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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="osalg_1.html#NM3">OrderSortedSign</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be  <a href="msualg_1.html#V5">non-empty</a> <a href="osalg_1.html#NM5">OSAlgebra</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="msualg_3.html#NM1">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="E1:17"/><i><font color="Green">E13</font></i>: 
<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="osalg_1.html#NM2">OperSymbol</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E2:17"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="osalg_1.html#R6">&lt;=</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>7</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>8</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/>



<b>assume </b><a NAME="E3:17"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/>

<a NAME="E4:17"/><i><font color="Green">E16</font></i>: 
(  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> &amp;  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="msualg_3.html#T6">MSUALG_3:6</a></i>;<br/>
<a NAME="E5:17"/><b>then </b><i><font color="Green">E17</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="msualg_3.html#K6">#</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="ref" href="msualg_3.html#T6">MSUALG_3:6</a></i>;<br/>
<a NAME="E6:17"/><i><font color="Green">E18</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="msualg_3.html#K6">#</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="osalg_3.html#E4:17"><i><font color="Green">E16</font></i></a>, <a class="ref" href="msualg_3.html#T6">MSUALG_3:6</a></i>;<br/>
<a NAME="E7:17"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>7</sub></font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="msualg_3.html#K6">#</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</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#K6">#</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:17_1"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>9</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="osalg_3.html#E1:17_1"><i><font color="Green">E19</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>12</sub></font> = <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>10</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E4:17_1"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 
;<br/>
<a NAME="E5:17_1"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E6:17_1"/><b>then </b><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="osalg_3.html#E4:17"><i><font color="Green">E16</font></i></a>, <a class="txt" href="osalg_3.html#E1:17_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>;<br/>
 <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>11</sub></font></span>)</span> = 
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font>
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="osalg_3.html#E4:17"><i><font color="Green">E16</font></i></a>, <a class="txt" href="osalg_3.html#E1:17_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="txt" href="osalg_3.html#E4:17"><i><font color="Green">E16</font></i></a>, <a class="txt" href="osalg_3.html#E1:17_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
;<br/>
<a NAME="E8:17_1"/><b>then </b><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</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>11</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="osalg_3.html#E3:17"><i><font color="Green">E15</font></i></a>, <a class="txt" href="osalg_3.html#E6:17_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="msualg_3.html#T6">MSUALG_3:6</a></i>;<br/>
<a NAME="E9:17_1"/><i><font color="Green">E22</font></i>: 
(  <a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>5</sub></font> <a href="osalg_1.html#R2">&lt;=</a>  <a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>6</sub></font> &amp;  <a href="osalg_1.html#K1">the_result_sort_of</a> <font color="Maroon">c<sub>5</sub></font> <a href="orders_2.html#R3">&lt;=</a>  <a href="osalg_1.html#K1">the_result_sort_of</a> <font color="Maroon">c<sub>6</sub></font> )
 
<b>by </b><i><a class="txt" href="osalg_3.html#E2:17"><i><font color="Green">E14</font></i></a>, <a class="ref" href="osalg_1.html#D22">OSALG_1:def 22</a></i>;<br/>
<a NAME="E10:17_1"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="osalg_1.html#D7">OSALG_1:def 7</a></i>;<br/>
<a NAME="E11:17_1"/><b>then </b>
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E12:17_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> &amp; <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> )
 
<b>by </b><i><a class="txt" href="osalg_3.html#E4:17"><i><font color="Green">E16</font></i></a>, <a class="txt" href="osalg_3.html#E1:17_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E13:17_1"/><b>then </b><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="txt" href="osalg_3.html#E4:17"><i><font color="Green">E16</font></i></a>, <a class="txt" href="osalg_3.html#E1:17_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="osalg_3.html#E9:17_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="osalg_1.html#D7">OSALG_1:def 7</a></i>;<br/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="msualg_3.html#K6">#</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="msualg_3.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="osalg_3.html#E3:17"><i><font color="Green">E15</font></i></a>, <a class="txt" href="osalg_3.html#E1:17_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="msualg_3.html#D8">MSUALG_3:def 8</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>12</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="osalg_3.html#E1:17"><i><font color="Green">E13</font></i></a>, <a class="txt" href="osalg_3.html#E8:17_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="osalg_3.html#E13:17_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="osalg_3.html#D1" target="_self">Def1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="msualg_3.html#K6">#</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="txt" href="osalg_3.html#E3:17"><i><font color="Green">E15</font></i></a>, <a class="txt" href="osalg_3.html#E1:17_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="msualg_3.html#D8">MSUALG_3:def 8</a></i>
;<br/>
<b>hence </b><a NAME="E15:17_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="msualg_3.html#K6">#</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</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#K6">#</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:17"/>
<font color="Maroon">c<sub>4</sub></font> <a href="msualg_3.html#K6">#</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="msualg_3.html#K6">#</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="osalg_3.html#E3:17"><i><font color="Green">E15</font></i></a>, <a class="txt" href="osalg_3.html#E5:17"><i><font color="Green">E17</font></i></a>, <a class="txt" href="osalg_3.html#E6:17"><i><font color="Green">E18</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


</div>
