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<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be  <a href="relat_1.html#V2">V2</a>  <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="pboole.html#M3">ManySortedFunction</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>;<br/>





<b>assume </b><a NAME="E1:18"/><i><font color="Green">E14</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> is <a href="msualg_3.html#V1">"1-1"</a> &amp; <font color="Maroon">c<sub>4</sub></font> is <a href="msualg_3.html#V2">"onto"</a> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:18_1"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> as   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <b>by </b><i><a class="ref" href="pboole.html#D18">PBOOLE:def 18</a></i>;<br/><a NAME="E3:18_1"/>
<font color="Maroon">c<sub>6</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i><a class="txt" href="autalg_1.html#E1:18"><i><font color="Green">E14</font></i></a>, <a class="txt" href="autalg_1.html#E1:18_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="msualg_3.html#T1">MSUALG_3:1</a></i>;<br/><a NAME="E4:18_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K2">"</a>  is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i><a class="ref" href="funct_1.html#T62">FUNCT_1:62</a></i>;<br/><b>hence </b><a NAME="E5:18_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="msualg_3.html#K4">""</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i><a class="txt" href="autalg_1.html#E1:18"><i><font color="Green">E14</font></i></a>, <a class="txt" href="autalg_1.html#E1:18_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="msualg_3.html#D5">MSUALG_3:def 5</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:18"/>
<font color="Maroon">c<sub>4</sub></font> <a href="msualg_3.html#K4">""</a>  is <a href="msualg_3.html#V1">"1-1"</a>
 <b>by </b><i><a class="ref" href="msualg_3.html#T1">MSUALG_3:1</a></i>;<br/>

<b>thus </b><a NAME="E4:18"/>
<font color="Maroon">c<sub>4</sub></font> <a href="msualg_3.html#K4">""</a>  is <a href="msualg_3.html#V2">"onto"</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="msualg_3.html#D3">MSUALG_3:def 3</a><br/>

<b>assume </b><a NAME="E1:18_2"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E2:18_2"/><b>then </b><i><font color="Green">E16</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i><a class="ref" href="pboole.html#D16">PBOOLE:def 16</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> as   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <b>by </b><i><a class="txt" href="autalg_1.html#E1:18_2"><i><font color="Green">E15</font></i></a>, <a class="ref" href="pboole.html#D18">PBOOLE:def 18</a></i>;<br/>
<a NAME="E4:18_2"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>6</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="autalg_1.html#E1:18"><i><font color="Green">E14</font></i></a>, <a class="txt" href="autalg_1.html#E1:18_2"><i><font color="Green">E15</font></i></a>, <a class="ref" href="msualg_3.html#T1">MSUALG_3:1</a></i>;<br/>
<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> = 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="txt" href="autalg_1.html#E2:18_2"><i><font color="Green">E16</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K2">"</a> </span>)</span>
<b>by </b><i><a class="txt" href="autalg_1.html#E4:18_2"><i><font color="Green">E17</font></i></a>, <a class="ref" href="funct_1.html#T55">FUNCT_1:55</a></i>
;<br/>
<b>hence </b><a NAME="E6:18_2"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="msualg_3.html#K4">""</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="autalg_1.html#E1:18"><i><font color="Green">E14</font></i></a>, <a class="txt" href="autalg_1.html#E1:18_2"><i><font color="Green">E15</font></i></a>, <a class="ref" href="msualg_3.html#D5">MSUALG_3:def 5</a></i>;<br/>


</div><b>end;</b></div>


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