<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="orders_2.html#NM1">Poset</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="osalg_1.html#NM4">OrderSortedSet</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="E1:8"/><i><font color="Green">E7</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> &amp; <font color="Maroon">c<sub>4</sub></font> is <a href="osalg_3.html#V1" target="_self">order-sorted</a> )
 ;<br/>

<b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="osalg_3.html#D1" target="_self">OSALG_3:def 1</a><br/>

<b>assume </b><a NAME="E2:8"/><i><font color="Green">E8</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/>

<a NAME="E3:8"/><i><font color="Green">E9</font></i>: 
( <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="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> &amp; <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="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> )
 
<b>by </b><i><a class="txt" href="osalg_3.html#E2:8"><i><font color="Green">E8</font></i></a>, <a class="ref" href="osalg_1.html#D18">OSALG_1:def 18</a></i>;<br/>
<a NAME="E4:8"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> )
 
;<br/>
<a NAME="E5:8"/><b>then </b><i><font color="Green">E10</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<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="E6:8"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</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="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 ;<br/>

<a NAME="E7:8"/><i><font color="Green">E12</font></i>: 
(  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <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>5</sub></font> &amp;  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <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>6</sub></font> )
 
<b>by </b><i><a class="txt" href="osalg_3.html#E1:8"><i><font color="Green">E7</font></i></a>, <a class="ref" href="msualg_3.html#D3">MSUALG_3:def 3</a></i>;<br/>
<a NAME="E8:8"/><i><font color="Green">E13</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> is <a href="funct_1.html#V2">one-to-one</a> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="msualg_3.html#K1">.</a> <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="osalg_3.html#E1:8"><i><font color="Green">E7</font></i></a>, <a class="txt" href="osalg_3.html#E5:8"><i><font color="Green">E10</font></i></a>, <a class="ref" href="msualg_3.html#D2">MSUALG_3:def 2</a></i>;<br/>
<a NAME="E9:8"/><b>then </b><i><font color="Green">E14</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K2">"</a>  is   <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>5</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_1.html#K2">"</a>  is   <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>6</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> )
 
<b>by </b><i><a class="txt" href="osalg_3.html#E7:8"><i><font color="Green">E12</font></i></a>, <a class="ref" href="funct_2.html#T31">FUNCT_2:31</a></i>;<br/>
<a NAME="E10:8"/><i><font color="Green">E15</font></i>: 
( <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="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K2">"</a>  &amp; <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="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_1.html#K2">"</a>  )
 
<b>by </b><i><a class="txt" href="osalg_3.html#E1:8"><i><font color="Green">E7</font></i></a>, <a class="ref" href="msualg_3.html#D5">MSUALG_3:def 5</a></i>;<br/>
<a NAME="E11:8"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <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="osalg_3.html#E6:8"><i><font color="Green">E11</font></i></a></i>;<br/>
<a NAME="E12:8"/><b>then </b><i><font color="Green">E17</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</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="txt" href="osalg_3.html#E3:8"><i><font color="Green">E9</font></i></a></i>;<br/>
<a NAME="E13:8"/><i><font color="Green">E18</font></i>: 
<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#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <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#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="osalg_3.html#E6:8"><i><font color="Green">E11</font></i></a>, <a class="txt" href="osalg_3.html#E10:8"><i><font color="Green">E15</font></i></a>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<a NAME="E14:8"/><i><font color="Green">E19</font></i>: 
 <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#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="tarski.html#R1">c=</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="osalg_3.html#E9:8"><i><font color="Green">E14</font></i></a>, <a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/>
<a NAME="E15:8"/><b>then </b><i><font color="Green">E20</font></i>: 
( <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#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> &amp; <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>  )
 
<b>by </b><i><a class="txt" href="osalg_3.html#E13:8"><i><font color="Green">E18</font></i></a></i>;<br/>
<b>hence </b><a NAME="E16:8"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</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="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="osalg_3.html#E3:8"><i><font color="Green">E9</font></i></a>, <a class="txt" href="osalg_3.html#E12:8"><i><font color="Green">E17</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>

<a NAME="E17:8"/><i><font color="Green">E21</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="msualg_3.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> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>6</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>6</sub></font> )
 
<b>by </b><i><a class="txt" href="osalg_3.html#E3:8"><i><font color="Green">E9</font></i></a>, <a class="txt" href="osalg_3.html#E11:8"><i><font color="Green">E16</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>8</sub></font> = <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#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>9</sub></font> = <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#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font>;<br/>
<a NAME="E18:8"/><i><font color="Green">E22</font></i>: 
( <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#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> &amp; <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#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="osalg_3.html#E3:8"><i><font color="Green">E9</font></i></a>, <a class="txt" href="osalg_3.html#E9:8"><i><font color="Green">E14</font></i></a>, <a class="txt" href="osalg_3.html#E11:8"><i><font color="Green">E16</font></i></a>, <a class="txt" href="osalg_3.html#E15:8"><i><font color="Green">E20</font></i></a>, <a class="txt" href="osalg_3.html#E17:8"><i><font color="Green">E21</font></i></a>, <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> = 
<font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="txt" href="osalg_3.html#E3:8"><i><font color="Green">E9</font></i></a>, <a class="txt" href="osalg_3.html#E7:8"><i><font color="Green">E12</font></i></a>, <a class="txt" href="osalg_3.html#E8:8"><i><font color="Green">E13</font></i></a>, <a class="txt" href="osalg_3.html#E11:8"><i><font color="Green">E16</font></i></a>, <a class="ref" href="funct_1.html#T57">FUNCT_1:57</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="osalg_3.html#E6:8"><i><font color="Green">E11</font></i></a>, <a class="txt" href="osalg_3.html#E7:8"><i><font color="Green">E12</font></i></a>, <a class="txt" href="osalg_3.html#E8:8"><i><font color="Green">E13</font></i></a>, <a class="ref" href="funct_1.html#T57">FUNCT_1:57</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="osalg_3.html#E1:8"><i><font color="Green">E7</font></i></a>, <a class="txt" href="osalg_3.html#E2:8"><i><font color="Green">E8</font></i></a>, <a class="txt" href="osalg_3.html#E13:8"><i><font color="Green">E18</font></i></a>, <a class="txt" href="osalg_3.html#E14:8"><i><font color="Green">E19</font></i></a>, <a class="txt" href="osalg_3.html#E17:8"><i><font color="Green">E21</font></i></a>, <a class="ref" href="osalg_3.html#D1" target="_self">Def1</a></i>
;<br/>
<b>hence </b><a NAME="E20:8"/>
<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="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>4</sub></font> <a href="msualg_3.html#K4">""</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></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="osalg_3.html#E8:8"><i><font color="Green">E13</font></i></a>, <a class="txt" href="osalg_3.html#E10:8"><i><font color="Green">E15</font></i></a>, <a class="txt" href="osalg_3.html#E18:8"><i><font color="Green">E22</font></i></a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/>


</div>
