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<b>let </b><font color="Maroon" title="c2">f</font>, <font color="Maroon" title="c3">g</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:37_1_2"/><i><font color="Green" title="E26">A24</font></i>: 
for <font color="Olive" title="b1">h</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> holds <br/> ( <font color="Maroon" title="c2">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">h</font>,0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c1">G</font> &amp; ( for <font color="Olive" title="b2">n</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>  holds  <font color="Maroon" title="c2">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">h</font>,<span class="p1">(<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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">h</font>,<font color="Olive" title="b2">n</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b1">h</font> ) )
 <b>and </b><br/><a NAME="E2:37_1_2"/><i><font color="Green" title="E27">A25</font></i>: 
for <font color="Olive" title="b1">h</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> holds <br/> ( <font color="Maroon" title="c3">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">h</font>,0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c1">G</font> &amp; ( for <font color="Olive" title="b2">n</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>  holds  <font color="Maroon" title="c3">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">h</font>,<span class="p1">(<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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">h</font>,<font color="Olive" title="b2">n</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b1">h</font> ) )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">h</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font>;<br/><b>let </b><font color="Maroon" title="c5">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> ;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <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> ] <b>means </b><font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">h</font>,$1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">h</font>,$1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>;<br/><font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">h</font>,0</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> = 
<font color="Maroon" title="c2">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c4">h</font>,0

<br/>.= 
 <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c1">G</font>
<b>by </b><i><a class="txt" href="group_1.html#E1:37_1_2"><i><font color="Green" title="E26">A24</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c3">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c4">h</font>,0
<b>by </b><i><a class="txt" href="group_1.html#E2:37_1_2"><i><font color="Green" title="E27">A25</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">h</font>,0</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>

;<br/><a NAME="E2:37_1_2_1"/><b>then </b><i><font color="Green" title="E28">A26</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 ;<br/><div><i><font color="Green" title="E29">A27</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_2_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">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> ;<br/><b>assume </b><a NAME="E1:37_1_2_1_2"/><i><font color="Green" title="E30">A28</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">n</font>]
 ;<br/><a NAME="E2:37_1_2_1_2"/><i><font color="Green" title="E31">A29</font></i>: 
( <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">h</font>,<font color="Maroon" title="c6">n</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c4">h</font>,<font color="Maroon" title="c6">n</font> &amp; <font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">h</font>,<font color="Maroon" title="c6">n</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c4">h</font>,<font color="Maroon" title="c6">n</font> )
 ;<br/><font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">h</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> = 
<font color="Maroon" title="c2">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c4">h</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c4">h</font>,<font color="Maroon" title="c6">n</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c4">h</font>
<b>by </b><i><a class="txt" href="group_1.html#E1:37_1_2"><i><font color="Green" title="E26">A24</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c3">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c4">h</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="group_1.html#E2:37_1_2"><i><font color="Green" title="E27">A25</font></i></a>, <a class="txt" href="group_1.html#E1:37_1_2_1_2"><i><font color="Green" title="E30">A28</font></i></a>, <a class="txt" href="group_1.html#E2:37_1_2_1_2"><i><font color="Green" title="E31">A29</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">h</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>

;<br/><b>hence </b><a NAME="E4:37_1_2_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 ;<br/></div><b>end;</b></div><a NAME="E4:37_1_2_1"/>
for <font color="Olive" title="b1">n</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>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="group_1.html#E2:37_1_2_1"><i><font color="Green" title="E28">A26</font></i></a>, <a class="txt" href="group_1.html#E3:37_1_2_1"><i><font color="Green" title="E29">A27</font></i></a>);<br/></i><b>hence </b><a NAME="E5:37_1_2_1"/>
<font color="Maroon" title="c2">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c4">h</font>,<font color="Maroon" title="c5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c4">h</font>,<font color="Maroon" title="c5">n</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:37_1_2"/>
<font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g</font>
 <b>by </b><i><a class="ref" href="binop_1.html#T2" title="BINOP_1:th.2">BINOP_1:2</a></i>;<br/>


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