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<b>consider </b><font color="Maroon">c<sub>1</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E2:25_1"/><i><font color="Green">E16</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a> 
 <b>and </b><br/><a NAME="E3:25_1"/><i><font color="Green">E17</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>() &amp; <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>() &amp; <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>() &amp; <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> 3 <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>5</sub></font>() )
 <b>and </b><br/><a NAME="E4:25_1"/><i><font color="Green">E18</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>6</sub></font>(<font color="Olive">b<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 3</span>)</span></span>)</span>)
 <b>from </b><i><a class="ref" href="recdef_2.html#S12" target="_self">RECDEF_2:sch 12</a>();<br/></i>
<a NAME="E5:25_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F<sub>1</sub></font>()
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:25_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>3</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:25_1_1"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E4:25_1_1"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="recdef_2.html#E2:25_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="recdef_2.html#E3:25_1_1"><i><font color="Green">E19</font></i></a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:25_1_1_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 3 or <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 3 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1_1_1"><b>suppose </b></a><a NAME="E1:25_1_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 3
 ;<br/><div class="add"><a NAME="E2:25_1_1_1_1"/><b>then </b>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 2 or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 3 )
 <b>by </b><i><a class="ref" href="cqc_the1.html#T4">CQC_THE1:4</a></i>;<br/><b>hence </b><a NAME="E3:25_1_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>()
 <b>by </b><i><a class="txt" href="recdef_2.html#E3:25_1"><i><font color="Green">E17</font></i></a>, <a class="txt" href="recdef_2.html#E4:25_1_1"><i><font color="Green">E20</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1_1_2"><b>suppose </b></a><a NAME="E1:25_1_1_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 3
 ;<br/><div class="add"><a NAME="E2:25_1_1_1_2"/><b>then </b>
3 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:25_1_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="xcmplx_0.html#K6">-</a> 4 is   <a href="nat_1.html#NM1">Nat</a>
 <b>by </b><i><a class="ref" href="int_1.html#T18">INT_1:18</a></i>;<br/><a NAME="E4:25_1_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="xcmplx_0.html#K6">-</a> 4</span>)</span> <a href="xcmplx_0.html#K2">+</a> 4</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>6</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="xcmplx_0.html#K6">-</a> 4</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="xcmplx_0.html#K6">-</a> 4</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="xcmplx_0.html#K6">-</a> 4</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="xcmplx_0.html#K6">-</a> 4</span>)</span> <a href="xcmplx_0.html#K2">+</a> 2</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="xcmplx_0.html#K6">-</a> 4</span>)</span> <a href="xcmplx_0.html#K2">+</a> 3</span>)</span></span>)</span>)
 <b>by </b><i><a class="txt" href="recdef_2.html#E4:25_1"><i><font color="Green">E18</font></i></a></i>;<br/><b>hence </b><a NAME="E5:25_1_1_1_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>()
 <b>by </b><i><a class="txt" href="recdef_2.html#E4:25_1_1"><i><font color="Green">E20</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E6:25_1"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> is   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> ,<font color="Maroon">F<sub>1</sub></font>()
 
<b>by </b><i><a class="txt" href="recdef_2.html#E2:25_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T10">RELSET_1:10</a></i>;<br/>
<b>hence </b><a NAME="E7:25_1"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> ,<font color="Maroon">F<sub>1</sub></font>() st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>() &amp; <font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K3">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>() &amp; <font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K3">.</a> 2 <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>() &amp; <font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K3">.</a> 3 <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>5</sub></font>() &amp; ( for <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 4</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>6</sub></font>(<font color="Olive">b<sub>2</sub></font>,<span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 3</span>)</span></span>)</span>) ) )
 <b>by </b><i><a class="txt" href="recdef_2.html#E3:25_1"><i><font color="Green">E17</font></i></a>, <a class="txt" href="recdef_2.html#E4:25_1"><i><font color="Green">E18</font></i></a></i>;<br/>


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