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<span class="kw">let </span><font color="Maroon" title="c1">G</font> be   <a href="ndiff_5.html#V1" title="NDIFF_5:attr.1">non-trivial</a>  <a href="prvect_2.html#NM3" title="PRVECT_2:NM.3">RealNormSpace-Sequence</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">F</font> being   non  <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a>  <a href="normsp_1.html#NM1" title="NORMSP_1:NM.1">RealNormSpace</a><br/>  for <font color="Olive" title="b2">f</font> being   <a href="struct_0.html#NM12" title="STRUCT_0:NM.12">PartFunc</a> of <span class="p1">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">G</font></span>)</span>,<font color="Olive" title="b1">F</font><br/>  for <font color="Olive" title="b3">X</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">G</font></span>)</span>  st <font color="Olive" title="b3">X</font> is  <a href="nfcont_1.html#V3" title="NFCONT_1:attr.3">open</a>  holds <br/>( (  for <font color="Olive" title="b4">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">G</font> holds <br/>( <font color="Olive" title="b2">f</font> <a href="ndiff_5.html#R2" title="NDIFF_5:pred.2">is_partial_differentiable_on</a> <font color="Olive" title="b3">X</font>,<font color="Olive" title="b4">i</font> &amp; <font color="Olive" title="b2">f</font> <a href="ndiff_5.html#K7" title="NDIFF_5:func.7">`partial|</a> (<font color="Olive" title="b3">X</font>,<font color="Olive" title="b4">i</font>) <a href="nfcont_1.html#R3" title="NFCONT_1:pred.3">is_continuous_on</a> <font color="Olive" title="b3">X</font> ) ) iff ( <font color="Olive" title="b2">f</font> <a href="ndiff_1.html#R2" title="NDIFF_1:pred.2">is_differentiable_on</a> <font color="Olive" title="b3">X</font> &amp; <font color="Olive" title="b2">f</font> <a href="ndiff_1.html#K4" title="NDIFF_1:func.4">`|</a> <font color="Olive" title="b3">X</font> <a href="nfcont_1.html#R3" title="NFCONT_1:pred.3">is_continuous_on</a> <font color="Olive" title="b3">X</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">F</font> be   non  <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a>  <a href="normsp_1.html#NM1" title="NORMSP_1:NM.1">RealNormSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">f</font> being   <a href="struct_0.html#NM12" title="STRUCT_0:NM.12">PartFunc</a> of <span class="p1">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">G</font></span>)</span>,<font color="Maroon" title="c2">F</font><br/>  for <font color="Olive" title="b2">X</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">G</font></span>)</span>  st <font color="Olive" title="b2">X</font> is  <a href="nfcont_1.html#V3" title="NFCONT_1:attr.3">open</a>  holds <br/>( (  for <font color="Olive" title="b3">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b3">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">G</font> holds <br/>( <font color="Olive" title="b1">f</font> <a href="ndiff_5.html#R2" title="NDIFF_5:pred.2">is_partial_differentiable_on</a> <font color="Olive" title="b2">X</font>,<font color="Olive" title="b3">i</font> &amp; <font color="Olive" title="b1">f</font> <a href="ndiff_5.html#K7" title="NDIFF_5:func.7">`partial|</a> (<font color="Olive" title="b2">X</font>,<font color="Olive" title="b3">i</font>) <a href="nfcont_1.html#R3" title="NFCONT_1:pred.3">is_continuous_on</a> <font color="Olive" title="b2">X</font> ) ) iff ( <font color="Olive" title="b1">f</font> <a href="ndiff_1.html#R2" title="NDIFF_1:pred.2">is_differentiable_on</a> <font color="Olive" title="b2">X</font> &amp; <font color="Olive" title="b1">f</font> <a href="ndiff_1.html#K4" title="NDIFF_1:func.4">`|</a> <font color="Olive" title="b2">X</font> <a href="nfcont_1.html#R3" title="NFCONT_1:pred.3">is_continuous_on</a> <font color="Olive" title="b2">X</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">f</font> be   <a href="struct_0.html#NM12" title="STRUCT_0:NM.12">PartFunc</a> of <span class="p1">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">G</font></span>)</span>,<font color="Maroon" title="c2">F</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">X</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">G</font></span>)</span>  st <font color="Olive" title="b1">X</font> is  <a href="nfcont_1.html#V3" title="NFCONT_1:attr.3">open</a>  holds <br/>( (  for <font color="Olive" title="b2">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">G</font> holds <br/>( <font color="Maroon" title="c3">f</font> <a href="ndiff_5.html#R2" title="NDIFF_5:pred.2">is_partial_differentiable_on</a> <font color="Olive" title="b1">X</font>,<font color="Olive" title="b2">i</font> &amp; <font color="Maroon" title="c3">f</font> <a href="ndiff_5.html#K7" title="NDIFF_5:func.7">`partial|</a> (<font color="Olive" title="b1">X</font>,<font color="Olive" title="b2">i</font>) <a href="nfcont_1.html#R3" title="NFCONT_1:pred.3">is_continuous_on</a> <font color="Olive" title="b1">X</font> ) ) iff ( <font color="Maroon" title="c3">f</font> <a href="ndiff_1.html#R2" title="NDIFF_1:pred.2">is_differentiable_on</a> <font color="Olive" title="b1">X</font> &amp; <font color="Maroon" title="c3">f</font> <a href="ndiff_1.html#K4" title="NDIFF_1:func.4">`|</a> <font color="Olive" title="b1">X</font> <a href="nfcont_1.html#R3" title="NFCONT_1:pred.3">is_continuous_on</a> <font color="Olive" title="b1">X</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">X</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">G</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">X</font> is  <a href="nfcont_1.html#V3" title="NFCONT_1:attr.3">open</a>  implies ( (  for <font color="Olive" title="b1">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">G</font> holds <br/>( <font color="Maroon" title="c3">f</font> <a href="ndiff_5.html#R2" title="NDIFF_5:pred.2">is_partial_differentiable_on</a> <font color="Maroon" title="c4">X</font>,<font color="Olive" title="b1">i</font> &amp; <font color="Maroon" title="c3">f</font> <a href="ndiff_5.html#K7" title="NDIFF_5:func.7">`partial|</a> (<font color="Maroon" title="c4">X</font>,<font color="Olive" title="b1">i</font>) <a href="nfcont_1.html#R3" title="NFCONT_1:pred.3">is_continuous_on</a> <font color="Maroon" title="c4">X</font> ) ) iff ( <font color="Maroon" title="c3">f</font> <a href="ndiff_1.html#R2" title="NDIFF_1:pred.2">is_differentiable_on</a> <font color="Maroon" title="c4">X</font> &amp; <font color="Maroon" title="c3">f</font> <a href="ndiff_1.html#K4" title="NDIFF_1:func.4">`|</a> <font color="Maroon" title="c4">X</font> <a href="nfcont_1.html#R3" title="NFCONT_1:pred.3">is_continuous_on</a> <font color="Maroon" title="c4">X</font> ) ) )</span><br/>







<span class="kw">assume </span><a NAME="E1:77"/><span class="lab"><font color="Green" title="E63">A1</font></span>: 
<font color="Maroon" title="c4">X</font> is  <a href="nfcont_1.html#V3" title="NFCONT_1:attr.3">open</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">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">G</font> holds <br/>( <font color="Maroon" title="c3">f</font> <a href="ndiff_5.html#R2" title="NDIFF_5:pred.2">is_partial_differentiable_on</a> <font color="Maroon" title="c4">X</font>,<font color="Olive" title="b1">i</font> &amp; <font color="Maroon" title="c3">f</font> <a href="ndiff_5.html#K7" title="NDIFF_5:func.7">`partial|</a> (<font color="Maroon" title="c4">X</font>,<font color="Olive" title="b1">i</font>) <a href="nfcont_1.html#R3" title="NFCONT_1:pred.3">is_continuous_on</a> <font color="Maroon" title="c4">X</font> ) ) iff ( <font color="Maroon" title="c3">f</font> <a href="ndiff_1.html#R2" title="NDIFF_1:pred.2">is_differentiable_on</a> <font color="Maroon" title="c4">X</font> &amp; <font color="Maroon" title="c3">f</font> <a href="ndiff_1.html#K4" title="NDIFF_1:func.4">`|</a> <font color="Maroon" title="c4">X</font> <a href="nfcont_1.html#R3" title="NFCONT_1:pred.3">is_continuous_on</a> <font color="Maroon" title="c4">X</font> ) )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c5">m</font> =  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">G</font>;<br/>
<a NAME="E2:77"/><span class="lab"><font color="Green" title="E64">CG2</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">G</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">f</font> <a href="ndiff_1.html#R2" title="NDIFF_1:pred.2">is_differentiable_on</a> <font color="Maroon" title="c4">X</font> &amp; <font color="Maroon" title="c3">f</font> <a href="ndiff_1.html#K4" title="NDIFF_1:func.4">`|</a> <font color="Maroon" title="c4">X</font> <a href="nfcont_1.html#R3" title="NFCONT_1:pred.3">is_continuous_on</a> <font color="Maroon" title="c4">X</font> implies  for <font color="Olive" title="b1">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">G</font> holds <br/>( <font color="Maroon" title="c3">f</font> <a href="ndiff_5.html#R2" title="NDIFF_5:pred.2">is_partial_differentiable_on</a> <font color="Maroon" title="c4">X</font>,<font color="Olive" title="b1">i</font> &amp; <font color="Maroon" title="c3">f</font> <a href="ndiff_5.html#K7" title="NDIFF_5:func.7">`partial|</a> (<font color="Maroon" title="c4">X</font>,<font color="Olive" title="b1">i</font>) <a href="nfcont_1.html#R3" title="NFCONT_1:pred.3">is_continuous_on</a> <font color="Maroon" title="c4">X</font> ) )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:77_1"/><span class="lab"><font color="Green" title="E65">A2</font></span>: 
 for <font color="Olive" title="b1">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">G</font> holds <br/>( <font color="Maroon" title="c3">f</font> <a href="ndiff_5.html#R2" title="NDIFF_5:pred.2">is_partial_differentiable_on</a> <font color="Maroon" title="c4">X</font>,<font color="Olive" title="b1">i</font> &amp; <font color="Maroon" title="c3">f</font> <a href="ndiff_5.html#K7" title="NDIFF_5:func.7">`partial|</a> (<font color="Maroon" title="c4">X</font>,<font color="Olive" title="b1">i</font>) <a href="nfcont_1.html#R3" title="NFCONT_1:pred.3">is_continuous_on</a> <font color="Maroon" title="c4">X</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">f</font> <a href="ndiff_1.html#R2" title="NDIFF_1:pred.2">is_differentiable_on</a> <font color="Maroon" title="c4">X</font> &amp; <font color="Maroon" title="c3">f</font> <a href="ndiff_1.html#K4" title="NDIFF_1:func.4">`|</a> <font color="Maroon" title="c4">X</font> <a href="nfcont_1.html#R3" title="NFCONT_1:pred.3">is_continuous_on</a> <font color="Maroon" title="c4">X</font> )</span><br/><div><span class="lab"><font color="Green" title="E66">A3</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c6">i</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">i</font> &amp; <font color="Maroon" title="c6">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">G</font> implies ( <font color="Maroon" title="c4">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="ndiff_5.html#K7" title="NDIFF_5:func.7">`partial|</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c6">i</font>)</span>)</span> &amp; (  for <font color="Olive" title="b1">y0</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">G</font></span>)</span><br/>  for <font color="Olive" title="b2">r</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">y0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</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="b2">r</font> holds <br/> ex <font color="Olive" title="b3">s</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>(  <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="b3">s</font> &amp; (  for <font color="Olive" title="b4">y1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">G</font></span>)</span>  st <font color="Olive" title="b4">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> &amp; <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b4">y1</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Olive" title="b1">y0</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b3">s</font> holds <br/><span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="ndiff_5.html#K7" title="NDIFF_5:func.7">`partial|</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c6">i</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b4">y1</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="ndiff_5.html#K7" title="NDIFF_5:func.7">`partial|</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c6">i</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">y0</font></span>)</span></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">r</font> ) ) ) ) )</span><br/><span class="kw">assume </span><a NAME="E1:77_1_1"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">i</font> &amp; <font color="Maroon" title="c6">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">G</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="ndiff_5.html#K7" title="NDIFF_5:func.7">`partial|</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c6">i</font>)</span>)</span> &amp; (  for <font color="Olive" title="b1">y0</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">G</font></span>)</span><br/>  for <font color="Olive" title="b2">r</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">y0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</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="b2">r</font> holds <br/> ex <font color="Olive" title="b3">s</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>(  <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="b3">s</font> &amp; (  for <font color="Olive" title="b4">y1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">G</font></span>)</span>  st <font color="Olive" title="b4">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> &amp; <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b4">y1</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Olive" title="b1">y0</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b3">s</font> holds <br/><span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="ndiff_5.html#K7" title="NDIFF_5:func.7">`partial|</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c6">i</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b4">y1</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="ndiff_5.html#K7" title="NDIFF_5:func.7">`partial|</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c6">i</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">y0</font></span>)</span></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">r</font> ) ) ) )</span><br/><a NAME="E2:77_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">G</font></span>)</span>
 ;<br/><a NAME="E3:77_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">f</font> <a href="ndiff_5.html#K7" title="NDIFF_5:func.7">`partial|</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c6">i</font>) <a href="nfcont_1.html#R3" title="NFCONT_1:pred.3">is_continuous_on</a> <font color="Maroon" title="c4">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:77_1"><span class="lab"><font color="Green" title="E65">A2</font></span></a>, <a class="txt" href="#E2:77"><span class="lab"><font color="Green" title="E64">CG2</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:77_1_1"/>
( <font color="Maroon" title="c4">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="ndiff_5.html#K7" title="NDIFF_5:func.7">`partial|</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c6">i</font>)</span>)</span> &amp; (  for <font color="Olive" title="b1">y0</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">G</font></span>)</span><br/>  for <font color="Olive" title="b2">r</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">y0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</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="b2">r</font> holds <br/> ex <font color="Olive" title="b3">s</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>(  <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="b3">s</font> &amp; (  for <font color="Olive" title="b4">y1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">G</font></span>)</span>  st <font color="Olive" title="b4">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> &amp; <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b4">y1</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Olive" title="b1">y0</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b3">s</font> holds <br/><span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="ndiff_5.html#K7" title="NDIFF_5:func.7">`partial|</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c6">i</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b4">y1</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="ndiff_5.html#K7" title="NDIFF_5:func.7">`partial|</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c6">i</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">y0</font></span>)</span></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">r</font> ) ) ) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nfcont_1.html#T19" title="NFCONT_1:th.19">NFCONT_1:19</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E3:77_1"/><span class="lab"><font color="Green" title="E67">V1</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></span>;<br/><a NAME="E4:77_1"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:77"><span class="lab"><font color="Green" title="E64">CG2</font></span></a></span>;<br/><a NAME="E5:77_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">f</font> <a href="ndiff_5.html#R2" title="NDIFF_5:pred.2">is_partial_differentiable_on</a> <font color="Maroon" title="c4">X</font>, <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:77_1"><span class="lab"><font color="Green" title="E65">A2</font></span></a></span>;<br/><a NAME="E6:77_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A4</font></span>: 
<font color="Maroon" title="c4">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">f</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ndiff_5.html#D8" target="_self" title="NDIFF_5:def.8">Def19</a></span>;<br/><a NAME="E7:77_1"/>
 for <font color="Olive" title="b1">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">G</font></span>)</span>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> 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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:77"><span class="lab"><font color="Green" title="E63">A1</font></span></a>, <a class="txt" href="#E1:77_1"><span class="lab"><font color="Green" title="E65">A2</font></span></a>, <a class="ref" href="ndiff_5.html#T56" target="_self" title="NDIFF_5:th.56">Th48X</a></span>;<br/><span class="kw">hence </span><a NAME="E8:77_1"/><span class="lab"><font color="Green" title="E69">A5</font></span>: 
<font color="Maroon" title="c3">f</font> <a href="ndiff_1.html#R2" title="NDIFF_1:pred.2">is_differentiable_on</a> <font color="Maroon" title="c4">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:77"><span class="lab"><font color="Green" title="E63">A1</font></span></a>, <a class="txt" href="#E6:77_1"><span class="lab"><font color="Green" title="E68">A4</font></span></a>, <a class="ref" href="ndiff_1.html#T31" title="NDIFF_1:th.31">NDIFF_1:31</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">f</font> <a href="ndiff_1.html#K4" title="NDIFF_1:func.4">`|</a> <font color="Maroon" title="c4">X</font> <a href="nfcont_1.html#R3" title="NFCONT_1:pred.3">is_continuous_on</a> <font color="Maroon" title="c4">X</font></span><br/><a NAME="E9:77_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A6</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="ndiff_1.html#K4" title="NDIFF_1:func.4">`|</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">X</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ndiff_1.html#D9" title="NDIFF_1:def.9">NDIFF_1:def 9</a></span>;<br/><a NAME="E10:77_1"/>
 for <font color="Olive" title="b1">y0</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">G</font></span>)</span><br/>  for <font color="Olive" title="b2">r</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">y0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</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="b2">r</font> holds <br/> ex <font color="Olive" title="b3">s</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>(  <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="b3">s</font> &amp; (  for <font color="Olive" title="b4">y1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">G</font></span>)</span>  st <font color="Olive" title="b4">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> &amp; <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b4">y1</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Olive" title="b1">y0</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b3">s</font> holds <br/><span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="ndiff_1.html#K4" title="NDIFF_1:func.4">`|</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b4">y1</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="ndiff_1.html#K4" title="NDIFF_1:func.4">`|</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">y0</font></span>)</span></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">r</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">y0</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">G</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">r</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Maroon" title="c6">y0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</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> holds <br/> ex <font color="Olive" title="b2">s</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>(  <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="b2">s</font> &amp; (  for <font color="Olive" title="b3">y1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">G</font></span>)</span>  st <font color="Olive" title="b3">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> &amp; <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">y1</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c6">y0</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">s</font> holds <br/><span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="ndiff_1.html#K4" title="NDIFF_1:func.4">`|</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b3">y1</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="ndiff_1.html#K4" title="NDIFF_1:func.4">`|</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">y0</font></span>)</span></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">r</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c7">r</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="c6">y0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</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="Maroon" title="c7">r</font> implies  ex <font color="Olive" title="b1">s</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>(  <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">s</font> &amp; (  for <font color="Olive" title="b2">y1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">G</font></span>)</span>  st <font color="Olive" title="b2">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> &amp; <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">y1</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c6">y0</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">s</font> holds <br/><span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="ndiff_1.html#K4" title="NDIFF_1:func.4">`|</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">y1</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="ndiff_1.html#K4" title="NDIFF_1:func.4">`|</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">y0</font></span>)</span></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">r</font> ) ) )</span><br/>



<span class="kw">assume </span><a NAME="E1:77_1_2"/><span class="lab"><font color="Green" title="E71">A7</font></span>: 
( <font color="Maroon" title="c6">y0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</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="Maroon" title="c7">r</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">s</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>(  <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">s</font> &amp; (  for <font color="Olive" title="b2">y1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">G</font></span>)</span>  st <font color="Olive" title="b2">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> &amp; <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">y1</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c6">y0</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">s</font> holds <br/><span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="ndiff_1.html#K4" title="NDIFF_1:func.4">`|</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">y1</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="ndiff_1.html#K4" title="NDIFF_1:func.4">`|</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">y0</font></span>)</span></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">r</font> ) )</span><br/>

<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>,  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>] <span class="kw">means </span> for <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 holds <br/>(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> $2 &amp; (  for <font color="Olive" title="b2">y1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">G</font></span>)</span>  st <font color="Olive" title="b2">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> &amp; <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">y1</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c6">y0</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> $2 holds <br/><span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="ndiff_5.html#K7" title="NDIFF_5:func.7">`partial|</a> (<font color="Maroon" title="c4">X</font>,<font color="Olive" title="b1">i</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">y1</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="ndiff_5.html#K7" title="NDIFF_5:func.7">`partial|</a> (<font color="Maroon" title="c4">X</font>,<font color="Olive" title="b1">i</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">y0</font></span>)</span></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">r</font> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span> ) );<br/>
<div><span class="lab"><font color="Green" title="E72">A8</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c8">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">G</font></span>)</span> implies  ex <font color="Olive" title="b1">s</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>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">i</font>,<font color="Olive" title="b1">s</font>] )</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c9">j</font> = <font color="Maroon" title="c8">i</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/><span class="kw">assume </span><a NAME="E2:77_1_2_1"/>
<font color="Maroon" title="c8">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">G</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">s</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>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">i</font>,<font color="Olive" title="b1">s</font>]</span><br/><a NAME="E3:77_1_2_1"/><span class="kw">then </span>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">j</font> &amp; <font color="Maroon" title="c9">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">G</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">s</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E5:77_1_2_1"/><span class="lab"><font color="Green" title="E73">A9</font></span>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">s</font> &amp; (  for <font color="Olive" title="b1">y1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">G</font></span>)</span>  st <font color="Olive" title="b1">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> &amp; <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="b1">y1</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c6">y0</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">s</font> holds <br/><span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="ndiff_5.html#K7" title="NDIFF_5:func.7">`partial|</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c9">j</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">y1</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="ndiff_5.html#K7" title="NDIFF_5:func.7">`partial|</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c9">j</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">y0</font></span>)</span></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">r</font> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:77_1_2"><span class="lab"><font color="Green" title="E71">A7</font></span></a>, <a class="txt" href="#E2:77_1"><span class="lab"><font color="Green" title="E66">A3</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c11">s</font> = <font color="Maroon" title="c10">s</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>  ;<br/><span class="kw">take </span><font color="Maroon" title="c12">s</font> = <font color="Maroon" title="c11">s</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c12">s</font>]</span><br/><span class="kw">thus </span><a NAME="E7:77_1_2_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c12">s</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:77_1_2_1"><span class="lab"><font color="Green" title="E73">A9</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>
<span class="kw">consider </span><font color="Maroon" title="c8">S</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <span class="kw"> such that </span><br/><a NAME="E4:77_1_2"/><span class="lab"><font color="Green" title="E73">A10</font></span>: 
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">G</font></span>)</span> &amp; (  for <font color="Olive" title="b1">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">G</font></span>)</span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>,<font color="Maroon" title="c8">S</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font>] ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="finseq_1.html#S5" title="FINSEQ_1:sch.5">FINSEQ_1:sch 5</a>(<a class="txt" href="#E2:77_1_2"><span class="lab"><font color="Green" title="E72">A8</font></span></a>);<br/></span>
<span class="kw">take </span><font color="Maroon" title="c9">s</font> =  <a href="rfinseq2.html#K4" title="RFINSEQ2:func.4">min</a> <font color="Maroon" title="c8">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">s</font> &amp; (  for <font color="Olive" title="b1">y1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">G</font></span>)</span>  st <font color="Olive" title="b1">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> &amp; <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="b1">y1</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c6">y0</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">s</font> holds <br/><span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="ndiff_1.html#K4" title="NDIFF_1:func.4">`|</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">y1</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="ndiff_1.html#K4" title="NDIFF_1:func.4">`|</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">y0</font></span>)</span></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">r</font> ) )</span><br/>

<a NAME="E5:77_1_2"/><span class="lab"><font color="Green" title="E74">A11</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:77_1_2"><span class="lab"><font color="Green" title="E73">A10</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E6:77_1_2"/><span class="kw">then </span>
 <a href="rfinseq2.html#K2" title="RFINSEQ2:func.2">min_p</a> <font color="Maroon" title="c8">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rfinseq2.html#D2" title="RFINSEQ2:def.2">RFINSEQ2:def 2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:77_1_2"/>
<font color="Maroon" title="c9">s</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:77_1_2"><span class="lab"><font color="Green" title="E73">A10</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">  for <font color="Olive" title="b1">y1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">G</font></span>)</span>  st <font color="Olive" title="b1">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> &amp; <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="b1">y1</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c6">y0</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">s</font> holds <br/><span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="ndiff_1.html#K4" title="NDIFF_1:func.4">`|</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">y1</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="ndiff_1.html#K4" title="NDIFF_1:func.4">`|</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">y0</font></span>)</span></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">r</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c10">y1</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">G</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c10">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> &amp; <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">y1</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c6">y0</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">s</font> implies <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="ndiff_1.html#K4" title="NDIFF_1:func.4">`|</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">y1</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="ndiff_1.html#K4" title="NDIFF_1:func.4">`|</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">y0</font></span>)</span></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">r</font> )</span><br/>

<span class="kw">assume </span><a NAME="E8:77_1_2"/><span class="lab"><font color="Green" title="E75">A12</font></span>: 
( <font color="Maroon" title="c10">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> &amp; <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">y1</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c6">y0</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">s</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="ndiff_1.html#K4" title="NDIFF_1:func.4">`|</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">y1</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="ndiff_1.html#K4" title="NDIFF_1:func.4">`|</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">y0</font></span>)</span></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">r</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c11">DD</font> = <span class="p1">(<span class="default"><a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c10">y1</font>)</span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c6">y0</font>)</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 <span class="p1">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">G</font></span>)</span>,<font color="Maroon" title="c2">F</font> <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="E10:77_1_2"/><span class="lab"><font color="Green" title="E76">A13</font></span>: 
 <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><a href="lopban_1.html#K14" title="LOPBAN_1:func.14">PreNorms</a> <font color="Maroon" title="c11">DD</font></span>)</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>,<font color="Maroon" title="c10">y1</font>)</span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p3">(<span class="default"><a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c6">y0</font>)</span>)</span></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lopban_1.html#T30" title="LOPBAN_1:th.30">LOPBAN_1:30</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1_2_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c12">mt</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">mt</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lopban_1.html#K14" title="LOPBAN_1:func.14">PreNorms</a> <font color="Maroon" title="c11">DD</font> implies <font color="Maroon" title="c12">mt</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">r</font> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 2 )</span><br/><span class="kw">assume </span><a NAME="E1:77_1_2_2"/>
<font color="Maroon" title="c12">mt</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lopban_1.html#K14" title="LOPBAN_1:func.14">PreNorms</a> <font color="Maroon" title="c11">DD</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="c12">mt</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">r</font> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 2</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">t</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <span class="p1">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">G</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E3:77_1_2_2"/><span class="lab"><font color="Green" title="E77">A14</font></span>: 
( <font color="Maroon" title="c12">mt</font> <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"><font color="Maroon" title="c11">DD</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">t</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> &amp; <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c13">t</font></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> 1 )
 ;<br/><span class="kw">consider </span><font color="Maroon" title="c14">w0</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c2">F</font><span class="kw"> such that </span><br/><a NAME="E5:77_1_2_2"/><span class="lab"><font color="Green" title="E78">A15</font></span>: 
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c14">w0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">G</font> &amp; (  for <font color="Olive" title="b1">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">G</font> holds <br/><font color="Maroon" title="c14">w0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ndiff_5.html#K6" title="NDIFF_5:func.6">partdiff</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c6">y0</font>,<font color="Olive" title="b1">i</font>)</span>)</span> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ndiff_5.html#K3" title="NDIFF_5:func.3">proj</a> <span class="p3">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Olive" title="b1">i</font>)</span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">t</font></span>)</span> ) &amp; <span class="p1">(<span class="default"><a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c6">y0</font>)</span>)</span> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Maroon" title="c13">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Maroon" title="c14">w0</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:77"><span class="lab"><font color="Green" title="E63">A1</font></span></a>, <a class="txt" href="#E1:77_1"><span class="lab"><font color="Green" title="E65">A2</font></span></a>, <a class="ref" href="ndiff_5.html#T56" target="_self" title="NDIFF_5:th.56">Th48X</a>, <a class="txt" href="#E1:77_1_2"><span class="lab"><font color="Green" title="E71">A7</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c15">Sw0</font> =  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Maroon" title="c14">w0</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">F</font> ;<br/><span class="kw">consider </span><font color="Maroon" title="c16">w1</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c2">F</font><span class="kw"> such that </span><br/><a NAME="E8:77_1_2_2"/><span class="lab"><font color="Green" title="E79">A16</font></span>: 
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">G</font> &amp; (  for <font color="Olive" title="b1">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">G</font> holds <br/><font color="Maroon" title="c16">w1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ndiff_5.html#K6" title="NDIFF_5:func.6">partdiff</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c10">y1</font>,<font color="Olive" title="b1">i</font>)</span>)</span> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ndiff_5.html#K3" title="NDIFF_5:func.3">proj</a> <span class="p3">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Olive" title="b1">i</font>)</span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">t</font></span>)</span> ) &amp; <span class="p1">(<span class="default"><a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c10">y1</font>)</span>)</span> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Maroon" title="c13">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Maroon" title="c16">w1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:77"><span class="lab"><font color="Green" title="E63">A1</font></span></a>, <a class="txt" href="#E1:77_1"><span class="lab"><font color="Green" title="E65">A2</font></span></a>, <a class="ref" href="ndiff_5.html#T56" target="_self" title="NDIFF_5:th.56">Th48X</a>, <a class="txt" href="#E8:77_1_2"><span class="lab"><font color="Green" title="E75">A12</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c17">Sw1</font> =  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Maroon" title="c16">w1</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">F</font> ;<br/><span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</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="c2">F</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c16">w1</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="p1">(<span class="default"><font color="Maroon" title="c14">w0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> $1</span>)</span>;<br/><span class="kw">consider </span><font color="Maroon" title="c18">w2</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c2">F</font><span class="kw"> such that </span><br/><a NAME="E11:77_1_2_2"/><span class="lab"><font color="Green" title="E80">A17</font></span>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">w2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">G</font> &amp; (  for <font color="Olive" title="b1">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c18">w2</font> holds <br/><font color="Maroon" title="c18">w2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">i</font>) ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="finseq_2.html#S1" title="FINSEQ_2:sch.1">FINSEQ_2:sch 1</a>();<br/></span><a NAME="E12:77_1_2_2"/><span class="lab"><font color="Green" title="E81">A18</font></span>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">G</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">w0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">G</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:77"><span class="lab"><font color="Green" title="E64">CG2</font></span></a>, <a class="txt" href="#E5:77_1_2_2"><span class="lab"><font color="Green" title="E78">A15</font></span></a>, <a class="txt" href="#E8:77_1_2_2"><span class="lab"><font color="Green" title="E79">A16</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1_2_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c19">i</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</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="c19">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">w1</font> implies <font color="Maroon" title="c18">w2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c19">i</font>) )</span><br/><span class="kw">assume </span><a NAME="E1:77_1_2_2_1"/>
<font color="Maroon" title="c19">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">w1</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="c18">w2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c19">i</font>)</span><br/><a NAME="E2:77_1_2_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c19">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c18">w2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:77_1_2_2"><span class="lab"><font color="Green" title="E80">A17</font></span></a>, <a class="txt" href="#E2:77"><span class="lab"><font color="Green" title="E64">CG2</font></span></a>, <a class="txt" href="#E8:77_1_2_2"><span class="lab"><font color="Green" title="E79">A16</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><span class="kw">hence </span><a NAME="E3:77_1_2_2_1"/>
<font color="Maroon" title="c18">w2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c19">i</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:77_1_2_2"><span class="lab"><font color="Green" title="E80">A17</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="E14:77_1_2_2"/><span class="kw">then </span>
 <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Maroon" title="c18">w2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Maroon" title="c16">w1</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Maroon" title="c14">w0</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:77_1_2_2"><span class="lab"><font color="Green" title="E80">A17</font></span></a>, <a class="txt" href="#E12:77_1_2_2"><span class="lab"><font color="Green" title="E81">A18</font></span></a>, <a class="ref" href="rlvect_2.html#T5" title="RLVECT_2:th.5">RLVECT_2:5</a></span>;<br/><a NAME="E15:77_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E82">A21</font></span>: 
<font color="Maroon" title="c12">mt</font> <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"><a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Maroon" title="c18">w2</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="#E3:77_1_2_2"><span class="lab"><font color="Green" title="E77">A14</font></span></a>, <a class="txt" href="#E8:77_1_2_2"><span class="lab"><font color="Green" title="E79">A16</font></span></a>, <a class="txt" href="#E5:77_1_2_2"><span class="lab"><font color="Green" title="E78">A15</font></span></a>, <a class="ref" href="lopban_1.html#T40" title="LOPBAN_1:th.40">LOPBAN_1:40</a></span>;<br/><span class="kw">deffunc </span><font color="Maroon">H<sub>2</sub></font>(  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</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"><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="c18">w2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> $1</span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c19">ys</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <span class="kw"> such that </span><br/><a NAME="E17:77_1_2_2"/><span class="lab"><font color="Green" title="E83">A22</font></span>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c19">ys</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">G</font> &amp; (  for <font color="Olive" title="b1">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c19">ys</font> holds <br/><font color="Maroon" title="c19">ys</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">j</font>) ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="finseq_2.html#S1" title="FINSEQ_2:sch.1">FINSEQ_2:sch 1</a>();<br/></span><div><span class="lab"><font color="Green" title="E84">A23</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1_2_2_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c20">i</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</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="c20">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c18">w2</font> implies <font color="Maroon" title="c19">ys</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">i</font> <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"><font color="Maroon" title="c18">w2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c20">i</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="E1:77_1_2_2_2"/>
<font color="Maroon" title="c20">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c18">w2</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="c19">ys</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">i</font> <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"><font color="Maroon" title="c18">w2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c20">i</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span></span><br/><a NAME="E2:77_1_2_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c20">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">G</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:77_1_2_2"><span class="lab"><font color="Green" title="E80">A17</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E3:77_1_2_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c20">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c19">ys</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:77_1_2_2"><span class="lab"><font color="Green" title="E83">A22</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><span class="kw">hence </span><a NAME="E4:77_1_2_2_2"/>
<font color="Maroon" title="c19">ys</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">i</font> <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"><font color="Maroon" title="c18">w2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c20">i</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="#E17:77_1_2_2"><span class="lab"><font color="Green" title="E83">A22</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="E19:77_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E85">A25</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="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Maroon" title="c18">w2</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="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c19">ys</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:77_1_2_2"><span class="lab"><font color="Green" title="E80">A17</font></span></a>, <a class="txt" href="#E17:77_1_2_2"><span class="lab"><font color="Green" title="E83">A22</font></span></a>, <a class="ref" href="ndiff_5.html#T7" target="_self" title="NDIFF_5:th.7">PDIFF617</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c20">rm</font> = <font color="Maroon" title="c7">r</font> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span> 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>  ;<br/><span class="kw">deffunc </span><font color="Maroon">H<sub>3</sub></font>(  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</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>  = <font color="Maroon" title="c20">rm</font>;<br/><span class="kw">consider </span><font color="Maroon" title="c21">rs</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <span class="kw"> such that </span><br/><a NAME="E22:77_1_2_2"/><span class="lab"><font color="Green" title="E86">A26</font></span>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">rs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">G</font> &amp; (  for <font color="Olive" title="b1">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">rs</font> holds <br/><font color="Maroon" title="c21">rs</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive" title="b1">j</font>) ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="finseq_2.html#S1" title="FINSEQ_2:sch.1">FINSEQ_2:sch 1</a>();<br/></span><a NAME="E23:77_1_2_2"/><span class="lab"><font color="Green" title="E87">A27</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">rs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">G</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:77_1_2_2"><span class="lab"><font color="Green" title="E86">A26</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1_2_2_3"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c22">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c22">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c21">rs</font> implies <font color="Maroon" title="c22">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><font color="Maroon" title="c20">rm</font></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span> )</span><br/><span class="kw">assume </span><a NAME="E1:77_1_2_2_3"/>
<font color="Maroon" title="c22">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c21">rs</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="c22">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><font color="Maroon" title="c20">rm</font></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c23">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:77_1_2_2_3"/><span class="lab"><font color="Green" title="E88">A28</font></span>: 
( <font color="Maroon" title="c23">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">rs</font> &amp; <font color="Maroon" title="c22">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">rs</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c23">b</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/><span class="kw">reconsider </span><font color="Maroon" title="c24">b</font> = <font color="Maroon" title="c23">b</font> as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:77_1_2_2_3"><span class="lab"><font color="Green" title="E88">A28</font></span></a></span>;<br/><a NAME="E5:77_1_2_2_3"/>
<font color="Maroon" title="c21">rs</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c24">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">rm</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:77_1_2_2_3"><span class="lab"><font color="Green" title="E88">A28</font></span></a>, <a class="txt" href="#E22:77_1_2_2"><span class="lab"><font color="Green" title="E86">A26</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E6:77_1_2_2_3"/>
<font color="Maroon" title="c22">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><font color="Maroon" title="c20">rm</font></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:77_1_2_2_3"><span class="lab"><font color="Green" title="E88">A28</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</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="E25:77_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E88">A270</font></span>: 
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c21">rs</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><font color="Maroon" title="c20">rm</font></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1_2_2_4"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c22">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c22">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><font color="Maroon" title="c20">rm</font></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span> implies <font color="Maroon" title="c22">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c21">rs</font> )</span><br/><span class="kw">assume </span><a NAME="E1:77_1_2_2_4"/>
<font color="Maroon" title="c22">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><font color="Maroon" title="c20">rm</font></span><a href="seq_4.html#K1" title="SEQ_4: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="c22">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c21">rs</font></span><br/><a NAME="E2:77_1_2_2_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E89">A29</font></span>: 
<font color="Maroon" title="c22">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">rm</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E3:77_1_2_2_4"/><span class="lab"><font color="Green" title="E90">A30</font></span>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">rs</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:77_1"><span class="lab"><font color="Green" title="E67">V1</font></span></a>, <a class="txt" href="#E23:77_1_2_2"><span class="lab"><font color="Green" title="E87">A27</font></span></a></span>;<br/><a NAME="E4:77_1_2_2_4"/><span class="kw">then </span>
<font color="Maroon" title="c22">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">rs</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="#E2:77_1_2_2_4"><span class="lab"><font color="Green" title="E89">A29</font></span></a>, <a class="txt" href="#E22:77_1_2_2"><span class="lab"><font color="Green" title="E86">A26</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E5:77_1_2_2_4"/>
<font color="Maroon" title="c22">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c21">rs</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:77_1_2_2_4"><span class="lab"><font color="Green" title="E90">A30</font></span></a>, <a class="ref" href="funct_1.html#T3" title="FUNCT_1:th.3">FUNCT_1:3</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="E27:77_1_2_2"/><span class="kw">then </span>
<span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><font color="Maroon" title="c20">rm</font></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c21">rs</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/><a NAME="E28:77_1_2_2"/><span class="kw">then </span>
<span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><font color="Maroon" title="c20">rm</font></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c21">rs</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:77_1_2_2"><span class="lab"><font color="Green" title="E88">A270</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/><a NAME="E29:77_1_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c21">rs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="prvect_1.html#K4" title="PRVECT_1:func.4">|-&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p2">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:77_1_2_2"><span class="lab"><font color="Green" title="E87">A27</font></span></a>, <a class="ref" href="funcop_1.html#T9" title="FUNCOP_1:th.9">FUNCOP_1:9</a></span>;<br/><span class="kw">then </span> <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c21">rs</font> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p2">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T80" title="RVSUM_1:th.80">RVSUM_1:80</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 2</span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T78" title="XCMPLX_1:th.78">XCMPLX_1:78</a></span>
;<br/><a NAME="E31:77_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E89">A31</font></span>: 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c21">rs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">r</font> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T87" title="XCMPLX_1:th.87">XCMPLX_1:87</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1_2_2_6"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c22">i</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</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="c22">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c19">ys</font> implies <font color="Maroon" title="c19">ys</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c22">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c21">rs</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c22">i</font> )</span><br/><span class="kw">assume </span><a NAME="E1:77_1_2_2_6"/>
<font color="Maroon" title="c22">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c19">ys</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="c19">ys</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c22">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c21">rs</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c22">i</font></span><br/><a NAME="E2:77_1_2_2_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E90">A32</font></span>: 
<font color="Maroon" title="c22">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">G</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:77_1_2_2"><span class="lab"><font color="Green" title="E83">A22</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E3:77_1_2_2_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E91">A33</font></span>: 
( <font color="Maroon" title="c22">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c18">w2</font> &amp; <font color="Maroon" title="c22">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">w1</font> &amp; <font color="Maroon" title="c22">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c14">w0</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:77_1_2_2"><span class="lab"><font color="Green" title="E78">A15</font></span></a>, <a class="txt" href="#E8:77_1_2_2"><span class="lab"><font color="Green" title="E79">A16</font></span></a>, <a class="txt" href="#E11:77_1_2_2"><span class="lab"><font color="Green" title="E80">A17</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E4:77_1_2_2_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E92">A34</font></span>: 
( <font color="Maroon" title="c19">ys</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c22">i</font> <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"><font color="Maroon" title="c18">w2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">i</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> &amp; <font color="Maroon" title="c18">w2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">w2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:77_1_2_2"><span class="lab"><font color="Green" title="E84">A23</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>;<br/><a NAME="E5:77_1_2_2_6"/><span class="lab"><font color="Green" title="E93">A35</font></span>: 
<font color="Maroon" title="c22">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">rs</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:77_1_2_2"><span class="lab"><font color="Green" title="E86">A26</font></span></a>, <a class="txt" href="#E2:77_1_2_2_6"><span class="lab"><font color="Green" title="E90">A32</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c23">p1</font> =  <a href="ndiff_5.html#K6" title="NDIFF_5:func.6">partdiff</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c10">y1</font>,<font color="Maroon" title="c22">i</font>), <font color="Maroon" title="c24">p0</font> =  <a href="ndiff_5.html#K6" title="NDIFF_5:func.6">partdiff</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c6">y0</font>,<font color="Maroon" title="c22">i</font>) as   <a href="lopban_1.html#V2" title="LOPBAN_1:attr.2">bounded</a>  <a href="lopban_1.html#NM1" title="LOPBAN_1:NM.1">LinearOperator</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="prvect_2.html#K11" title="PRVECT_2:func.11">.</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c22">i</font>)</span>)</span></span>)</span>,<font color="Maroon" title="c2">F</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="lopban_1.html#D9" title="LOPBAN_1:def.9">LOPBAN_1:def 9</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c25">P1</font> = <font color="Maroon" title="c23">p1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ndiff_5.html#K3" title="NDIFF_5:func.3">proj</a> <span class="p3">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c22">i</font>)</span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">t</font></span>)</span> as   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c2">F</font> ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c26">P0</font> = <font color="Maroon" title="c24">p0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ndiff_5.html#K3" title="NDIFF_5:func.3">proj</a> <span class="p3">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c22">i</font>)</span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">t</font></span>)</span> as   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c2">F</font> ;<br/><a NAME="E9:77_1_2_2_6"/>
( <font color="Maroon" title="c14">w0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">w0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font> &amp; <font color="Maroon" title="c16">w1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">w1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:77_1_2_2_6"><span class="lab"><font color="Green" title="E91">A33</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>;<br/><a NAME="E10:77_1_2_2_6"/><span class="kw">then </span>
( <font color="Maroon" title="c14">w0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">P0</font> &amp; <font color="Maroon" title="c16">w1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">P1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:77"><span class="lab"><font color="Green" title="E64">CG2</font></span></a>, <a class="txt" href="#E5:77_1_2_2"><span class="lab"><font color="Green" title="E78">A15</font></span></a>, <a class="txt" href="#E8:77_1_2_2"><span class="lab"><font color="Green" title="E79">A16</font></span></a>, <a class="txt" href="#E2:77_1_2_2_6"><span class="lab"><font color="Green" title="E90">A32</font></span></a></span>;<br/><a NAME="E11:77_1_2_2_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E94">A40</font></span>: 
<font color="Maroon" title="c18">w2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">P1</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c26">P0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:77_1_2_2_6"><span class="lab"><font color="Green" title="E91">A33</font></span></a>, <a class="txt" href="#E11:77_1_2_2"><span class="lab"><font color="Green" title="E80">A17</font></span></a></span>;<br/><a NAME="E12:77_1_2_2_6"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c22">i</font> &amp; <font color="Maroon" title="c22">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">S</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:77_1_2"><span class="lab"><font color="Green" title="E74">A11</font></span></a>, <a class="txt" href="#E2:77_1_2_2_6"><span class="lab"><font color="Green" title="E90">A32</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E13:77_1_2_2_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E95">A41</font></span>: 
( <font color="Maroon" title="c9">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">S</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c22">i</font> &amp; <font color="Maroon" title="c3">f</font> <a href="ndiff_5.html#R2" title="NDIFF_5:pred.2">is_partial_differentiable_on</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c22">i</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:77"><span class="lab"><font color="Green" title="E64">CG2</font></span></a>, <a class="txt" href="#E2:77_1_2_2_6"><span class="lab"><font color="Green" title="E90">A32</font></span></a>, <a class="txt" href="#E1:77_1"><span class="lab"><font color="Green" title="E65">A2</font></span></a>, <a class="ref" href="rfinseq2.html#T2" title="RFINSEQ2:th.2">RFINSEQ2:2</a></span>;<br/><a NAME="E14:77_1_2_2_6"/><span class="kw">then </span>
<span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">y1</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c6">y0</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">S</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c22">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:77_1_2"><span class="lab"><font color="Green" title="E75">A12</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E15:77_1_2_2_6"/><span class="kw">then </span>
<span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="ndiff_5.html#K7" title="NDIFF_5:func.7">`partial|</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c22">i</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">y1</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="ndiff_5.html#K7" title="NDIFF_5:func.7">`partial|</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c22">i</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">y0</font></span>)</span></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">r</font> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:77_1_2"><span class="lab"><font color="Green" title="E73">A10</font></span></a>, <a class="txt" href="#E2:77_1_2_2_6"><span class="lab"><font color="Green" title="E90">A32</font></span></a>, <a class="txt" href="#E8:77_1_2"><span class="lab"><font color="Green" title="E75">A12</font></span></a></span>;<br/><a NAME="E16:77_1_2_2_6"/><span class="kw">then </span>
<span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ndiff_5.html#K6" title="NDIFF_5:func.6">partdiff</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c10">y1</font>,<font color="Maroon" title="c22">i</font>)</span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="ndiff_5.html#K7" title="NDIFF_5:func.7">`partial|</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c22">i</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">y0</font></span>)</span></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">r</font> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ndiff_5.html#D9" target="_self" title="NDIFF_5:def.9">Def19f</a>, <a class="txt" href="#E8:77_1_2"><span class="lab"><font color="Green" title="E75">A12</font></span></a>, <a class="txt" href="#E13:77_1_2_2_6"><span class="lab"><font color="Green" title="E95">A41</font></span></a>, <a class="txt" href="#E2:77"><span class="lab"><font color="Green" title="E64">CG2</font></span></a>, <a class="txt" href="#E2:77_1_2_2_6"><span class="lab"><font color="Green" title="E90">A32</font></span></a></span>;<br/><a NAME="E17:77_1_2_2_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E96">A42</font></span>: 
<span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ndiff_5.html#K6" title="NDIFF_5:func.6">partdiff</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c10">y1</font>,<font color="Maroon" title="c22">i</font>)</span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p3">(<span class="default"><a href="ndiff_5.html#K6" title="NDIFF_5:func.6">partdiff</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c6">y0</font>,<font color="Maroon" title="c22">i</font>)</span>)</span></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">r</font> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ndiff_5.html#D9" target="_self" title="NDIFF_5:def.9">Def19f</a>, <a class="txt" href="#E1:77_1_2"><span class="lab"><font color="Green" title="E71">A7</font></span></a>, <a class="txt" href="#E13:77_1_2_2_6"><span class="lab"><font color="Green" title="E95">A41</font></span></a>, <a class="txt" href="#E2:77"><span class="lab"><font color="Green" title="E64">CG2</font></span></a>, <a class="txt" href="#E2:77_1_2_2_6"><span class="lab"><font color="Green" title="E90">A32</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c27">PP</font> = <span class="p1">(<span class="default"><a href="ndiff_5.html#K6" title="NDIFF_5:func.6">partdiff</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c10">y1</font>,<font color="Maroon" title="c22">i</font>)</span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><a href="ndiff_5.html#K6" title="NDIFF_5:func.6">partdiff</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c6">y0</font>,<font color="Maroon" title="c22">i</font>)</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 <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="prvect_2.html#K11" title="PRVECT_2:func.11">.</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c22">i</font>)</span>)</span></span>)</span>,<font color="Maroon" title="c2">F</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="E19:77_1_2_2_6"/><span class="lab"><font color="Green" title="E97">A43</font></span>: 
 <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><a href="lopban_1.html#K14" title="LOPBAN_1:func.14">PreNorms</a> <font color="Maroon" title="c27">PP</font></span>)</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_5.html#K6" title="NDIFF_5:func.6">partdiff</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c10">y1</font>,<font color="Maroon" title="c22">i</font>)</span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p3">(<span class="default"><a href="ndiff_5.html#K6" title="NDIFF_5:func.6">partdiff</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c6">y0</font>,<font color="Maroon" title="c22">i</font>)</span>)</span></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lopban_1.html#T30" title="LOPBAN_1:th.30">LOPBAN_1:30</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c28">pt</font> = <span class="p1">(<span class="default"><a href="ndiff_5.html#K3" title="NDIFF_5:func.3">proj</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c22">i</font>)</span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">t</font> as   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="prvect_2.html#K11" title="PRVECT_2:func.11">.</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c22">i</font>)</span>)</span></span>)</span> ;<br/><a NAME="E21:77_1_2_2_6"/><span class="lab"><font color="Green" title="E98">A44</font></span>: 
<font color="Maroon" title="c27">PP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c28">pt</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">P1</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c26">P0</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lopban_1.html#T40" title="LOPBAN_1:th.40">LOPBAN_1:40</a></span>;<br/><a NAME="E22:77_1_2_2_6"/>
<span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c28">pt</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c13">t</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ndiff_5.html#T31" target="_self" title="NDIFF_5:th.31">XTh50</a></span>;<br/><a NAME="E23:77_1_2_2_6"/><span class="kw">then </span>
<span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c28">pt</font></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> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:77_1_2_2"><span class="lab"><font color="Green" title="E77">A14</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E24:77_1_2_2_6"/><span class="kw">then </span>
( <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c27">PP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c28">pt</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lopban_1.html#K14" title="LOPBAN_1:func.14">PreNorms</a> <font color="Maroon" title="c27">PP</font> &amp;  not  <a href="lopban_1.html#K14" title="LOPBAN_1:func.14">PreNorms</a> <font color="Maroon" title="c27">PP</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp;  <a href="lopban_1.html#K14" title="LOPBAN_1:func.14">PreNorms</a> <font color="Maroon" title="c27">PP</font> is  <a href="xxreal_2.html#V4" title="XXREAL_2:attr.4">bounded_above</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lopban_1.html#T27" title="LOPBAN_1:th.27">LOPBAN_1:27</a></span>;<br/><a NAME="E25:77_1_2_2_6"/><span class="kw">then </span>
<span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c27">PP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c28">pt</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="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><a href="lopban_1.html#K14" title="LOPBAN_1:func.14">PreNorms</a> <font color="Maroon" title="c27">PP</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="seq_4.html#D1" title="SEQ_4:def.1">SEQ_4:def 1</a></span>;<br/><a NAME="E26:77_1_2_2_6"/><span class="kw">then </span>
<span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c25">P1</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c26">P0</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="c7">r</font> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:77_1_2_2_6"><span class="lab"><font color="Green" title="E98">A44</font></span></a>, <a class="txt" href="#E17:77_1_2_2_6"><span class="lab"><font color="Green" title="E96">A42</font></span></a>, <a class="txt" href="#E19:77_1_2_2_6"><span class="lab"><font color="Green" title="E97">A43</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><span class="kw">hence </span><a NAME="E27:77_1_2_2_6"/>
<font color="Maroon" title="c19">ys</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c22">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c21">rs</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c22">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:77_1_2_2_6"><span class="lab"><font color="Green" title="E92">A34</font></span></a>, <a class="txt" href="#E22:77_1_2_2"><span class="lab"><font color="Green" title="E86">A26</font></span></a>, <a class="txt" href="#E5:77_1_2_2_6"><span class="lab"><font color="Green" title="E93">A35</font></span></a>, <a class="txt" href="#E11:77_1_2_2_6"><span class="lab"><font color="Green" title="E94">A40</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="E33:77_1_2_2"/><span class="kw">then </span>
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c19">ys</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">r</font> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:77_1_2_2"><span class="lab"><font color="Green" title="E89">A31</font></span></a>, <a class="txt" href="#E22:77_1_2_2"><span class="lab"><font color="Green" title="E86">A26</font></span></a>, <a class="txt" href="#E17:77_1_2_2"><span class="lab"><font color="Green" title="E83">A22</font></span></a>, <a class="ref" href="integra5.html#T3" title="INTEGRA5:th.3">INTEGRA5:3</a></span>;<br/><span class="kw">hence </span><a NAME="E34:77_1_2_2"/>
<font color="Maroon" title="c12">mt</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">r</font> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:77_1_2_2"><span class="lab"><font color="Green" title="E82">A21</font></span></a>, <a class="txt" href="#E19:77_1_2_2"><span class="lab"><font color="Green" title="E85">A25</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="E12:77_1_2"/><span class="kw">then </span>
( <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c10">y1</font>)</span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p3">(<span class="default"><a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c6">y0</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="c7">r</font> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 2 &amp; <font color="Maroon" title="c7">r</font> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">r</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:77_1_2"><span class="lab"><font color="Green" title="E76">A13</font></span></a>, <a class="txt" href="#E1:77_1_2"><span class="lab"><font color="Green" title="E71">A7</font></span></a>, <a class="ref" href="seq_4.html#T45" title="SEQ_4:th.45">SEQ_4:45</a>, <a class="ref" href="xreal_1.html#T216" title="XREAL_1:th.216">XREAL_1:216</a></span>;<br/>
<a NAME="E13:77_1_2"/><span class="kw">then </span>
<span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c10">y1</font>)</span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p3">(<span class="default"><a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c6">y0</font>)</span>)</span></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E14:77_1_2"/><span class="kw">then </span>
<span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c10">y1</font>)</span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="ndiff_1.html#K4" title="NDIFF_1:func.4">`|</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">y0</font></span>)</span></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:77_1"><span class="lab"><font color="Green" title="E69">A5</font></span></a>, <a class="txt" href="#E1:77_1_2"><span class="lab"><font color="Green" title="E71">A7</font></span></a>, <a class="ref" href="ndiff_1.html#D9" title="NDIFF_1:def.9">NDIFF_1:def 9</a></span>;<br/>
<span class="kw">hence </span><a NAME="E15:77_1_2"/>
<span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="ndiff_1.html#K4" title="NDIFF_1:func.4">`|</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">y1</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="ndiff_1.html#K4" title="NDIFF_1:func.4">`|</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">y0</font></span>)</span></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:77_1"><span class="lab"><font color="Green" title="E69">A5</font></span></a>, <a class="txt" href="#E8:77_1_2"><span class="lab"><font color="Green" title="E75">A12</font></span></a>, <a class="ref" href="ndiff_1.html#D9" title="NDIFF_1:def.9">NDIFF_1:def 9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E11:77_1"/>
<font color="Maroon" title="c3">f</font> <a href="ndiff_1.html#K4" title="NDIFF_1:func.4">`|</a> <font color="Maroon" title="c4">X</font> <a href="nfcont_1.html#R3" title="NFCONT_1:pred.3">is_continuous_on</a> <font color="Maroon" title="c4">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:77_1"><span class="lab"><font color="Green" title="E70">A6</font></span></a>, <a class="ref" href="nfcont_1.html#T19" title="NFCONT_1:th.19">NFCONT_1:19</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E4:77"/><span class="lab"><font color="Green" title="E65">A52</font></span>: 
( <font color="Maroon" title="c3">f</font> <a href="ndiff_1.html#R2" title="NDIFF_1:pred.2">is_differentiable_on</a> <font color="Maroon" title="c4">X</font> &amp; <font color="Maroon" title="c3">f</font> <a href="ndiff_1.html#K4" title="NDIFF_1:func.4">`|</a> <font color="Maroon" title="c4">X</font> <a href="nfcont_1.html#R3" title="NFCONT_1:pred.3">is_continuous_on</a> <font color="Maroon" title="c4">X</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">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">G</font> holds <br/>( <font color="Maroon" title="c3">f</font> <a href="ndiff_5.html#R2" title="NDIFF_5:pred.2">is_partial_differentiable_on</a> <font color="Maroon" title="c4">X</font>,<font color="Olive" title="b1">i</font> &amp; <font color="Maroon" title="c3">f</font> <a href="ndiff_5.html#K7" title="NDIFF_5:func.7">`partial|</a> (<font color="Maroon" title="c4">X</font>,<font color="Olive" title="b1">i</font>) <a href="nfcont_1.html#R3" title="NFCONT_1:pred.3">is_continuous_on</a> <font color="Maroon" title="c4">X</font> )</span><br/>

<a NAME="E5:77"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A53</font></span>: 
( <font color="Maroon" title="c4">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="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 <span class="p1">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">G</font></span>)</span>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> 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> ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:77"><span class="lab"><font color="Green" title="E63">A1</font></span></a>, <a class="ref" href="ndiff_1.html#T31" title="NDIFF_1:th.31">NDIFF_1:31</a></span>;<br/>
<span class="kw">thus </span><a NAME="E6:77"/>
 for <font color="Olive" title="b1">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">G</font> holds <br/>( <font color="Maroon" title="c3">f</font> <a href="ndiff_5.html#R2" title="NDIFF_5:pred.2">is_partial_differentiable_on</a> <font color="Maroon" title="c4">X</font>,<font color="Olive" title="b1">i</font> &amp; <font color="Maroon" title="c3">f</font> <a href="ndiff_5.html#K7" title="NDIFF_5:func.7">`partial|</a> (<font color="Maroon" title="c4">X</font>,<font color="Olive" title="b1">i</font>) <a href="nfcont_1.html#R3" title="NFCONT_1:pred.3">is_continuous_on</a> <font color="Maroon" title="c4">X</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">G</font> implies ( <font color="Maroon" title="c3">f</font> <a href="ndiff_5.html#R2" title="NDIFF_5:pred.2">is_partial_differentiable_on</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c6">i</font> &amp; <font color="Maroon" title="c3">f</font> <a href="ndiff_5.html#K7" title="NDIFF_5:func.7">`partial|</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c6">i</font>) <a href="nfcont_1.html#R3" title="NFCONT_1:pred.3">is_continuous_on</a> <font color="Maroon" title="c4">X</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:77_2"/><span class="lab"><font color="Green" title="E67">A540</font></span>: 
<font color="Maroon" title="c6">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">G</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">f</font> <a href="ndiff_5.html#R2" title="NDIFF_5:pred.2">is_partial_differentiable_on</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c6">i</font> &amp; <font color="Maroon" title="c3">f</font> <a href="ndiff_5.html#K7" title="NDIFF_5:func.7">`partial|</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c6">i</font>) <a href="nfcont_1.html#R3" title="NFCONT_1:pred.3">is_continuous_on</a> <font color="Maroon" title="c4">X</font> )</span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">i0</font> = <font color="Maroon" title="c6">i</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c8">x</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">G</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> implies ( <font color="Maroon" title="c3">f</font> <a href="ndiff_5.html#R1" title="NDIFF_5:pred.1">is_partial_differentiable_in</a> <font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c6">i</font> &amp;  <a href="ndiff_5.html#K6" title="NDIFF_5:func.6">partdiff</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c6">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c8">x</font>)</span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p2">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span>)</span>)</span> ) )</span><br/><span class="kw">assume </span><a NAME="E1:77_2_1"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">f</font> <a href="ndiff_5.html#R1" title="NDIFF_5:pred.1">is_partial_differentiable_in</a> <font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c6">i</font> &amp;  <a href="ndiff_5.html#K6" title="NDIFF_5:func.6">partdiff</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c6">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c8">x</font>)</span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p2">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span>)</span>)</span> )</span><br/><a NAME="E2:77_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">f</font> <a href="ndiff_1.html#R1" title="NDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c8">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:77"><span class="lab"><font color="Green" title="E65">A52</font></span></a>, <a class="txt" href="#E1:77"><span class="lab"><font color="Green" title="E63">A1</font></span></a>, <a class="ref" href="ndiff_1.html#T31" title="NDIFF_1:th.31">NDIFF_1:31</a></span>;<br/><span class="kw">hence </span><a NAME="E3:77_2_1"/>
( <font color="Maroon" title="c3">f</font> <a href="ndiff_5.html#R1" title="NDIFF_5:pred.1">is_partial_differentiable_in</a> <font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c6">i</font> &amp;  <a href="ndiff_5.html#K6" title="NDIFF_5:func.6">partdiff</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c6">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c8">x</font>)</span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p2">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span>)</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ndiff_5.html#T41" target="_self" title="NDIFF_5:th.41">Th21X</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="E4:77_2"/><span class="kw">then </span>
 for <font color="Olive" title="b1">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">G</font></span>)</span>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> holds <br/><font color="Maroon" title="c3">f</font> <a href="ndiff_5.html#R1" title="NDIFF_5:pred.1">is_partial_differentiable_in</a> <font color="Olive" title="b1">x</font>,<font color="Maroon" title="c6">i</font>
 
;<br/>
<span class="kw">hence </span><a NAME="E5:77_2"/><span class="lab"><font color="Green" title="E68">A55</font></span>: 
<font color="Maroon" title="c3">f</font> <a href="ndiff_5.html#R2" title="NDIFF_5:pred.2">is_partial_differentiable_on</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c6">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:77"><span class="lab"><font color="Green" title="E63">A1</font></span></a>, <a class="ref" href="ndiff_5.html#T32" target="_self" title="NDIFF_5:th.32">XTh8</a>, <a class="txt" href="#E5:77"><span class="lab"><font color="Green" title="E66">A53</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"> <font color="Maroon" title="c3">f</font> <a href="ndiff_5.html#K7" title="NDIFF_5:func.7">`partial|</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c6">i</font>) <a href="nfcont_1.html#R3" title="NFCONT_1:pred.3">is_continuous_on</a> <font color="Maroon" title="c4">X</font></span><br/>

<a NAME="E6:77_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A56</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="ndiff_5.html#K7" title="NDIFF_5:func.7">`partial|</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c6">i</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ndiff_5.html#D9" target="_self" title="NDIFF_5:def.9">Def19f</a>, <a class="txt" href="#E1:77_2"><span class="lab"><font color="Green" title="E67">A540</font></span></a></span>;<br/>
<a NAME="E7:77_2"/>
 for <font color="Olive" title="b1">y0</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">G</font></span>)</span><br/>  for <font color="Olive" title="b2">r</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">y0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</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="b2">r</font> holds <br/> ex <font color="Olive" title="b3">s</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>(  <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="b3">s</font> &amp; (  for <font color="Olive" title="b4">y1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">G</font></span>)</span>  st <font color="Olive" title="b4">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> &amp; <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b4">y1</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Olive" title="b1">y0</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b3">s</font> holds <br/><span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="ndiff_5.html#K7" title="NDIFF_5:func.7">`partial|</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c6">i</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b4">y1</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="ndiff_5.html#K7" title="NDIFF_5:func.7">`partial|</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c6">i</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">y0</font></span>)</span></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">r</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">y0</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">G</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">r</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Maroon" title="c8">y0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</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> holds <br/> ex <font color="Olive" title="b2">s</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>(  <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="b2">s</font> &amp; (  for <font color="Olive" title="b3">y1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">G</font></span>)</span>  st <font color="Olive" title="b3">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> &amp; <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">y1</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c8">y0</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">s</font> holds <br/><span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="ndiff_5.html#K7" title="NDIFF_5:func.7">`partial|</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c6">i</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b3">y1</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="ndiff_5.html#K7" title="NDIFF_5:func.7">`partial|</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c6">i</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c8">y0</font></span>)</span></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">r</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c9">r</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="c8">y0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</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="Maroon" title="c9">r</font> implies  ex <font color="Olive" title="b1">s</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>(  <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">s</font> &amp; (  for <font color="Olive" title="b2">y1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">G</font></span>)</span>  st <font color="Olive" title="b2">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> &amp; <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">y1</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c8">y0</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">s</font> holds <br/><span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="ndiff_5.html#K7" title="NDIFF_5:func.7">`partial|</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c6">i</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">y1</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="ndiff_5.html#K7" title="NDIFF_5:func.7">`partial|</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c6">i</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c8">y0</font></span>)</span></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">r</font> ) ) )</span><br/>



<span class="kw">assume </span><a NAME="E1:77_2_2"/><span class="lab"><font color="Green" title="E70">A57</font></span>: 
( <font color="Maroon" title="c8">y0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</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="Maroon" title="c9">r</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">s</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>(  <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">s</font> &amp; (  for <font color="Olive" title="b2">y1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">G</font></span>)</span>  st <font color="Olive" title="b2">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> &amp; <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">y1</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c8">y0</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">s</font> holds <br/><span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="ndiff_5.html#K7" title="NDIFF_5:func.7">`partial|</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c6">i</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">y1</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="ndiff_5.html#K7" title="NDIFF_5:func.7">`partial|</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c6">i</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c8">y0</font></span>)</span></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">r</font> ) )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">s</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E3:77_2_2"/><span class="lab"><font color="Green" title="E71">A58</font></span>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">s</font> &amp; (  for <font color="Olive" title="b1">y1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">G</font></span>)</span>  st <font color="Olive" title="b1">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> &amp; <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="b1">y1</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c8">y0</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">s</font> holds <br/><span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="ndiff_1.html#K4" title="NDIFF_1:func.4">`|</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">y1</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="ndiff_1.html#K4" title="NDIFF_1:func.4">`|</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c8">y0</font></span>)</span></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">r</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:77"><span class="lab"><font color="Green" title="E65">A52</font></span></a>, <a class="ref" href="nfcont_1.html#T19" title="NFCONT_1:th.19">NFCONT_1:19</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c10">s</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">s</font> &amp; (  for <font color="Olive" title="b1">y1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">G</font></span>)</span>  st <font color="Olive" title="b1">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> &amp; <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="b1">y1</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c8">y0</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">s</font> holds <br/><span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="ndiff_5.html#K7" title="NDIFF_5:func.7">`partial|</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c6">i</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">y1</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="ndiff_5.html#K7" title="NDIFF_5:func.7">`partial|</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c6">i</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c8">y0</font></span>)</span></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">r</font> ) )</span><br/>

<span class="kw">thus </span><a NAME="E4:77_2_2"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:77_2_2"><span class="lab"><font color="Green" title="E71">A58</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">  for <font color="Olive" title="b1">y1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">G</font></span>)</span>  st <font color="Olive" title="b1">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> &amp; <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="b1">y1</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c8">y0</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">s</font> holds <br/><span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="ndiff_5.html#K7" title="NDIFF_5:func.7">`partial|</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c6">i</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">y1</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="ndiff_5.html#K7" title="NDIFF_5:func.7">`partial|</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c6">i</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c8">y0</font></span>)</span></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">r</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c11">y1</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">G</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> &amp; <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="c11">y1</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c8">y0</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">s</font> implies <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="ndiff_5.html#K7" title="NDIFF_5:func.7">`partial|</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c6">i</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">y1</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="ndiff_5.html#K7" title="NDIFF_5:func.7">`partial|</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c6">i</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c8">y0</font></span>)</span></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">r</font> )</span><br/>

<span class="kw">assume </span><a NAME="E5:77_2_2"/><span class="lab"><font color="Green" title="E72">A59</font></span>: 
( <font color="Maroon" title="c11">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> &amp; <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="c11">y1</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c8">y0</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">s</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="ndiff_5.html#K7" title="NDIFF_5:func.7">`partial|</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c6">i</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">y1</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="ndiff_5.html#K7" title="NDIFF_5:func.7">`partial|</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c6">i</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c8">y0</font></span>)</span></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">r</font></span><br/>

<a NAME="E6:77_2_2"/><span class="kw">then </span>
<span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="ndiff_1.html#K4" title="NDIFF_1:func.4">`|</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">y1</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="ndiff_1.html#K4" title="NDIFF_1:func.4">`|</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c8">y0</font></span>)</span></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:77_2_2"><span class="lab"><font color="Green" title="E71">A58</font></span></a></span>;<br/>
<a NAME="E7:77_2_2"/><span class="kw">then </span>
<span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c11">y1</font>)</span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="ndiff_1.html#K4" title="NDIFF_1:func.4">`|</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c8">y0</font></span>)</span></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:77_2_2"><span class="lab"><font color="Green" title="E72">A59</font></span></a>, <a class="txt" href="#E4:77"><span class="lab"><font color="Green" title="E65">A52</font></span></a>, <a class="ref" href="ndiff_1.html#D9" title="NDIFF_1:def.9">NDIFF_1:def 9</a></span>;<br/>
<a NAME="E8:77_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E73">A60</font></span>: 
<span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c11">y1</font>)</span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p3">(<span class="default"><a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c8">y0</font>)</span>)</span></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:77_2_2"><span class="lab"><font color="Green" title="E70">A57</font></span></a>, <a class="txt" href="#E4:77"><span class="lab"><font color="Green" title="E65">A52</font></span></a>, <a class="ref" href="ndiff_1.html#D9" title="NDIFF_1:def.9">NDIFF_1:def 9</a></span>;<br/>
<a NAME="E9:77_2_2"/>
( <font color="Maroon" title="c3">f</font> <a href="ndiff_1.html#R1" title="NDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c11">y1</font> &amp; <font color="Maroon" title="c3">f</font> <a href="ndiff_1.html#R1" title="NDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c8">y0</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:77"><span class="lab"><font color="Green" title="E65">A52</font></span></a>, <a class="txt" href="#E1:77"><span class="lab"><font color="Green" title="E63">A1</font></span></a>, <a class="txt" href="#E5:77_2_2"><span class="lab"><font color="Green" title="E72">A59</font></span></a>, <a class="txt" href="#E1:77_2_2"><span class="lab"><font color="Green" title="E70">A57</font></span></a>, <a class="ref" href="ndiff_1.html#T31" title="NDIFF_1:th.31">NDIFF_1:31</a></span>;<br/>
<a NAME="E10:77_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A61</font></span>: 
(  <a href="ndiff_5.html#K6" title="NDIFF_5:func.6">partdiff</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c11">y1</font>,<font color="Maroon" title="c6">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c11">y1</font>)</span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p2">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span>)</span>)</span> &amp;  <a href="ndiff_5.html#K6" title="NDIFF_5:func.6">partdiff</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c8">y0</font>,<font color="Maroon" title="c6">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c8">y0</font>)</span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p2">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span>)</span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ndiff_5.html#T41" target="_self" title="NDIFF_5:th.41">Th21X</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c12">PP</font> = <span class="p1">(<span class="default"><a href="ndiff_5.html#K6" title="NDIFF_5:func.6">partdiff</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c11">y1</font>,<font color="Maroon" title="c6">i</font>)</span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><a href="ndiff_5.html#K6" title="NDIFF_5:func.6">partdiff</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c8">y0</font>,<font color="Maroon" title="c6">i</font>)</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 <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="prvect_2.html#K11" title="PRVECT_2:func.11">.</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i</font>)</span>)</span></span>)</span>,<font color="Maroon" title="c2">F</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="lopban_1.html#D9" title="LOPBAN_1:def.9">LOPBAN_1:def 9</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c13">DD</font> = <span class="p1">(<span class="default"><a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c11">y1</font>)</span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c8">y0</font>)</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 <span class="p1">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">G</font></span>)</span>,<font color="Maroon" title="c2">F</font> <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="E13:77_2_2"/><span class="lab"><font color="Green" title="E75">A62</font></span>: 
(  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><a href="lopban_1.html#K14" title="LOPBAN_1:func.14">PreNorms</a> <font color="Maroon" title="c12">PP</font></span>)</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_5.html#K6" title="NDIFF_5:func.6">partdiff</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c11">y1</font>,<font color="Maroon" title="c6">i</font>)</span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p3">(<span class="default"><a href="ndiff_5.html#K6" title="NDIFF_5:func.6">partdiff</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c8">y0</font>,<font color="Maroon" title="c6">i</font>)</span>)</span></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> &amp;  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><a href="lopban_1.html#K14" title="LOPBAN_1:func.14">PreNorms</a> <font color="Maroon" title="c13">DD</font></span>)</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>,<font color="Maroon" title="c11">y1</font>)</span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p3">(<span class="default"><a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c8">y0</font>)</span>)</span></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lopban_1.html#T30" title="LOPBAN_1:th.30">LOPBAN_1:30</a></span>;<br/>
<a NAME="E14:77_2_2"/><span class="lab"><font color="Green" title="E76">A64</font></span>: 
(  <a href="lopban_1.html#K14" title="LOPBAN_1:func.14">PreNorms</a> <font color="Maroon" title="c12">PP</font> is  <a href="xxreal_2.html#V4" title="XXREAL_2:attr.4">bounded_above</a>  &amp;  <a href="lopban_1.html#K14" title="LOPBAN_1:func.14">PreNorms</a> <font color="Maroon" title="c13">DD</font> is  <a href="xxreal_2.html#V4" title="XXREAL_2:attr.4">bounded_above</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lopban_1.html#T28" title="LOPBAN_1:th.28">LOPBAN_1:28</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_2_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c14">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c14">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lopban_1.html#K14" title="LOPBAN_1:func.14">PreNorms</a> <font color="Maroon" title="c12">PP</font> implies <font color="Maroon" title="c14">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lopban_1.html#K14" title="LOPBAN_1:func.14">PreNorms</a> <font color="Maroon" title="c13">DD</font> )</span><br/><span class="kw">assume </span><a NAME="E1:77_2_2_1"/>
<font color="Maroon" title="c14">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lopban_1.html#K14" title="LOPBAN_1:func.14">PreNorms</a> <font color="Maroon" title="c12">PP</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c14">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lopban_1.html#K14" title="LOPBAN_1:func.14">PreNorms</a> <font color="Maroon" title="c13">DD</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c15">t</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="prvect_2.html#K11" title="PRVECT_2:func.11">.</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i</font>)</span>)</span></span>)</span><span class="kw"> such that </span><br/><a NAME="E3:77_2_2_1"/><span class="lab"><font color="Green" title="E77">A65</font></span>: 
( <font color="Maroon" title="c14">a</font> <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"><font color="Maroon" title="c12">PP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">t</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> &amp; <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c15">t</font></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> 1 )
 ;<br/><a NAME="E4:77_2_2_1"/><span class="lab"><font color="Green" title="E78">A66</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p2">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="prvect_2.html#K11" title="PRVECT_2:func.11">.</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i</font>)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c16">tm</font> = <span class="p1">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p2">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">t</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">G</font></span>)</span> ;<br/><a NAME="E6:77_2_2_1"/><span class="lab"><font color="Green" title="E79">A67</font></span>: 
<span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c16">tm</font></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> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:77_2_2_1"><span class="lab"><font color="Green" title="E77">A65</font></span></a>, <a class="ref" href="ndiff_5.html#T21" target="_self" title="NDIFF_5:th.21">XTh5</a></span>;<br/><a NAME="E7:77_2_2_1"/>
( <span class="p1">(<span class="default"><a href="ndiff_5.html#K6" title="NDIFF_5:func.6">partdiff</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c11">y1</font>,<font color="Maroon" title="c6">i</font>)</span>)</span> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Maroon" title="c15">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c11">y1</font>)</span>)</span> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Maroon" title="c16">tm</font> &amp; <span class="p1">(<span class="default"><a href="ndiff_5.html#K6" title="NDIFF_5:func.6">partdiff</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c8">y0</font>,<font color="Maroon" title="c6">i</font>)</span>)</span> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Maroon" title="c15">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c8">y0</font>)</span>)</span> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Maroon" title="c16">tm</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:77_2_2_1"><span class="lab"><font color="Green" title="E78">A66</font></span></a>, <a class="txt" href="#E10:77_2_2"><span class="lab"><font color="Green" title="E74">A61</font></span></a>, <a class="ref" href="funct_1.html#T13" title="FUNCT_1:th.13">FUNCT_1:13</a></span>;<br/><a NAME="E8:77_2_2_1"/><span class="kw">then </span>
<span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">PP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">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"><span class="p4">(<span class="default"><a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c11">y1</font>)</span>)</span> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Maroon" title="c16">tm</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c8">y0</font>)</span>)</span> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Maroon" title="c16">tm</font></span>)</span></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lopban_1.html#T40" title="LOPBAN_1:th.40">LOPBAN_1:40</a></span>;<br/><a NAME="E9:77_2_2_1"/><span class="kw">then </span>
<span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">PP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">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"><font color="Maroon" title="c13">DD</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">tm</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#T40" title="LOPBAN_1:th.40">LOPBAN_1:40</a></span>;<br/><span class="kw">hence </span><a NAME="E10:77_2_2_1"/>
<font color="Maroon" title="c14">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lopban_1.html#K14" title="LOPBAN_1:func.14">PreNorms</a> <font color="Maroon" title="c13">DD</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:77_2_2_1"><span class="lab"><font color="Green" title="E77">A65</font></span></a>, <a class="txt" href="#E6:77_2_2_1"><span class="lab"><font color="Green" title="E79">A67</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="E16:77_2_2"/><span class="kw">then </span>
 <a href="lopban_1.html#K14" title="LOPBAN_1:func.14">PreNorms</a> <font color="Maroon" title="c12">PP</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="lopban_1.html#K14" title="LOPBAN_1:func.14">PreNorms</a> <font color="Maroon" title="c13">DD</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/>
<a NAME="E17:77_2_2"/><span class="kw">then </span>
<span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ndiff_5.html#K6" title="NDIFF_5:func.6">partdiff</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c11">y1</font>,<font color="Maroon" title="c6">i</font>)</span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p3">(<span class="default"><a href="ndiff_5.html#K6" title="NDIFF_5:func.6">partdiff</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c8">y0</font>,<font color="Maroon" title="c6">i</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"><span class="p3">(<span class="default"><a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c11">y1</font>)</span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p3">(<span class="default"><a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c8">y0</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="#E14:77_2_2"><span class="lab"><font color="Green" title="E76">A64</font></span></a>, <a class="txt" href="#E13:77_2_2"><span class="lab"><font color="Green" title="E75">A62</font></span></a>, <a class="ref" href="seq_4.html#T48" title="SEQ_4:th.48">SEQ_4:48</a></span>;<br/>
<a NAME="E18:77_2_2"/><span class="kw">then </span>
<span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ndiff_5.html#K6" title="NDIFF_5:func.6">partdiff</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c11">y1</font>,<font color="Maroon" title="c6">i</font>)</span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p3">(<span class="default"><a href="ndiff_5.html#K6" title="NDIFF_5:func.6">partdiff</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c8">y0</font>,<font color="Maroon" title="c6">i</font>)</span>)</span></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:77_2_2"><span class="lab"><font color="Green" title="E73">A60</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E19:77_2_2"/><span class="kw">then </span>
<span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ndiff_5.html#K6" title="NDIFF_5:func.6">partdiff</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c11">y1</font>,<font color="Maroon" title="c6">i</font>)</span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="ndiff_5.html#K7" title="NDIFF_5:func.7">`partial|</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c6">i</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c8">y0</font></span>)</span></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ndiff_5.html#D9" target="_self" title="NDIFF_5:def.9">Def19f</a>, <a class="txt" href="#E1:77_2_2"><span class="lab"><font color="Green" title="E70">A57</font></span></a>, <a class="txt" href="#E5:77_2"><span class="lab"><font color="Green" title="E68">A55</font></span></a>, <a class="txt" href="#E1:77_2"><span class="lab"><font color="Green" title="E67">A540</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E20:77_2_2"/>
<span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="ndiff_5.html#K7" title="NDIFF_5:func.7">`partial|</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c6">i</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">y1</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="ndiff_5.html#K7" title="NDIFF_5:func.7">`partial|</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c6">i</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c8">y0</font></span>)</span></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">r</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ndiff_5.html#D9" target="_self" title="NDIFF_5:def.9">Def19f</a>, <a class="txt" href="#E5:77_2_2"><span class="lab"><font color="Green" title="E72">A59</font></span></a>, <a class="txt" href="#E5:77_2"><span class="lab"><font color="Green" title="E68">A55</font></span></a>, <a class="txt" href="#E1:77_2"><span class="lab"><font color="Green" title="E67">A540</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>
<span class="kw">hence </span><a NAME="E8:77_2"/>
<font color="Maroon" title="c3">f</font> <a href="ndiff_5.html#K7" title="NDIFF_5:func.7">`partial|</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c6">i</font>) <a href="nfcont_1.html#R3" title="NFCONT_1:pred.3">is_continuous_on</a> <font color="Maroon" title="c4">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:77_2"><span class="lab"><font color="Green" title="E69">A56</font></span></a>, <a class="ref" href="nfcont_1.html#T19" title="NFCONT_1:th.19">NFCONT_1:19</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>
