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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>, <font color="Maroon">c<sub>4</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>5</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>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="pboole.html#M3">ManySortedFunction</a> of <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:22"/><i><font color="Green">E16</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> is <a href="msualg_3.html#V1">"1-1"</a> &amp; <font color="Maroon">c<sub>5</sub></font> is <a href="msualg_3.html#V2">"onto"</a> )
 <b>and </b><br/><a NAME="E2:22"/><i><font color="Green">E17</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> is <a href="msualg_3.html#V1">"1-1"</a> &amp; <font color="Maroon">c<sub>6</sub></font> is <a href="msualg_3.html#V2">"onto"</a> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>7</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:22_1"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>7</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>8</sub></font> = <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</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>7</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <b>by </b><i><a class="ref" href="pboole.html#D18">PBOOLE:def 18</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> as   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <b>by </b><i><a class="txt" href="autalg_1.html#E1:22_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="pboole.html#D18">PBOOLE:def 18</a></i>;<br/><a NAME="E4:22_1"/><i><font color="Green">E19</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="msualg_3.html#K4">""</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K2">"</a> 
 <b>by </b><i><a class="txt" href="autalg_1.html#E1:22"><i><font color="Green">E16</font></i></a>, <a class="txt" href="autalg_1.html#E1:22_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="msualg_3.html#D5">MSUALG_3:def 5</a></i>;<br/><a NAME="E5:22_1"/><i><font color="Green">E20</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="msualg_3.html#K4">""</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K2">"</a> 
 <b>by </b><i><a class="txt" href="autalg_1.html#E2:22"><i><font color="Green">E17</font></i></a>, <a class="txt" href="autalg_1.html#E1:22_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="msualg_3.html#D5">MSUALG_3:def 5</a></i>;<br/><a NAME="E6:22_1"/><i><font color="Green">E21</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="msualg_3.html#K3">**</a> <font color="Maroon">c<sub>5</sub></font> is <a href="msualg_3.html#V1">"1-1"</a> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="msualg_3.html#K3">**</a> <font color="Maroon">c<sub>5</sub></font> is <a href="msualg_3.html#V2">"onto"</a> )
 <b>by </b><i><a class="txt" href="autalg_1.html#E1:22"><i><font color="Green">E16</font></i></a>, <a class="txt" href="autalg_1.html#E2:22"><i><font color="Green">E17</font></i></a>, <a class="ref" href="autalg_1.html#T16" target="_self">Th16</a>, <a class="ref" href="autalg_1.html#T17" target="_self">Th17</a></i>;<br/><a NAME="E7:22_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>8</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i><a class="txt" href="autalg_1.html#E1:22"><i><font color="Green">E16</font></i></a>, <a class="txt" href="autalg_1.html#E1:22_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="msualg_3.html#T1">MSUALG_3:1</a></i>;<br/><a NAME="E8:22_1"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>9</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i><a class="txt" href="autalg_1.html#E2:22"><i><font color="Green">E17</font></i></a>, <a class="txt" href="autalg_1.html#E1:22_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="msualg_3.html#T1">MSUALG_3:1</a></i>;<br/><a NAME="E9:22_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="msualg_3.html#K3">**</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="autalg_1.html#E1:22_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="msualg_3.html#T2">MSUALG_3:2</a></i>;<br/><a NAME="E10:22_1"/><b>then </b><i><font color="Green">E24</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="msualg_3.html#K3">**</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="msualg_3.html#K4">""</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="funct_1.html#K2">"</a> 
 <b>by </b><i><a class="txt" href="autalg_1.html#E1:22_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="autalg_1.html#E6:22_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="msualg_3.html#D5">MSUALG_3:def 5</a></i>;<br/><a NAME="E11:22_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="autalg_1.html#E1:22"><i><font color="Green">E16</font></i></a>, <a class="txt" href="autalg_1.html#E1:22_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="msualg_3.html#D3">MSUALG_3:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>10</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="msualg_3.html#K4">""</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> as   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <b>by </b><i><a class="txt" href="autalg_1.html#E4:22_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="autalg_1.html#E7:22_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="funct_2.html#T31">FUNCT_2:31</a></i>;<br/><a NAME="E13:22_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="autalg_1.html#E2:22"><i><font color="Green">E17</font></i></a>, <a class="txt" href="autalg_1.html#E1:22_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="msualg_3.html#D3">MSUALG_3:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>11</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="msualg_3.html#K4">""</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> as   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <b>by </b><i><a class="txt" href="autalg_1.html#E5:22_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="autalg_1.html#E8:22_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="funct_2.html#T31">FUNCT_2:31</a></i>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="msualg_3.html#K4">""</a> </span>)</span> <a href="msualg_3.html#K3">**</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="msualg_3.html#K4">""</a> </span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> = 
<font color="Maroon">c<sub>10</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>11</sub></font>
<b>by </b><i><a class="txt" href="autalg_1.html#E1:22_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="msualg_3.html#T2">MSUALG_3:2</a></i>
<br/>.= 
<font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K2">"</a> </span>)</span>
<b>by </b><i><a class="txt" href="autalg_1.html#E2:22"><i><font color="Green">E17</font></i></a>, <a class="txt" href="autalg_1.html#E1:22_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="msualg_3.html#D5">MSUALG_3:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K2">"</a> </span>)</span>
<b>by </b><i><a class="txt" href="autalg_1.html#E1:22"><i><font color="Green">E16</font></i></a>, <a class="txt" href="autalg_1.html#E1:22_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="msualg_3.html#D5">MSUALG_3:def 5</a></i>
;<br/><b>hence </b><a NAME="E16:22_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="msualg_3.html#K3">**</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="msualg_3.html#K4">""</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="msualg_3.html#K4">""</a> </span>)</span> <a href="msualg_3.html#K3">**</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="msualg_3.html#K4">""</a> </span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="autalg_1.html#E7:22_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="autalg_1.html#E8:22_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="autalg_1.html#E10:22_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="funct_1.html#T66">FUNCT_1:66</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:22"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="msualg_3.html#K3">**</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="msualg_3.html#K4">""</a>  <a href="pboole.html#R6">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="msualg_3.html#K4">""</a> </span>)</span> <a href="msualg_3.html#K3">**</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="msualg_3.html#K4">""</a> </span>)</span>
 <b>by </b><i><a class="ref" href="pboole.html#T3">PBOOLE:3</a></i>;<br/>


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