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<span class="kw">set </span><font color="Maroon" title="c1">seq</font> =  <a href="funct_3.html#K6" title="FUNCT_3:func.6">incl</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c2">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="funct_3.html#K6" title="FUNCT_3:func.6">incl</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="funct_3.html#K6" title="FUNCT_3:func.6">incl</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><br/><a NAME="E1:21_1"/>
( <span class="p1">(<span class="default"><a href="funct_3.html#K6" title="FUNCT_3:func.6">incl</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font> &amp; <span class="p1">(<span class="default"><a href="funct_3.html#K6" title="FUNCT_3:func.6">incl</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T18" title="FUNCT_1:th.18">FUNCT_1:18</a></span>;<br/><span class="kw">hence </span><a NAME="E2:21_1"/>
<span class="p1">(<span class="default"><a href="funct_3.html#K6" title="FUNCT_3:func.6">incl</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="funct_3.html#K6" title="FUNCT_3:func.6">incl</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:21"/>
(  <a href="funct_3.html#K6" title="FUNCT_3:func.6">incl</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> is  <a href="valued_0.html#V7" title="VALUED_0:attr.7">increasing</a>  &amp;  <a href="funct_3.html#K6" title="FUNCT_3:func.6">incl</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> is  <a href="valued_0.html#V6" title="VALUED_0:attr.6">natural-valued</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="seqm_3.html#E10"><span class="lab"><font color="Green" title="E10">Lm6</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/>


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