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<b>let </b><font color="Maroon" title="c1">k</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">S</font> being  <a href="normsp_1.html#NM1" title="NORMSP_1:NM.1">RealNormSpace</a><br/> for <font color="Olive" title="b2">seq</font> being  <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Olive" title="b1">S</font>  st <font color="Olive" title="b2">seq</font> <a href="ndiff_1.html#NR1" title="NDIFF_1:NR.1">is_not_0</a>  holds <br/><font color="Olive" title="b2">seq</font> <a href="lopban_3.html#K2" title="LOPBAN_3:func.2">^\</a> <font color="Maroon" title="c1">k</font> <a href="ndiff_1.html#NR1" title="NDIFF_1:NR.1">is_not_0</a> </span><br/><b>let </b><font color="Maroon" title="c2">S</font> be   <a href="normsp_1.html#NM1" title="NORMSP_1:NM.1">RealNormSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">seq</font> being  <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c2">S</font>  st <font color="Olive" title="b1">seq</font> <a href="ndiff_1.html#NR1" title="NDIFF_1:NR.1">is_not_0</a>  holds <br/><font color="Olive" title="b1">seq</font> <a href="lopban_3.html#K2" title="LOPBAN_3:func.2">^\</a> <font color="Maroon" title="c1">k</font> <a href="ndiff_1.html#NR1" title="NDIFF_1:NR.1">is_not_0</a> </span><br/><b>let </b><font color="Maroon" title="c3">seq</font> be   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c2">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">seq</font> <a href="ndiff_1.html#NR1" title="NDIFF_1:NR.1">is_not_0</a>  implies <font color="Maroon" title="c3">seq</font> <a href="lopban_3.html#K2" title="LOPBAN_3:func.2">^\</a> <font color="Maroon" title="c1">k</font> <a href="ndiff_1.html#NR1" title="NDIFF_1:NR.1">is_not_0</a>  )</span><br/>





<b>assume </b><a NAME="E1:20"/><i><font color="Green" title="E17">A1</font></i>: 
<font color="Maroon" title="c3">seq</font> <a href="ndiff_1.html#NR1" title="NDIFF_1:NR.1">is_not_0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">seq</font> <a href="lopban_3.html#K2" title="LOPBAN_3:func.2">^\</a> <font color="Maroon" title="c1">k</font> <a href="ndiff_1.html#NR1" title="NDIFF_1:NR.1">is_not_0</a> </span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">n</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c3">seq</font> <a href="lopban_3.html#K2" title="LOPBAN_3:func.2">^\</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c4">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">S</font></span><br/><a NAME="E1:20_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">seq</font> <a href="lopban_3.html#K2" title="LOPBAN_3:func.2">^\</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">seq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c1">k</font></span>)</span>
 <b>by </b><i><a class="ref" href="lopban_3.html#D6" title="LOPBAN_3:def.6">LOPBAN_3:def 6</a></i>;<br/><b>hence </b><a NAME="E2:20_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">seq</font> <a href="lopban_3.html#K2" title="LOPBAN_3:func.2">^\</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c4">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">S</font>
 <b>by </b><i><a class="txt" href="ndiff_1.html#E1:20"><i><font color="Green" title="E17">A1</font></i></a>, <a class="ref" href="ndiff_1.html#T7" target="_self" title="NDIFF_1:th.7">Th7</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:20"/>
<font color="Maroon" title="c3">seq</font> <a href="lopban_3.html#K2" title="LOPBAN_3:func.2">^\</a> <font color="Maroon" title="c1">k</font> <a href="ndiff_1.html#NR1" title="NDIFF_1:NR.1">is_not_0</a> 
 <b>by </b><i><a class="ref" href="ndiff_1.html#T7" target="_self" title="NDIFF_1:th.7">Th7</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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