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<b>let </b><font color="Maroon">D1</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>, <font color="Maroon">D2</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:30_2_2"/><i><font color="Green">E28</font></i>: 
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D1</font> iff ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp;  ex <font color="Olive">y</font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="mesfunc1.html#K4" target="_self">.</a> <font color="Olive">x</font> &amp; <font color="Olive">y</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> ) ) )
 <b>and </b><br/><a NAME="E2:30_2_2"/><i><font color="Green">E34</font></i>: 
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D2</font> iff ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp;  ex <font color="Olive">y</font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="mesfunc1.html#K4" target="_self">.</a> <font color="Olive">x</font> &amp; <font color="Olive">y</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> ) ) )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>;<br/><b>thus </b><a NAME="E1:30_2_2_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D1</font> implies <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D2</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2_2_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:30_2_2_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D1</font>
 ;<br/>

<a NAME="E2:30_2_2_1_1"/><b>then </b>
( <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> &amp;  ex <font color="Olive">y</font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="mesfunc1.html#K4" target="_self">.</a> <font color="Maroon">x</font> &amp; <font color="Olive">y</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> ) )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:30_2_2_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D2</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><b>assume </b><a NAME="E2:30_2_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D2</font>
 ;<br/><a NAME="E3:30_2_2_1"/><b>then </b>
( <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> &amp;  ex <font color="Olive">y</font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="mesfunc1.html#K4" target="_self">.</a> <font color="Maroon">x</font> &amp; <font color="Olive">y</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> ) )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:30_2_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D1</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:30_2_2"/>
<font color="Maroon">D1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D2</font>
 <b>by </b><i><a class="ref" href="subset_1.html#T8">SUBSET_1:8</a></i>;<br/>


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