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<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">L</font> be   <a href="lattices.html#NM1">Lattice</a>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>;<br/><b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>







<a NAME="E1:26"/><i><font color="Green">E17</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E2:26"/><b>then </b><i><font color="Green">E21</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b><span class="p1">(<span class="default"><a href="abian.html#K1">iter</a> <font color="Maroon">f</font>,<font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font>,<font color="Maroon">a<sub>1</sub></font> <a href="knaster.html#K4" target="_self">-.</a> <font color="Maroon">x</font>;<br/>
<span class="p1">(<span class="default"><a href="abian.html#K1">iter</a> <font color="Maroon">f</font>,0</span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="funct_7.html#T70">FUNCT_7:70</a></i>
<br/>.= 
<font color="Maroon">x</font>
<b>by </b><i><a class="txt" href="knaster.html#E2:26"><i><font color="Green">E21</font></i></a>, <a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>
<br/>.= 
<font color="Maroon">f</font>,0 <a href="knaster.html#K4" target="_self">-.</a> <font color="Maroon">x</font>
<b>by </b><i/>
;<br/>
<a NAME="E4:26"/><b>then </b><i><font color="Green">E22</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
;<br/>
<div><i><font color="Green">E23</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="26_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:26_2"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>]
 ;<br/><a NAME="E2:26_2"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 ;<br/><a NAME="E3:26_2"/><b>then </b><i><font color="Green">E25</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="abian.html#K1">iter</a> <font color="Maroon">f</font>,<font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 <b>by </b><i>, <a class="ref" href="funct_7.html#T76">FUNCT_7:76</a></i>;<br/><span class="p1">(<span class="default"><a href="abian.html#K1">iter</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="abian.html#K1">iter</a> <font color="Maroon">f</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="funct_7.html#T73">FUNCT_7:73</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font>,<font color="Maroon">n</font> <a href="knaster.html#K4" target="_self">-.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i>, <a class="txt" href="knaster.html#E1:26_2"><i><font color="Green">E24</font></i></a>, <a class="txt" href="knaster.html#E3:26_2"><i><font color="Green">E25</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">f</font>,<span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">n</font></span>)</span> <a href="knaster.html#K4" target="_self">-.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="txt" href="knaster.html#E1:26"><i><font color="Green">E17</font></i></a></i>
<br/>.= 
<font color="Maroon">f</font>,<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="knaster.html#K4" target="_self">-.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="card_1.html#T52">CARD_1:52</a></i>
;<br/><b>hence </b><a NAME="E5:26_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1]
 ;<br/></div><b>end;</b></div>
<a NAME="E6:26"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="knaster.html#E4:26"><i><font color="Green">E22</font></i></a>, <a class="txt" href="knaster.html#E5:26"><i><font color="Green">E23</font></i></a>);<br/></i>
<b>hence </b><a NAME="E7:26"/>
<span class="p1">(<span class="default"><a href="abian.html#K1">iter</a> <font color="Maroon">f</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font>,<font color="Maroon">n</font> <a href="knaster.html#K4" target="_self">-.</a> <font color="Maroon">x</font>
 ;<br/>


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