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<b>let </b><font color="Maroon">IT1</font> be  <a href="seqm_3.html#V6">natural-yielding</a> <a href="funct_1.html#NM1">Function</a>, <font color="Maroon">IT2</font> be  <a href="seqm_3.html#V6">natural-yielding</a> <a href="funct_1.html#NM1">Function</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:19_1_2"/><i><font color="Green">E48</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">IT1</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> &amp; ( for <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( ( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> implies <font color="Maroon">IT1</font> <a href="polynom1.html#K8">.</a> <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="polynom1.html#K8">.</a> <font color="Olive">y</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">k</font> ) &amp; ( not <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> implies <font color="Maroon">IT1</font> <a href="polynom1.html#K8">.</a> <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="polynom1.html#K8">.</a> <font color="Olive">y</font> ) ) ) )
 <b>and </b><br/><a NAME="E2:19_1_2"/><i><font color="Green">E49</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">IT2</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> &amp; ( for <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( ( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> implies <font color="Maroon">IT2</font> <a href="polynom1.html#K8">.</a> <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="polynom1.html#K8">.</a> <font color="Olive">y</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">k</font> ) &amp; ( not <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> implies <font color="Maroon">IT2</font> <a href="polynom1.html#K8">.</a> <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="polynom1.html#K8">.</a> <font color="Olive">y</font> ) ) ) )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:19_1_2_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">IT1</font>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:19_1_2_1_1"/>
( ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> ) or  not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> )
 </span><b>by </b><i>, </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:19_1_2_1_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> )
 ;<br/><div class="add"><a NAME="E2:19_1_2_1_1_1"/><b>then </b>
( <font color="Maroon">IT1</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">x</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">k</font> &amp; <font color="Maroon">IT2</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">x</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">k</font> )
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:19_1_2_1_1_1"/>
<font color="Maroon">IT1</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">IT2</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">x</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:19_1_2_1_1_2"/>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 ;<br/><div class="add"><a NAME="E2:19_1_2_1_1_2"/><b>then </b>
( <font color="Maroon">IT1</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">x</font> &amp; <font color="Maroon">IT2</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">x</font> )
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:19_1_2_1_1_2"/>
<font color="Maroon">IT1</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">IT2</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">x</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:19_1_2"/>
<font color="Maroon">IT1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">IT2</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


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