
<a NAME="E15:31"/><span class="lab"><font color="Green" title="E35">P1B</font></span>: 
 for <font color="Olive" title="b1">t</font> being   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b1">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> holds <br/><font color="Maroon" title="c9">phi</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">t</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</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></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">t</font> be   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c10">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> implies <font color="Maroon" title="c9">phi</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">t</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</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></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:31_4"/><span class="lab"><font color="Green" title="E36">X1</font></span>: 
<font color="Maroon" title="c10">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</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="c9">phi</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">t</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</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></span>)</span></span><br/>

<span class="kw">then </span><span class="lab"><font color="Green" title="E37">X2</font></span>: <font color="Maroon" title="c9">phi</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">t</font> = 
<font color="Maroon" title="c9">phi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">t</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:31"><span class="lab"><font color="Green" title="E34">P1A</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>
<br/>.= 
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">pt0</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">t</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31_4"><span class="lab"><font color="Green" title="E36">X1</font></span></a>, <a class="txt" href="#E14:31"><span class="lab"><font color="Green" title="E34">P1A</font></span></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>
;<br/>
<a NAME="E3:31_4"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">pt0</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">pt0</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31_4"><span class="lab"><font color="Green" title="E36">X1</font></span></a>, <a class="txt" href="#E5:31"><span class="lab"><font color="Green" title="E28">D3</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
<a NAME="E4:31_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">X3</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">pt0</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">t</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="txt" href="#E12:31"><span class="lab"><font color="Green" title="E33">D7</font></span></a></span>;<br/>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">pt0</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">t</font> = 
<font color="Maroon" title="c7">pt0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">t</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31_4"><span class="lab"><font color="Green" title="E36">X1</font></span></a>, <a class="txt" href="#E5:31"><span class="lab"><font color="Green" title="E28">D3</font></span></a>, <a class="ref" href="funct_1.html#T47" title="FUNCT_1:th.47">FUNCT_1:47</a></span>
<br/>.= 
<font color="Maroon" title="c4">p</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">t</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</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="#E3:31"><span class="lab"><font color="Green" title="E26">D0</font></span></a>, <a class="txt" href="#E1:31_4"><span class="lab"><font color="Green" title="E36">X1</font></span></a></span>
;<br/>
<span class="kw">hence </span><a NAME="E6:31_4"/>
<font color="Maroon" title="c9">phi</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">t</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</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></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:31_4"><span class="lab"><font color="Green" title="E37">X2</font></span></a>, <a class="txt" href="#E4:31_4"><span class="lab"><font color="Green" title="E38">X3</font></span></a>, <a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E25">AS1</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1: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"> verum</span><br/>


</div><span class="kw">end;</span></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_5"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c10">x0</font> be   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c10">x0</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">phi</font> implies <font color="Maroon" title="c9">phi</font> <a href="nfcont_3.html#R1" title="NFCONT_3:pred.1">is_continuous_in</a> <font color="Maroon" title="c10">x0</font> )</span><br/><span class="kw">assume </span><a NAME="E1:31_5"/><span class="lab"><font color="Green" title="E36">X1</font></span>: 
<font color="Maroon" title="c10">x0</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">phi</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">phi</font> <a href="nfcont_3.html#R1" title="NFCONT_3:pred.1">is_continuous_in</a> <font color="Maroon" title="c10">x0</font></span><br/><a NAME="E2:31_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">X3</font></span>: 
<font color="Maroon" title="c7">pt0</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="nfcont_3.html#R1" title="NFCONT_3:pred.1">is_continuous_in</a> <font color="Maroon" title="c10">x0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:31"><span class="lab"><font color="Green" title="E28">D3</font></span></a>, <a class="txt" href="#E7:31"><span class="lab"><font color="Green" title="E29">D2</font></span></a>, <a class="txt" href="#E14:31"><span class="lab"><font color="Green" title="E34">P1A</font></span></a>, <a class="ref" href="nfcont_3.html#D2" title="NFCONT_3:def.2">NFCONT_3:def 2</a></span>;<br/><a NAME="E3:31_5"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">pt0</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">pt0</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:31"><span class="lab"><font color="Green" title="E28">D3</font></span></a>, <a class="txt" href="#E1:31_5"><span class="lab"><font color="Green" title="E36">X1</font></span></a>, <a class="txt" href="#E14:31"><span class="lab"><font color="Green" title="E34">P1A</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/><a NAME="E4:31_5"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">pt0</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="ndiff_5.html#NK3" title="NDIFF_5:NK.3">[.</a><span class="default"><font color="Maroon" title="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="txt" href="#E12:31"><span class="lab"><font color="Green" title="E33">D7</font></span></a></span>;<br/><a NAME="E5:31_5"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">pt0</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="ndiff_5.html#NK3" title="NDIFF_5:NK.3">[.</a><span class="default"><font color="Maroon" title="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="txt" href="#E1:31_5"><span class="lab"><font color="Green" title="E36">X1</font></span></a>, <a class="txt" href="#E14:31"><span class="lab"><font color="Green" title="E34">P1A</font></span></a>, <a class="txt" href="#E5:31"><span class="lab"><font color="Green" title="E28">D3</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>;<br/><span class="kw">hence </span><a NAME="E6:31_5"/>
<font color="Maroon" title="c9">phi</font> <a href="nfcont_3.html#R1" title="NFCONT_3:pred.1">is_continuous_in</a> <font color="Maroon" title="c10">x0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E25">AS1</font></span></a>, <a class="txt" href="#E1:31_5"><span class="lab"><font color="Green" title="E36">X1</font></span></a>, <a class="txt" href="#E2:31_5"><span class="lab"><font color="Green" title="E37">X3</font></span></a>, <a class="ref" href="nfcont_3.html#T15" title="NFCONT_3:th.15">NFCONT_3:15</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="E17:31"/><span class="kw">then </span>
<font color="Maroon" title="c9">phi</font> is  <a href="nfcont_3.html#V1" title="NFCONT_3:attr.1">continuous</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nfcont_3.html#D2" title="NFCONT_3:def.2">NFCONT_3:def 2</a></span>;<br/>
<a NAME="E18:31"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">P2</font></span>: 
<font color="Maroon" title="c9">phi</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> is  <a href="nfcont_3.html#V1" title="NFCONT_3:attr.1">continuous</a> 
 
;<br/>
<div><span class="lab"><font color="Green" title="E37">P40</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="31_6"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c10">x</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"> ( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> implies ( <font color="Maroon" title="c9">phi</font> <a href="ndiff_3.html#R1" title="NDIFF_3:pred.1">is_differentiable_in</a> <font color="Maroon" title="c10">x</font> &amp;  <a href="ndiff_3.html#K1" title="NDIFF_3:func.1">diff</a> (<font color="Maroon" title="c9">phi</font>,<font color="Maroon" title="c10">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>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">x</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</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>)</span> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <span class="p1">(<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><br/><span class="kw">assume </span><a NAME="E1:31_6"/><span class="lab"><font color="Green" title="E38">X1</font></span>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="rcomp_1.html#K2" title="RCOMP_1: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="c9">phi</font> <a href="ndiff_3.html#R1" title="NDIFF_3:pred.1">is_differentiable_in</a> <font color="Maroon" title="c10">x</font> &amp;  <a href="ndiff_3.html#K1" title="NDIFF_3:func.1">diff</a> (<font color="Maroon" title="c9">phi</font>,<font color="Maroon" title="c10">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>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">x</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</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>)</span> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <span class="p1">(<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><br/><a NAME="E2:31_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">X2</font></span>: 
<font color="Maroon" title="c7">pt0</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="ndiff_3.html#R1" title="NDIFF_3:pred.1">is_differentiable_in</a> <font color="Maroon" title="c10">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:31"><span class="lab"><font color="Green" title="E32">D6</font></span></a>, <a class="ref" href="ndiff_3.html#T10" title="NDIFF_3:th.10">NDIFF_3:10</a></span>;<br/><a NAME="E3:31_6"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">pt0</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span> <a href="ndiff_3.html#K2" title="NDIFF_3:func.2">`|</a> <span class="p2"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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 class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31_6"><span class="lab"><font color="Green" title="E38">X1</font></span></a>, <a class="txt" href="#E9:31"><span class="lab"><font color="Green" title="E31">D5</font></span></a>, <a class="txt" href="#E5:31"><span class="lab"><font color="Green" title="E28">D3</font></span></a>, <a class="txt" href="#E8:31"><span class="lab"><font color="Green" title="E30">D4</font></span></a>, <a class="ref" href="ndiff_3.html#T21" title="NDIFF_3:th.21">NDIFF_3:21</a></span>;<br/><a NAME="E4:31_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">X3</font></span>: 
 <a href="ndiff_3.html#K1" title="NDIFF_3:func.1">diff</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">pt0</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span>,<font color="Maroon" title="c10">x</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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 class="kw">by</span> <span class="lab"><a class="txt" href="#E10:31"><span class="lab"><font color="Green" title="E32">D6</font></span></a>, <a class="txt" href="#E1:31_6"><span class="lab"><font color="Green" title="E38">X1</font></span></a>, <a class="ref" href="ndiff_3.html#D6" title="NDIFF_3:def.6">NDIFF_3:def 6</a></span>;<br/><a NAME="E5:31_6"/><span class="lab"><font color="Green" title="E41">X5</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">pt0</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">pt0</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31_6"><span class="lab"><font color="Green" title="E38">X1</font></span></a>, <a class="txt" href="#E8:31"><span class="lab"><font color="Green" title="E30">D4</font></span></a>, <a class="txt" href="#E5:31"><span class="lab"><font color="Green" title="E28">D3</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>;<br/><a NAME="E6:31_6"/><span class="lab"><font color="Green" title="E42">Y3</font></span>: 
 ex <font color="Olive" title="b1">r</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">r</font> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31_6"><span class="lab"><font color="Green" title="E38">X1</font></span></a></span>;<br/><a NAME="E7:31_6"/><span class="lab"><font color="Green" title="E43">Y4</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">pt0</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">pt0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31_6"><span class="lab"><font color="Green" title="E38">X1</font></span></a>, <a class="txt" href="#E8:31"><span class="lab"><font color="Green" title="E30">D4</font></span></a>, <a class="txt" href="#E5:31"><span class="lab"><font color="Green" title="E28">D3</font></span></a>, <a class="ref" href="funct_1.html#T47" title="FUNCT_1:th.47">FUNCT_1:47</a></span>;<br/><a NAME="E8:31_6"/><span class="lab"><font color="Green" title="E44">Y5</font></span>: 
<font color="Maroon" title="c7">pt0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">x</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</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="#E3:31"><span class="lab"><font color="Green" title="E26">D0</font></span></a></span>;<br/><a NAME="E9:31_6"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">pt0</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:31_6"><span class="lab"><font color="Green" title="E42">Y3</font></span></a>, <a class="txt" href="#E7:31_6"><span class="lab"><font color="Green" title="E43">Y4</font></span></a></span>;<br/><a NAME="E10:31_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">X6</font></span>: 
<font color="Maroon" title="c3">f</font> <a href="ndiff_1.html#R1" title="NDIFF_1:pred.1">is_differentiable_in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">pt0</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:31_6"><span class="lab"><font color="Green" title="E41">X5</font></span></a>, <a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E25">AS1</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E11:31_6"/>
<font color="Maroon" title="c9">phi</font> <a href="ndiff_3.html#R1" title="NDIFF_3:pred.1">is_differentiable_in</a> <font color="Maroon" title="c10">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:31_6"><span class="lab"><font color="Green" title="E39">X2</font></span></a>, <a class="ref" href="ndiff_5.html#T6" target="_self" title="NDIFF_5:th.6">NDIFF213</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_3.html#K1" title="NDIFF_3:func.1">diff</a> (<font color="Maroon" title="c9">phi</font>,<font color="Maroon" title="c10">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>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">x</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</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>)</span> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <span class="p1">(<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><br/><span class="kw">thus </span><a NAME="E12:31_6"/>
 <a href="ndiff_3.html#K1" title="NDIFF_3:func.1">diff</a> (<font color="Maroon" title="c9">phi</font>,<font color="Maroon" title="c10">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>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">x</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</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>)</span> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <span class="p1">(<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 class="kw">by</span> <span class="lab"><a class="txt" href="#E4:31_6"><span class="lab"><font color="Green" title="E40">X3</font></span></a>, <a class="txt" href="#E5:31_6"><span class="lab"><font color="Green" title="E41">X5</font></span></a>, <a class="txt" href="#E7:31_6"><span class="lab"><font color="Green" title="E43">Y4</font></span></a>, <a class="txt" href="#E8:31_6"><span class="lab"><font color="Green" title="E44">Y5</font></span></a>, <a class="txt" href="#E10:31_6"><span class="lab"><font color="Green" title="E45">X6</font></span></a>, <a class="txt" href="#E2:31_6"><span class="lab"><font color="Green" title="E39">X2</font></span></a>, <a class="ref" href="ndiff_5.html#T6" target="_self" title="NDIFF_5:th.6">NDIFF213</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="E20:31"/><span class="kw">then </span>
( <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</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="c9">phi</font> &amp; (  for <font color="Olive" title="b1">x</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> holds <br/><font color="Maroon" title="c9">phi</font> <a href="ndiff_3.html#R1" title="NDIFF_3:pred.1">is_differentiable_in</a> <font color="Olive" title="b1">x</font> ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:31"><span class="lab"><font color="Green" title="E34">P1A</font></span></a>, <a class="ref" href="xxreal_1.html#T25" title="XXREAL_1:th.25">XXREAL_1:25</a></span>;<br/>
<a NAME="E21:31"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">P3</font></span>: 
<font color="Maroon" title="c9">phi</font> <a href="ndiff_3.html#R2" title="NDIFF_3:pred.2">is_differentiable_on</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ndiff_3.html#T10" title="NDIFF_3:th.10">NDIFF_3:10</a></span>;<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>2</sub></font>(   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ) <span class="kw">-&gt; </span>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  = <span class="p1">(<span class="default"><font color="Maroon" title="c6">M</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p2"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><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><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> $1;<br/>
<span class="kw">consider </span><font color="Maroon" title="c10">g0</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a><span class="kw"> such that </span><br/><a NAME="E23:31"/><span class="lab"><font color="Green" title="E39">Q0</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  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  holds  <font color="Maroon" title="c10">g0</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</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>
<span class="kw">set </span><font color="Maroon" title="c11">g</font> = <font color="Maroon" title="c10">g0</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>;<br/>
<a NAME="E24:31"/>
 for <font color="Olive" title="b1">t</font> being   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b1">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> holds <br/><font color="Maroon" title="c10">g0</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">t</font> <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="c6">M</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p3"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><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><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Olive" title="b1">t</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:31"><span class="lab"><font color="Green" title="E39">Q0</font></span></a></span>;<br/>
<a NAME="E25:31"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">Q9</font></span>: 
<font color="Maroon" title="c10">g0</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> is  <a href="fcont_1.html#V1" title="FCONT_1:attr.1">continuous</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="fcont_1.html#T41" title="FCONT_1:th.41">FCONT_1:41</a></span>;<br/>
<a NAME="E26:31"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c10">g0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E27:31"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">Q3</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="c10">g0</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T62" title="RELAT_1:th.62">RELAT_1:62</a></span>;<br/>
<a NAME="E28:31"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">Q2</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">g0</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> is  <a href="fcont_1.html#V1" title="FCONT_1:attr.1">continuous</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:31"><span class="lab"><font color="Green" title="E40">Q9</font></span></a>, <a class="ref" href="relat_1.html#T68" title="RELAT_1:th.68">RELAT_1:68</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E43">Q5</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="31_7"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c12">t</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"> ( <font color="Maroon" title="c12">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c10">g0</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">t</font> <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="c6">M</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p3"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><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><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c12">t</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> )</span><br/><span class="kw">assume </span><a NAME="E1:31_7"/>
<font color="Maroon" title="c12">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="rcomp_1.html#K2" title="RCOMP_1: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">(<span class="default"><font color="Maroon" title="c10">g0</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">t</font> <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="c6">M</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p3"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><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><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c12">t</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><br/><span class="kw">hence </span><span class="p1">(<span class="default"><font color="Maroon" title="c10">g0</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">t</font> = 
<font color="Maroon" title="c10">g0</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">t</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:31"><span class="lab"><font color="Green" title="E41">Q3</font></span></a>, <a class="txt" href="#E8:31"><span class="lab"><font color="Green" title="E30">D4</font></span></a>, <a class="ref" href="funct_1.html#T47" title="FUNCT_1:th.47">FUNCT_1:47</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">M</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p3"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><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><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c12">t</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:31"><span class="lab"><font color="Green" title="E39">Q0</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="E30:31"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">Q6</font></span>: 
( <font color="Maroon" title="c10">g0</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> &amp; (  for <font color="Olive" title="b1">t</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">g0</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <span class="p2"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="#E27:31"><span class="lab"><font color="Green" title="E41">Q3</font></span></a>, <a class="txt" href="#E8:31"><span class="lab"><font color="Green" title="E30">D4</font></span></a>, <a class="ref" href="fdiff_1.html#T23" title="FDIFF_1:th.23">FDIFF_1:23</a></span>;<br/>
<a NAME="E31:31"/>
 for <font color="Olive" title="b1">t</font> being   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b1">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="rcomp_1.html#K2" title="RCOMP_1: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_3.html#K1" title="NDIFF_3:func.1">diff</a> (<font color="Maroon" title="c9">phi</font>,<font color="Olive" title="b1">t</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>  <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c10">g0</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span>,<font color="Olive" title="b1">t</font>)
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_8"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">t</font> be   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c12">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="rcomp_1.html#K2" title="RCOMP_1: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_3.html#K1" title="NDIFF_3:func.1">diff</a> (<font color="Maroon" title="c9">phi</font>,<font color="Maroon" title="c12">t</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>  <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c10">g0</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span>,<font color="Maroon" title="c12">t</font>) )</span><br/>

<span class="kw">assume </span><a NAME="E1:31_8"/><span class="lab"><font color="Green" title="E45">X1</font></span>: 
<font color="Maroon" title="c12">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="rcomp_1.html#K2" title="RCOMP_1: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_3.html#K1" title="NDIFF_3:func.1">diff</a> (<font color="Maroon" title="c9">phi</font>,<font color="Maroon" title="c12">t</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>  <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c10">g0</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span>,<font color="Maroon" title="c12">t</font>)</span><br/>

<a NAME="E2:31_8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">X2</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"><a href="ndiff_3.html#K1" title="NDIFF_3:func.1">diff</a> (<font color="Maroon" title="c9">phi</font>,<font color="Maroon" title="c12">t</font>)</span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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"><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>,<span class="p4">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p5">(<span class="default"><font color="Maroon" title="c12">t</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p0">(<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>)</span> <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></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="#E19:31"><span class="lab"><font color="Green" title="E37">P40</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c13">L</font> =  <a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> (<font color="Maroon" title="c3">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">t</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</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></span>)</span>) 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/>
<a NAME="E4:31_8"/><span class="lab"><font color="Green" title="E47">X3</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"><font color="Maroon" title="c13">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</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></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"><span class="p2">(<span class="default"><a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> (<font color="Maroon" title="c3">f</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c12">t</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p5">(<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>)</span></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="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="lopban_1.html#T32" title="LOPBAN_1:th.32">LOPBAN_1:32</a></span>;<br/>
<a NAME="E5:31_8"/>
 ex <font color="Olive" title="b1">r</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Maroon" title="c12">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">r</font> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31_8"><span class="lab"><font color="Green" title="E45">X1</font></span></a></span>;<br/>
<a NAME="E6:31_8"/><span class="kw">then </span>
<font color="Maroon" title="c4">p</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">t</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</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> <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>
 
;<br/>
<a NAME="E7:31_8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">Y3</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"><a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> (<font color="Maroon" title="c3">f</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c12">t</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p5">(<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>)</span></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="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> <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="#E1:31"><span class="lab"><font color="Green" title="E25">AS1</font></span></a>, <a class="ref" href="xreal_1.html#T64" title="XREAL_1:th.64">XREAL_1:64</a></span>;<br/>
<a NAME="E8:31_8"/>
 <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c10">g0</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span>,<font color="Maroon" title="c12">t</font>) <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="c10">g0</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <span class="p2"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">t</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31_8"><span class="lab"><font color="Green" title="E45">X1</font></span></a>, <a class="txt" href="#E30:31"><span class="lab"><font color="Green" title="E44">Q6</font></span></a>, <a class="ref" href="fdiff_1.html#D7" title="FDIFF_1:def.7">FDIFF_1:def 7</a></span>;<br/>
<a NAME="E9:31_8"/><span class="kw">then </span>
 <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c10">g0</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span>,<font color="Maroon" title="c12">t</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="#E1:31_8"><span class="lab"><font color="Green" title="E45">X1</font></span></a>, <a class="txt" href="#E29:31"><span class="lab"><font color="Green" title="E43">Q5</font></span></a>, <a class="txt" href="#E27:31"><span class="lab"><font color="Green" title="E41">Q3</font></span></a>, <a class="txt" href="#E8:31"><span class="lab"><font color="Green" title="E30">D4</font></span></a>, <a class="ref" href="fdiff_1.html#T23" title="FDIFF_1:th.23">FDIFF_1:23</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:31_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_3.html#K1" title="NDIFF_3:func.1">diff</a> (<font color="Maroon" title="c9">phi</font>,<font color="Maroon" title="c12">t</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>  <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c10">g0</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span>,<font color="Maroon" title="c12">t</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:31_8"><span class="lab"><font color="Green" title="E48">Y3</font></span></a>, <a class="txt" href="#E4:31_8"><span class="lab"><font color="Green" title="E47">X3</font></span></a>, <a class="txt" href="#E2:31_8"><span class="lab"><font color="Green" title="E46">X2</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="E32:31"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">R2</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">phi</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</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">phi</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></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">(<span class="default"><font color="Maroon" title="c10">g0</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">g0</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="ndiff_5.html#E24"><span class="lab"><font color="Green" title="E23">LMFDAF10</font></span></a>, <a class="txt" href="#E14:31"><span class="lab"><font color="Green" title="E34">P1A</font></span></a>, <a class="txt" href="#E18:31"><span class="lab"><font color="Green" title="E36">P2</font></span></a>, <a class="txt" href="#E21:31"><span class="lab"><font color="Green" title="E38">P3</font></span></a>, <a class="txt" href="#E27:31"><span class="lab"><font color="Green" title="E41">Q3</font></span></a>, <a class="txt" href="#E28:31"><span class="lab"><font color="Green" title="E42">Q2</font></span></a>, <a class="txt" href="#E30:31"><span class="lab"><font color="Green" title="E44">Q6</font></span></a></span>;<br/>
<a NAME="E33:31"/><span class="lab"><font color="Green" title="E46">R31</font></span>: 
( 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> )
 
;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E47">R3</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c10">g0</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 = 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">g0</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:31"><span class="lab"><font color="Green" title="E41">Q3</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>
<br/>.= 
<font color="Maroon" title="c10">g0</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:31"><span class="lab"><font color="Green" title="E41">Q3</font></span></a>, <a class="txt" href="#E33:31"><span class="lab"><font color="Green" title="E46">R31</font></span></a>, <a class="ref" href="funct_1.html#T47" title="FUNCT_1:th.47">FUNCT_1:47</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">M</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p2"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><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><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:31"><span class="lab"><font color="Green" title="E39">Q0</font></span></a></span>
;<br/>
<span class="lab"><font color="Green" title="E48">R4</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c10">g0</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">g0</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:31"><span class="lab"><font color="Green" title="E41">Q3</font></span></a>, <a class="txt" href="#E33:31"><span class="lab"><font color="Green" title="E46">R31</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>
<br/>.= 
<font color="Maroon" title="c10">g0</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:31"><span class="lab"><font color="Green" title="E41">Q3</font></span></a>, <a class="txt" href="#E33:31"><span class="lab"><font color="Green" title="E46">R31</font></span></a>, <a class="ref" href="funct_1.html#T47" title="FUNCT_1:th.47">FUNCT_1:47</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">M</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p2"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><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><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:31"><span class="lab"><font color="Green" title="E39">Q0</font></span></a></span>
;<br/>
<span class="lab"><font color="Green" title="E49">R5</font></span>: <font color="Maroon" title="c9">phi</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 = 
<font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default">1 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</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></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:31"><span class="lab"><font color="Green" title="E35">P1B</font></span></a>, <a class="txt" href="#E33:31"><span class="lab"><font color="Green" title="E46">R31</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</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="ref" href="rlvect_1.html#D8" title="RLVECT_1:def.8">RLVECT_1:def 8</a></span>
<br/>.= 
<font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</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="ref" href="rlvect_1.html#T29" title="RLVECT_1:th.29">RLVECT_1:29</a></span>
<br/>.= 
<font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</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/>.= 
<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 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/>
<font color="Maroon" title="c9">phi</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> = 
<font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="rlvect_1.html#K3" title="RLVECT_1: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> <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></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:31"><span class="lab"><font color="Green" title="E35">P1B</font></span></a>, <a class="txt" href="#E33:31"><span class="lab"><font color="Green" title="E46">R31</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="rlvect_1.html#K3" title="RLVECT_1: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#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#T4" title="RLVECT_1:th.4">RLVECT_1:4</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E38:31"/>
<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="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="p3">(<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></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="#E32:31"><span class="lab"><font color="Green" title="E45">R2</font></span></a>, <a class="txt" href="#E34:31"><span class="lab"><font color="Green" title="E47">R3</font></span></a>, <a class="txt" href="#E35:31"><span class="lab"><font color="Green" title="E48">R4</font></span></a>, <a class="txt" href="#E36:31"><span class="lab"><font color="Green" title="E49">R5</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>
