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<span class="kw">let </span><font color="Maroon" title="c1">F</font>, <font color="Maroon" title="c2">G</font> be   <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Field</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">V</font> being   <a href="vectsp_1.html#NM9" title="VECTSP_1:NM.9">VectSp</a> of <font color="Maroon" title="c1">F</font><br/>  for <font color="Olive" title="b2">W</font> being   <a href="vectsp_1.html#NM9" title="VECTSP_1:NM.9">VectSp</a> of <font color="Maroon" title="c2">G</font><br/>  for <font color="Olive" title="b3">f</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Olive" title="b1">V</font>,<font color="Olive" title="b2">W</font><br/>  for <font color="Olive" title="b4">x</font>, <font color="Olive" title="b5">h</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b1">V</font><br/>  for <font color="Olive" title="b6">r</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font><br/>  for <font color="Olive" title="b7">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="vsdiff_1.html#NK9" title="VSDIFF_1:NK.9">fdif</a> (<span class="p3">(<span class="default"><font color="Olive" title="b6">r</font> <a href="vsdiff_1.html#K1" title="VSDIFF_1:func.1">(#)</a> <font color="Olive" title="b3">f</font></span>)</span>,<font color="Olive" title="b5">h</font>)</span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">r</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="vsdiff_1.html#NK9" title="VSDIFF_1:NK.9">fdif</a> (<font color="Olive" title="b3">f</font>,<font color="Olive" title="b5">h</font>)</span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b4">x</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">V</font> be   <a href="vectsp_1.html#NM9" title="VECTSP_1:NM.9">VectSp</a> of <font color="Maroon" title="c1">F</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">W</font> being   <a href="vectsp_1.html#NM9" title="VECTSP_1:NM.9">VectSp</a> of <font color="Maroon" title="c2">G</font><br/>  for <font color="Olive" title="b2">f</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c3">V</font>,<font color="Olive" title="b1">W</font><br/>  for <font color="Olive" title="b3">x</font>, <font color="Olive" title="b4">h</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">V</font><br/>  for <font color="Olive" title="b5">r</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font><br/>  for <font color="Olive" title="b6">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="vsdiff_1.html#NK9" title="VSDIFF_1:NK.9">fdif</a> (<span class="p3">(<span class="default"><font color="Olive" title="b5">r</font> <a href="vsdiff_1.html#K1" title="VSDIFF_1:func.1">(#)</a> <font color="Olive" title="b2">f</font></span>)</span>,<font color="Olive" title="b4">h</font>)</span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">r</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="vsdiff_1.html#NK9" title="VSDIFF_1:NK.9">fdif</a> (<font color="Olive" title="b2">f</font>,<font color="Olive" title="b4">h</font>)</span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b3">x</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c4">W</font> be   <a href="vectsp_1.html#NM9" title="VECTSP_1:NM.9">VectSp</a> of <font color="Maroon" title="c2">G</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c3">V</font>,<font color="Maroon" title="c4">W</font><br/>  for <font color="Olive" title="b2">x</font>, <font color="Olive" title="b3">h</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">V</font><br/>  for <font color="Olive" title="b4">r</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font><br/>  for <font color="Olive" title="b5">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="vsdiff_1.html#NK9" title="VSDIFF_1:NK.9">fdif</a> (<span class="p3">(<span class="default"><font color="Olive" title="b4">r</font> <a href="vsdiff_1.html#K1" title="VSDIFF_1:func.1">(#)</a> <font color="Olive" title="b1">f</font></span>)</span>,<font color="Olive" title="b3">h</font>)</span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">r</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="vsdiff_1.html#NK9" title="VSDIFF_1:NK.9">fdif</a> (<font color="Olive" title="b1">f</font>,<font color="Olive" title="b3">h</font>)</span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">x</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c5">f</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c3">V</font>,<font color="Maroon" title="c4">W</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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>, <font color="Olive" title="b2">h</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">V</font><br/>  for <font color="Olive" title="b3">r</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font><br/>  for <font color="Olive" title="b4">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="vsdiff_1.html#NK9" title="VSDIFF_1:NK.9">fdif</a> (<span class="p3">(<span class="default"><font color="Olive" title="b3">r</font> <a href="vsdiff_1.html#K1" title="VSDIFF_1:func.1">(#)</a> <font color="Maroon" title="c5">f</font></span>)</span>,<font color="Olive" title="b2">h</font>)</span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">r</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="vsdiff_1.html#NK9" title="VSDIFF_1:NK.9">fdif</a> (<font color="Maroon" title="c5">f</font>,<font color="Olive" title="b2">h</font>)</span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">x</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c7">h</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">V</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font><br/>  for <font color="Olive" title="b2">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="vsdiff_1.html#NK9" title="VSDIFF_1:NK.9">fdif</a> (<span class="p3">(<span class="default"><font color="Olive" title="b1">r</font> <a href="vsdiff_1.html#K1" title="VSDIFF_1:func.1">(#)</a> <font color="Maroon" title="c5">f</font></span>)</span>,<font color="Maroon" title="c7">h</font>)</span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">r</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="vsdiff_1.html#NK9" title="VSDIFF_1:NK.9">fdif</a> (<font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c7">h</font>)</span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">x</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c8">r</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="vsdiff_1.html#NK9" title="VSDIFF_1:NK.9">fdif</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="vsdiff_1.html#K1" title="VSDIFF_1:func.1">(#)</a> <font color="Maroon" title="c5">f</font></span>)</span>,<font color="Maroon" title="c7">h</font>)</span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">r</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="vsdiff_1.html#NK9" title="VSDIFF_1:NK.9">fdif</a> (<font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c7">h</font>)</span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">x</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c9">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="vsdiff_1.html#NK9" title="VSDIFF_1:NK.9">fdif</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="vsdiff_1.html#K1" title="VSDIFF_1:func.1">(#)</a> <font color="Maroon" title="c5">f</font></span>)</span>,<font color="Maroon" title="c7">h</font>)</span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">r</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="vsdiff_1.html#NK9" title="VSDIFF_1:NK.9">fdif</a> (<font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c7">h</font>)</span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">x</font></span>)</span></span><br/>

















<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>] <span class="kw">means </span> for <font color="Olive" title="b1">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">V</font> holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="vsdiff_1.html#NK9" title="VSDIFF_1:NK.9">fdif</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="vsdiff_1.html#K1" title="VSDIFF_1:func.1">(#)</a> <font color="Maroon" title="c5">f</font></span>)</span>,<font color="Maroon" title="c7">h</font>)</span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p2">(<span class="default">$1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">r</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="vsdiff_1.html#NK9" title="VSDIFF_1:NK.9">fdif</a> (<font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c7">h</font>)</span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p3">(<span class="default">$1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">x</font></span>)</span>;<br/>
<a NAME="E1:21"/><span class="lab"><font color="Green" title="E12">A1</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="21_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">k</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="c10">k</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:21_1"/><span class="lab"><font color="Green" title="E13">A2</font></span>: 
 for <font color="Olive" title="b1">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">V</font> holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="vsdiff_1.html#NK9" title="VSDIFF_1:NK.9">fdif</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="vsdiff_1.html#K1" title="VSDIFF_1:func.1">(#)</a> <font color="Maroon" title="c5">f</font></span>)</span>,<font color="Maroon" title="c7">h</font>)</span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">r</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="vsdiff_1.html#NK9" title="VSDIFF_1:NK.9">fdif</a> (<font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c7">h</font>)</span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">x</font></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="c10">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<span class="kw">let </span><font color="Maroon" title="c11">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">V</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="vsdiff_1.html#NK9" title="VSDIFF_1:NK.9">fdif</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="vsdiff_1.html#K1" title="VSDIFF_1:func.1">(#)</a> <font color="Maroon" title="c5">f</font></span>)</span>,<font color="Maroon" title="c7">h</font>)</span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">r</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="vsdiff_1.html#NK9" title="VSDIFF_1:NK.9">fdif</a> (<font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c7">h</font>)</span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">x</font></span>)</span></span><br/>

<a NAME="E2:21_1"/><span class="lab"><font color="Green" title="E14">A3</font></span>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="vsdiff_1.html#NK9" title="VSDIFF_1:NK.9">fdif</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="vsdiff_1.html#K1" title="VSDIFF_1:func.1">(#)</a> <font color="Maroon" title="c5">f</font></span>)</span>,<font color="Maroon" title="c7">h</font>)</span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">r</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="vsdiff_1.html#NK9" title="VSDIFF_1:NK.9">fdif</a> (<font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c7">h</font>)</span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">x</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="vsdiff_1.html#NK9" title="VSDIFF_1:NK.9">fdif</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="vsdiff_1.html#K1" title="VSDIFF_1:func.1">(#)</a> <font color="Maroon" title="c5">f</font></span>)</span>,<font color="Maroon" title="c7">h</font>)</span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c7">h</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">r</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="vsdiff_1.html#NK9" title="VSDIFF_1:NK.9">fdif</a> (<font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c7">h</font>)</span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c7">h</font></span>)</span></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_1"><span class="lab"><font color="Green" title="E13">A2</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c12">rfdk</font> = <span class="p1">(<span class="default"><a href="vsdiff_1.html#NK9" title="VSDIFF_1:NK.9">fdif</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="vsdiff_1.html#K1" title="VSDIFF_1:func.1">(#)</a> <font color="Maroon" title="c5">f</font></span>)</span>,<font color="Maroon" title="c7">h</font>)</span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c3">V</font>,<font color="Maroon" title="c4">W</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="vsdiff_1.html#T3" target="_self" onmouseover="rs('vsdiff_1/T3')" onmouseout="rh()">Th2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c13">fdk</font> = <span class="p1">(<span class="default"><a href="vsdiff_1.html#NK9" title="VSDIFF_1:NK.9">fdif</a> (<font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c7">h</font>)</span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c3">V</font>,<font color="Maroon" title="c4">W</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="vsdiff_1.html#T3" target="_self" onmouseover="rs('vsdiff_1/T3')" onmouseout="rh()">Th2</a></span>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="vsdiff_1.html#NK9" title="VSDIFF_1:NK.9">fdif</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="vsdiff_1.html#K1" title="VSDIFF_1:func.1">(#)</a> <font color="Maroon" title="c5">f</font></span>)</span>,<font color="Maroon" title="c7">h</font>)</span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">x</font> = 
<span class="p1">(<span class="default"><a href="vsdiff_1.html#K5" title="VSDIFF_1:func.5">fD</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="vsdiff_1.html#NK9" title="VSDIFF_1:NK.9">fdif</a> (<span class="p4">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="vsdiff_1.html#K1" title="VSDIFF_1:func.1">(#)</a> <font color="Maroon" title="c5">f</font></span>)</span>,<font color="Maroon" title="c7">h</font>)</span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">h</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">x</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vsdiff_1.html#D8" target="_self" onmouseover="rs('vsdiff_1/D8')" onmouseout="rh()">Def6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">rfdk</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c7">h</font></span>)</span></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">rfdk</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c11">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vsdiff_1.html#T4" target="_self" onmouseover="rs('vsdiff_1/T4')" onmouseout="rh()">Th3</a></span>
<br/>.= 
<font color="Maroon" title="c8">r</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">fdk</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c7">h</font></span>)</span></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">fdk</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c11">x</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#T23" onmouseover="rs('vectsp_1/T23')" onmouseout="rh()">VECTSP_1:23</a>, <a class="txt" href="#E2:21_1"><span class="lab"><font color="Green" title="E14">A3</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c8">r</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="vsdiff_1.html#K5" title="VSDIFF_1:func.5">fD</a> (<font color="Maroon" title="c13">fdk</font>,<font color="Maroon" title="c7">h</font>)</span>)</span> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c11">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vsdiff_1.html#T4" target="_self" onmouseover="rs('vsdiff_1/T4')" onmouseout="rh()">Th3</a></span>
<br/>.= 
<font color="Maroon" title="c8">r</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="vsdiff_1.html#NK9" title="VSDIFF_1:NK.9">fdif</a> (<font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c7">h</font>)</span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vsdiff_1.html#D8" target="_self" onmouseover="rs('vsdiff_1/D8')" onmouseout="rh()">Def6</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E6:21_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="vsdiff_1.html#NK9" title="VSDIFF_1:NK.9">fdif</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="vsdiff_1.html#K1" title="VSDIFF_1:func.1">(#)</a> <font color="Maroon" title="c5">f</font></span>)</span>,<font color="Maroon" title="c7">h</font>)</span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">r</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="vsdiff_1.html#NK9" title="VSDIFF_1:NK.9">fdif</a> (<font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c7">h</font>)</span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">x</font></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:21"/><span class="lab"><font color="Green" title="E13">A6</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> ]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="21_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">V</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="vsdiff_1.html#NK9" title="VSDIFF_1:NK.9">fdif</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="vsdiff_1.html#K1" title="VSDIFF_1:func.1">(#)</a> <font color="Maroon" title="c5">f</font></span>)</span>,<font color="Maroon" title="c7">h</font>)</span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p2">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">r</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="vsdiff_1.html#NK9" title="VSDIFF_1:NK.9">fdif</a> (<font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c7">h</font>)</span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p3">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">x</font></span>)</span></span><br/>

<a NAME="E1:21_2"/>
<font color="Maroon" title="c10">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">V</font>
 
;<br/>
<a NAME="E2:21_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E14">A7</font></span>: 
<font color="Maroon" title="c10">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="vsdiff_1.html#K1" title="VSDIFF_1:func.1">(#)</a> <font color="Maroon" title="c5">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E3:21_2"/>
<font color="Maroon" title="c10">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c7">h</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">V</font>
 
;<br/>
<a NAME="E4:21_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E15">A8</font></span>: 
<font color="Maroon" title="c10">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c7">h</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="vsdiff_1.html#K1" title="VSDIFF_1:func.1">(#)</a> <font color="Maroon" title="c5">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="vsdiff_1.html#NK9" title="VSDIFF_1:NK.9">fdif</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="vsdiff_1.html#K1" title="VSDIFF_1:func.1">(#)</a> <font color="Maroon" title="c5">f</font></span>)</span>,<font color="Maroon" title="c7">h</font>)</span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p2">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">x</font> = 
<span class="p1">(<span class="default"><a href="vsdiff_1.html#K5" title="VSDIFF_1:func.5">fD</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="vsdiff_1.html#NK9" title="VSDIFF_1:NK.9">fdif</a> (<span class="p4">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="vsdiff_1.html#K1" title="VSDIFF_1:func.1">(#)</a> <font color="Maroon" title="c5">f</font></span>)</span>,<font color="Maroon" title="c7">h</font>)</span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span>,<font color="Maroon" title="c7">h</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">x</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vsdiff_1.html#D8" target="_self" onmouseover="rs('vsdiff_1/D8')" onmouseout="rh()">Def6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="vsdiff_1.html#K5" title="VSDIFF_1:func.5">fD</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="vsdiff_1.html#K1" title="VSDIFF_1:func.1">(#)</a> <font color="Maroon" title="c5">f</font></span>)</span>,<font color="Maroon" title="c7">h</font>)</span>)</span> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c10">x</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vsdiff_1.html#D8" target="_self" onmouseover="rs('vsdiff_1/D8')" onmouseout="rh()">Def6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="vsdiff_1.html#K1" title="VSDIFF_1:func.1">(#)</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c7">h</font></span>)</span></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="vsdiff_1.html#K1" title="VSDIFF_1:func.1">(#)</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c10">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vsdiff_1.html#T4" target="_self" onmouseover="rs('vsdiff_1/T4')" onmouseout="rh()">Th3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c7">h</font></span>)</span></span>)</span></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="vsdiff_1.html#K1" title="VSDIFF_1:func.1">(#)</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c10">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:21_2"><span class="lab"><font color="Green" title="E15">A8</font></span></a>, <a class="ref" href="vsdiff_1.html#D1" target="_self" onmouseover="rs('vsdiff_1/D1')" onmouseout="rh()">Def4X</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c7">h</font></span>)</span></span>)</span></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c10">x</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:21_2"><span class="lab"><font color="Green" title="E14">A7</font></span></a>, <a class="ref" href="vsdiff_1.html#D1" target="_self" onmouseover="rs('vsdiff_1/D1')" onmouseout="rh()">Def4X</a></span>
<br/>.= 
<font color="Maroon" title="c8">r</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c7">h</font></span>)</span></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c10">x</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#T23" onmouseover="rs('vectsp_1/T23')" onmouseout="rh()">VECTSP_1:23</a></span>
<br/>.= 
<font color="Maroon" title="c8">r</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="vsdiff_1.html#K5" title="VSDIFF_1:func.5">fD</a> (<font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c7">h</font>)</span>)</span> <a href="funct_2.html#K10" title="FUNCT_2:func.10">/.</a> <font color="Maroon" title="c10">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vsdiff_1.html#T4" target="_self" onmouseover="rs('vsdiff_1/T4')" onmouseout="rh()">Th3</a></span>
<br/>.= 
<font color="Maroon" title="c8">r</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="vsdiff_1.html#K5" title="VSDIFF_1:func.5">fD</a> (<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="vsdiff_1.html#NK9" title="VSDIFF_1:NK.9">fdif</a> (<font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c7">h</font>)</span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span>,<font color="Maroon" title="c7">h</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vsdiff_1.html#D8" target="_self" onmouseover="rs('vsdiff_1/D8')" onmouseout="rh()">Def6</a></span>
<br/>.= 
<font color="Maroon" title="c8">r</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="vsdiff_1.html#NK9" title="VSDIFF_1:NK.9">fdif</a> (<font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c7">h</font>)</span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p3">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vsdiff_1.html#D8" target="_self" onmouseover="rs('vsdiff_1/D8')" onmouseout="rh()">Def6</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E6:21_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="vsdiff_1.html#NK9" title="VSDIFF_1:NK.9">fdif</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="vsdiff_1.html#K1" title="VSDIFF_1:func.1">(#)</a> <font color="Maroon" title="c5">f</font></span>)</span>,<font color="Maroon" title="c7">h</font>)</span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p2">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">r</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="vsdiff_1.html#NK9" title="VSDIFF_1:NK.9">fdif</a> (<font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c7">h</font>)</span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p3">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">x</font></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E3:21"/>
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S2" onmouseover="rs('nat_1/S2')" onmouseout="rh()">NAT_1:sch 2</a>(<a class="txt" href="#E2:21"><span class="lab"><font color="Green" title="E13">A6</font></span></a>, <a class="txt" href="#E1:21"><span class="lab"><font color="Green" title="E12">A1</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E4:21"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="vsdiff_1.html#NK9" title="VSDIFF_1:NK.9">fdif</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="vsdiff_1.html#K1" title="VSDIFF_1:func.1">(#)</a> <font color="Maroon" title="c5">f</font></span>)</span>,<font color="Maroon" title="c7">h</font>)</span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">r</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="vsdiff_1.html#NK9" title="VSDIFF_1:NK.9">fdif</a> (<font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c7">h</font>)</span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">x</font></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
