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<b>let </b><font color="Maroon">f</font> be    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">X</font>, <font color="Maroon">g</font> be    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">X</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:38_1_2"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( ( <font color="Maroon">b1</font> <a href="seq_1.html#K1">.</a> <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b2</font> <a href="seq_1.html#K1">.</a> <font color="Olive">i</font> implies <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b2</font> <a href="seq_1.html#K1">.</a> <font color="Olive">i</font> ) &amp; ( <font color="Maroon">b1</font> <a href="seq_1.html#K1">.</a> <font color="Olive">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">b2</font> <a href="seq_1.html#K1">.</a> <font color="Olive">i</font> implies <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b1</font> <a href="seq_1.html#K1">.</a> <font color="Olive">i</font> ) )
 <b>and </b><br/><a NAME="E2:38_1_2"/><i><font color="Green">E47</font></i>: 
for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( ( <font color="Maroon">b1</font> <a href="seq_1.html#K1">.</a> <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b2</font> <a href="seq_1.html#K1">.</a> <font color="Olive">i</font> implies <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b2</font> <a href="seq_1.html#K1">.</a> <font color="Olive">i</font> ) &amp; ( <font color="Maroon">b1</font> <a href="seq_1.html#K1">.</a> <font color="Olive">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">b2</font> <a href="seq_1.html#K1">.</a> <font color="Olive">i</font> implies <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b1</font> <a href="seq_1.html#K1">.</a> <font color="Olive">i</font> ) )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:38_1_2_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_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="E1:38_1_2_1_1"/>
( <font color="Maroon">b1</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b2</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">i</font> or <font color="Maroon">b1</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">b2</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">i</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:38_1_2_1_1_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">b1</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b2</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">i</font>
 ;<br/><div class="add"><b>hence </b><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> = 
<font color="Maroon">b2</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i>, </i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:38_1_2_1_1_2"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">b1</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">b2</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">i</font>
 ;<br/><div class="add"><b>hence </b><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> = 
<font color="Maroon">b1</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i>, </i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:38_1_2"/>
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font>
 <b>by </b><i><a class="ref" href="pboole.html#T3">PBOOLE:3</a></i>;<br/>


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