<?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> non <a href="msualg_1.html#V2">void</a>  <a href="msualg_1.html#L1">ManySortedSign</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>





<b>assume </b><a NAME="E1:34"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="msualg_2.html#K14">MSSub</a> <font color="Maroon">c<sub>2</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> </span>}</span> 
 ;<br/>

<b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="pralg_2.html#M2">MSAlgebra-Family</a> of <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E2:34"/><i><font color="Green">E19</font></i>: 
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> <font color="Maroon">c<sub>3</sub></font> holds <br/><font color="Olive">b<sub>1</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="Olive">b<sub>1</sub></font>
 ;<br/>

<b>set </b><font color="Maroon">c<sub>5</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>
<div><i><font color="Green">E20</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>6</sub></font> be  <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><a NAME="E1:34_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="msualg_2.html#K14">MSSub</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="msualg_2.html#D20">MSUALG_2:def 20</a></i>;<br/><b>hence </b><a NAME="E2:34_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="equation.html#E1:34"><i><font color="Green">E18</font></i></a></i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>3</sub></font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<b>consider </b><font color="Maroon">c<sub>7</sub></font> being  <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>4</sub></font> as    <a href="pralg_2.html#M2">MSAlgebra-Family</a> of <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>1</sub></font> ;<br/>
<b>set </b><font color="Maroon">c<sub>9</sub></font> =  <a href="pralg_2.html#K15">product</a> <font color="Maroon">c<sub>8</sub></font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] means ex <font color="Olive">b<sub>1</sub></font> being  <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">c<sub>1</sub></font>ex <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K11">SORTS</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>ex <font color="Olive">b<sub>3</sub></font> being <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>2</sub></font> &amp; ( for <font color="Olive">b<sub>4</sub></font> being <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>3</sub></font> is    <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Olive">b<sub>4</sub></font> holds <br/><font color="Olive">b<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>4</sub></font> ) );<br/>
<b>consider </b><font color="Maroon">c<sub>10</sub></font> being    <a href="pboole.html#M1">ManySortedSet</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E8:34"/><i><font color="Green">E21</font></i>: 
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> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> iff ( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K11">SORTS</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> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>] ) )
 <b>from </b><i><a class="ref" href="pboole.html#S2">PBOOLE:sch 2</a>();<br/></i>
<a NAME="E9:34"/>
<font color="Maroon">c<sub>10</sub></font> is   <a href="msualg_2.html#NM1">MSSubset</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>11</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="pboole.html#D5">PBOOLE:def 5</a>,<a class="ref" href="pboole.html#D23">PBOOLE:def 23</a><br/>

<b>assume </b><a NAME="E1:34_2"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>11</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/>

<b>let </b><font color="Maroon">c<sub>12</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:34_2"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font>
 ;<br/>

<a NAME="E3:34_2"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K11">SORTS</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="txt" href="equation.html#E8:34"><i><font color="Green">E21</font></i></a>, <a class="txt" href="equation.html#E1:34_2"><i><font color="Green">E22</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:34_2"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>10</sub></font> as   <a href="msualg_2.html#NM1">MSSubset</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> ;<br/>
<a NAME="E11:34"/>
<font color="Maroon">c<sub>11</sub></font> is <a href="msualg_2.html#V3">opers_closed</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>12</sub></font> be   <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">c<sub>1</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="msualg_2.html#D7">MSUALG_2:def 7</a><br/><b>let </b><font color="Maroon">c<sub>13</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a>,<a class="ref" href="msualg_2.html#D6">MSUALG_2:def 6</a><br/>



<b>assume </b><a NAME="E1:34_3"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">c<sub>12</sub></font>,<span class="p3">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="pboole.html#K6">#</a> </span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span>
 ;<br/>

<b>set </b><font color="Maroon">c<sub>14</sub></font> =  <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">c<sub>12</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>15</sub></font> =  <a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>12</sub></font>;<br/>
<a NAME="E2:34_3"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="pboole.html#R2">c=</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="pboole.html#D23">PBOOLE:def 23</a></i>;<br/>
<a NAME="E3:34_3"/>
the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> is   <a href="msualg_2.html#NM1">MSSubset</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="pboole.html#D23">PBOOLE:def 23</a></i>;<br/>
<a NAME="E4:34_3"/><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>11</sub></font> <a href="pboole.html#K6">#</a> </span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <span class="p3">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="pboole.html#K6">#</a> </span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="txt" href="equation.html#E2:34_3"><i><font color="Green">E23</font></i></a>, <a class="ref" href="msualg_2.html#T3">MSUALG_2:3</a></i>;<br/>
<a NAME="E5:34_3"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="pboole.html#K6">#</a> </span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="tarski.html#R1">c=</a>  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">c<sub>12</sub></font>,<span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 
;<br/>
<a NAME="E6:34_3"/><b>then </b><i><font color="Green">E25</font></i>: 
<span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">c<sub>12</sub></font>,<span class="p2">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="pboole.html#K6">#</a> </span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> is   <a href="funct_2.html#NM1">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="pboole.html#K6">#</a> </span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font>, <a href="msualg_1.html#K4">Result</a> <font color="Maroon">c<sub>12</sub></font>,<span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#T38">FUNCT_2:38</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>13</sub></font> as    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K11">SORTS</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <b>by </b><i><a class="txt" href="equation.html#E1:34_3"><i><font color="Green">E22</font></i></a>, <a class="ref" href="pralg_2.html#T10">PRALG_2:10</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>17</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E9:34_3"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">c<sub>17</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"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">c<sub>12</sub></font>,<span class="p3">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="pboole.html#K6">#</a> </span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">c<sub>12</sub></font>,<span class="p3">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="pboole.html#K6">#</a> </span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font> )
 <b>by </b><i><a class="txt" href="equation.html#E1:34_3"><i><font color="Green">E22</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E10:34_3"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">c<sub>12</sub></font>,<span class="p2">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font>
 
<b>by </b><i><a class="txt" href="equation.html#E9:34_3"><i><font color="Green">E26</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/>
<a NAME="E11:34_3"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="pboole.html#K6">#</a> </span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="txt" href="equation.html#E6:34_3"><i><font color="Green">E25</font></i></a>, <a class="txt" href="equation.html#E9:34_3"><i><font color="Green">E26</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>17</sub></font> as    <a href="subset_1.html#M1">Element</a> of  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">c<sub>12</sub></font>,<span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <b>by </b><i><a class="txt" href="equation.html#E4:34_3"><i><font color="Green">E24</font></i></a></i>;<br/>
<a NAME="E13:34_3"/>
<font color="Maroon">S<sub>1</sub></font>[ <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_3_1"><b>proof </b></a><div class="add">

<b>take </b>
 <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">c<sub>12</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>16</sub></font>
;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:34_3_1_1"/>
(  <a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or  <a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_3_1_1_1"><b>suppose </b></a><a NAME="E1:34_3_1_1_1"/><i><font color="Green">E29</font></i>: 
 <a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>consider </b><font color="Maroon">c<sub>19</sub></font> being  <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><b>take </b>
<font color="Maroon">c<sub>19</sub></font>
;<br/><b>thus </b><a NAME="E3:34_3_1_1_1"/>
 <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">c<sub>12</sub></font>
 ;<br/><b>thus </b><a NAME="E4:34_3_1_1_1"/>
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font>
 ;<br/><b>let </b><font color="Maroon">c<sub>20</sub></font> be  <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><b>assume </b><a NAME="E5:34_3_1_1_1"/>
<font color="Maroon">c<sub>19</sub></font> is    <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">c<sub>20</sub></font>
 ;<br/><a NAME="E6:34_3_1_1_1"/>
( <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a>  <a href="msualg_2.html#K14">MSSub</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a>  <a href="msualg_2.html#K14">MSSub</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="ref" href="msualg_2.html#D20">MSUALG_2:def 20</a></i>;<br/><a NAME="E7:34_3_1_1_1"/><b>then </b>
( <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="equation.html#E1:34"><i><font color="Green">E18</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>21</sub></font> = <font color="Maroon">c<sub>19</sub></font>, <font color="Maroon">c<sub>22</sub></font> = <font color="Maroon">c<sub>20</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>6</sub></font> ;<br/><a NAME="E9:34_3_1_1_1"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">c<sub>21</sub></font> &amp; <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">c<sub>22</sub></font> )
 <b>by </b><i><a class="txt" href="equation.html#E2:34"><i><font color="Green">E19</font></i></a></i>;<br/><a NAME="E10:34_3_1_1_1"/>
 <a href="msualg_1.html#K3">Args</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="equation.html#E1:34_3_1_1_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="pralg_2.html#T11">PRALG_2:11</a></i>;<br/><a NAME="E11:34_3_1_1_1"/><b>then </b><i><font color="Green">E31</font></i>: 
 <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a>  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>19</sub></font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E12:34_3_1_1_1"/>
 <a href="msualg_1.html#K3">Args</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="equation.html#E1:34_3_1_1_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="pralg_2.html#T11">PRALG_2:11</a></i>;<br/><a NAME="E13:34_3_1_1_1"/><b>then </b><i><font color="Green">E32</font></i>: 
 <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a>  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>20</sub></font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E14:34_3_1_1_1"/>
 <a href="msualg_1.html#K3">Args</a> <font color="Maroon">c<sub>12</sub></font>,<span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="equation.html#E1:34_3_1_1_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="pralg_2.html#T11">PRALG_2:11</a></i>;<br/><a NAME="E15:34_3_1_1_1"/><b>then </b><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><font color="Maroon">c<sub>16</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font> = 
<span class="p1">(<span class="default"><a href="pralg_3.html#K2">const</a> <font color="Maroon">c<sub>12</sub></font>,<span class="p2">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font>
<b>by </b><i><a class="txt" href="equation.html#E10:34_3"><i><font color="Green">E27</font></i></a>, <a class="ref" href="pralg_3.html#D1">PRALG_3:def 1</a></i>
<br/>.= 
 <a href="pralg_3.html#K2">const</a> <font color="Maroon">c<sub>12</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">c<sub>21</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="equation.html#E1:34_3_1_1_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="pralg_3.html#T10">PRALG_3:10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">c<sub>12</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">c<sub>21</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i><a class="ref" href="pralg_3.html#D1">PRALG_3:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i><a class="txt" href="equation.html#E9:34_3_1_1_1"><i><font color="Green">E30</font></i></a>, <a class="txt" href="equation.html#E11:34_3_1_1_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="equation.html#T21" target="_self">Th21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">c<sub>12</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">c<sub>22</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i><a class="txt" href="equation.html#E9:34_3_1_1_1"><i><font color="Green">E30</font></i></a>, <a class="txt" href="equation.html#E13:34_3_1_1_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="equation.html#T21" target="_self">Th21</a></i>
<br/>.= 
 <a href="pralg_3.html#K2">const</a> <font color="Maroon">c<sub>12</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">c<sub>22</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="pralg_3.html#D1">PRALG_3:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pralg_3.html#K2">const</a> <font color="Maroon">c<sub>12</sub></font>,<span class="p2">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font>
<b>by </b><i><a class="txt" href="equation.html#E1:34_3_1_1_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="pralg_3.html#T10">PRALG_3:10</a></i>
<br/>.= 
<font color="Maroon">c<sub>16</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font>
<b>by </b><i><a class="txt" href="equation.html#E10:34_3"><i><font color="Green">E27</font></i></a>, <a class="txt" href="equation.html#E15:34_3_1_1_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="pralg_3.html#D1">PRALG_3:def 1</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_3_1_1_2"><b>suppose </b></a><a NAME="E1:34_3_1_1_2"/><i><font color="Green">E29</font></i>: 
 <a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">c<sub>19</sub></font> =  <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>12</sub></font></span>)</span> as  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="ref" href="finseq_1.html#T26">FINSEQ_1:26</a></i>;<br/><a NAME="E3:34_3_1_1_2"/>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="pboole.html#K6">#</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="equation.html#E11:34_3"><i><font color="Green">E28</font></i></a>, <a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>;<br/><a NAME="E4:34_3_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="pboole.html#K6">#</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
 ;<br/><a NAME="E5:34_3_1_1_2"/><b>then </b><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="msualg_4.html#K6">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="pboole.html#D19">PBOOLE:def 19</a></i>;<br/><b>then </b><i><font color="Green">E31</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="msualg_4.html#K6">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> = 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>18</sub></font>
<b>by </b><i><a class="ref" href="card_3.html#T18">CARD_3:18</a></i>
<br/>.= 
<font color="Maroon">c<sub>19</sub></font>
<b>by </b><i><a class="ref" href="msualg_3.html#T6">MSUALG_3:6</a></i>
;<br/><b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] means ex <font color="Olive">b<sub>1</sub></font> being  <a href="funct_1.html#NM1">Function</a>ex <font color="Olive">b<sub>2</sub></font> being <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>3</sub></font> being <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>2</sub></font> is    <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Olive">b<sub>3</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>3</sub></font> ) &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> );<br/><a NAME="E7:34_3_1_1_2"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>19</sub></font>ex <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="msualg_2.html#K14">MSSub</a> <font color="Maroon">c<sub>2</sub></font> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_3_1_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>20</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>19</sub></font>;<br/>

<a NAME="E1:34_3_1_1_2_2"/><i><font color="Green">E33</font></i>: 
<span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>20</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>
 
<b>by </b><i><a class="ref" href="partfun1.html#T27">PARTFUN1:27</a></i>;<br/>
<a NAME="E2:34_3_1_1_2_2"/>
<font color="Maroon">c<sub>18</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="msualg_4.html#K6">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font>
 
<b>by </b><i><a class="txt" href="equation.html#E5:34_3_1_1_2"><i><font color="Green">E30</font></i></a>, <a class="txt" href="equation.html#E6:34_3_1_1_2"><i><font color="Green">E31</font></i></a>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/>
<a NAME="E3:34_3_1_1_2_2"/><b>then </b>
<font color="Maroon">c<sub>18</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</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>12</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>21</sub></font> being   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>22</sub></font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K11">SORTS</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font>, <font color="Maroon">c<sub>23</sub></font> being  <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E5:34_3_1_1_2_2"/>
<font color="Maroon">c<sub>21</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>12</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font>
 <b>and </b><br/><a NAME="E6:34_3_1_1_2_2"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font>
 <b>and </b><br/><a NAME="E7:34_3_1_1_2_2"/><i><font color="Green">E35</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Maroon">c<sub>23</sub></font> is    <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>22</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="equation.html#E8:34"><i><font color="Green">E21</font></i></a>, <a class="txt" href="equation.html#E1:34_3_1_1_2_2"><i><font color="Green">E33</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>24</sub></font> = <font color="Maroon">c<sub>23</sub></font> as    <a href="subset_1.html#M1">Element</a> of  <a href="msualg_2.html#K14">MSSub</a> <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="msualg_2.html#D20">MSUALG_2:def 20</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>24</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>22</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>23</sub></font>
;<br/>

<b>thus </b><a NAME="E9:34_3_1_1_2_2"/>
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font>
 <b>by </b><i><a class="txt" href="equation.html#E6:34_3_1_1_2_2"><i><font color="Green">E34</font></i></a></i>;<br/>

<b>thus </b><a NAME="E10:34_3_1_1_2_2"/>
for <font color="Olive">b<sub>1</sub></font> being <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Maroon">c<sub>23</sub></font> is    <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>22</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="equation.html#E7:34_3_1_1_2_2"><i><font color="Green">E35</font></i></a></i>;<br/>

<b>thus </b><a NAME="E11:34_3_1_1_2_2"/>
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font>
 ;<br/>


</div><b>end;</b></div><b>consider </b><font color="Maroon">c<sub>20</sub></font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>19</sub></font>, <a href="msualg_2.html#K14">MSSub</a> <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E9:34_3_1_1_2"/><i><font color="Green">E33</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>19</sub></font> holds <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>20</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>(<a class="txt" href="equation.html#E7:34_3_1_1_2"><i><font color="Green">E32</font></i></a>);<br/></i><a NAME="E10:34_3_1_1_2"/>
<font color="Maroon">c<sub>20</sub></font> is    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">c<sub>19</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_3_1_1_2_3"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:34_3_1_1_2_3"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a><br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">c<sub>21</sub></font> = <font color="Maroon">c<sub>20</sub></font> as    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">c<sub>19</sub></font> ;<br/><a NAME="E12:34_3_1_1_2"/>
<font color="Maroon">c<sub>21</sub></font> is    <a href="pralg_2.html#M2">MSAlgebra-Family</a> of <font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_3_1_1_2_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>22</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="pralg_2.html#D12">PRALG_2:def 12</a><br/>

<b>assume </b><a NAME="E1:34_3_1_1_2_4"/>
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>19</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>23</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E3:34_3_1_1_2_4"/><i><font color="Green">E34</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font> &amp; ( for <font color="Olive">b<sub>2</sub></font> being <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>1</sub></font> is    <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Olive">b<sub>2</sub></font> holds <br/><font color="Maroon">c<sub>23</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> ) &amp; <font color="Maroon">c<sub>21</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="equation.html#E9:34_3_1_1_2"><i><font color="Green">E33</font></i></a></i>;<br/>
<b>thus </b><a NAME="E4:34_3_1_1_2_4"/>
<font color="Maroon">c<sub>21</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font> is  <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="equation.html#E3:34_3_1_1_2_4"><i><font color="Green">E34</font></i></a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">c<sub>22</sub></font> = <font color="Maroon">c<sub>21</sub></font> as    <a href="pralg_2.html#M2">MSAlgebra-Family</a> of <font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>1</sub></font> ;<br/><a NAME="E14:34_3_1_1_2"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>19</sub></font>ex <font color="Olive">b<sub>2</sub></font> being <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> <a href="pralg_2.html#K6">.</a> <font color="Olive">b<sub>1</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_3_1_1_2_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>23</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>19</sub></font>;<br/>

<a NAME="E1:34_3_1_1_2_5"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="funct_1.html#NM1">Function</a>ex <font color="Olive">b<sub>2</sub></font> being <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>23</sub></font> &amp; ( for <font color="Olive">b<sub>3</sub></font> being <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>2</sub></font> is    <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Olive">b<sub>3</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>3</sub></font> ) &amp; <font color="Maroon">c<sub>22</sub></font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="equation.html#E9:34_3_1_1_2"><i><font color="Green">E33</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:34_3_1_1_2_5"/>
ex <font color="Olive">b<sub>1</sub></font> being <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">c<sub>23</sub></font>
 ;<br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon">c<sub>23</sub></font> being  <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E16:34_3_1_1_2"/><i><font color="Green">E34</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>19</sub></font> holds <font color="Maroon">c<sub>22</sub></font> <a href="pralg_2.html#K6">.</a> <font color="Olive">b<sub>1</sub></font> is    <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">c<sub>23</sub></font>
 <b>by </b><i><a class="ref" href="equation.html#T27" target="_self">Th27</a></i>;<br/><b>take </b>
<font color="Maroon">c<sub>23</sub></font>
;<br/><b>thus </b><a NAME="E17:34_3_1_1_2"/>
 <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">c<sub>12</sub></font>
 ;<br/><b>thus </b><a NAME="E18:34_3_1_1_2"/>
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font>
 ;<br/><b>let </b><font color="Maroon">c<sub>24</sub></font> be  <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><b>assume </b><a NAME="E19:34_3_1_1_2"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>23</sub></font> is    <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">c<sub>24</sub></font>
 ;<br/><a NAME="E20:34_3_1_1_2"/>
( <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a>  <a href="msualg_2.html#K14">MSSub</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a>  <a href="msualg_2.html#K14">MSSub</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="ref" href="msualg_2.html#D20">MSUALG_2:def 20</a></i>;<br/><a NAME="E21:34_3_1_1_2"/><b>then </b>
( <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="equation.html#E1:34"><i><font color="Green">E18</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>25</sub></font> = <font color="Maroon">c<sub>24</sub></font>, <font color="Maroon">c<sub>26</sub></font> = <font color="Maroon">c<sub>23</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>6</sub></font> ;<br/><a NAME="E23:34_3_1_1_2"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">c<sub>26</sub></font> &amp; <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">c<sub>25</sub></font> )
 <b>by </b><i><a class="txt" href="equation.html#E2:34"><i><font color="Green">E19</font></i></a></i>;<br/><a NAME="E24:34_3_1_1_2"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>6</sub></font>,<span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <span class="p5">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pralg_2.html#K6">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>6</sub></font>, B is    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> : verum </span>}</span> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="pralg_3.html#T15">PRALG_3:15</a></i>;<br/><a NAME="E25:34_3_1_1_2"/><b>then </b><i><font color="Green">E38</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_6.html#K10">commute</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font>
 <b>by </b><i><a class="ref" href="equation.html#T3" target="_self">Th3</a></i>;<br/><a NAME="E26:34_3_1_1_2"/><i><font color="Green">E39</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_6.html#K10">commute</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>26</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font>
 <b>by </b><i><a class="txt" href="equation.html#E24:34_3_1_1_2"><i><font color="Green">E37</font></i></a>, <a class="ref" href="equation.html#T3" target="_self">Th3</a></i>;<br/><a NAME="E27:34_3_1_1_2"/><i><font color="Green">E40</font></i>: 
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="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_6.html#K10">commute</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>26</sub></font></span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_6.html#K10">commute</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>25</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"><span class="p2">(<span class="default"><a href="funct_6.html#K10">commute</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>26</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="34_3_1_1_2_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>27</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:34_3_1_1_2_6"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_6.html#K10">commute</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>26</sub></font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>28</sub></font> being   <a href="funct_1.html#NM1">Function</a>, <font color="Maroon">c<sub>29</sub></font> being  <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E3:34_3_1_1_2_6"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>27</sub></font>
 <b>and </b><br/><a NAME="E4:34_3_1_1_2_6"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Maroon">c<sub>29</sub></font> is    <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>28</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>28</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 <b>and </b><br/><a NAME="E5:34_3_1_1_2_6"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>22</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>29</sub></font>
 <b>by </b><i><a class="txt" href="equation.html#E9:34_3_1_1_2"><i><font color="Green">E33</font></i></a>, <a class="txt" href="equation.html#E26:34_3_1_1_2"><i><font color="Green">E39</font></i></a></i>;<br/>
<a NAME="E6:34_3_1_1_2_6"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>22</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>27</sub></font> is    <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">c<sub>23</sub></font>
 
<b>by </b><i><a class="txt" href="equation.html#E16:34_3_1_1_2"><i><font color="Green">E34</font></i></a>, <a class="txt" href="equation.html#E26:34_3_1_1_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="equation.html#E1:34_3_1_1_2_6"><i><font color="Green">E41</font></i></a></i>;<br/>
<a NAME="E7:34_3_1_1_2_6"/><b>then </b><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>29</sub></font> is    <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">c<sub>24</sub></font>
 
<b>by </b><i><a class="txt" href="equation.html#E19:34_3_1_1_2"><i><font color="Green">E35</font></i></a>, <a class="txt" href="equation.html#E5:34_3_1_1_2_6"><i><font color="Green">E44</font></i></a>, <a class="ref" href="msualg_2.html#T7">MSUALG_2:7</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_6.html#K10">commute</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>27</sub></font> = 
<font color="Maroon">c<sub>28</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>25</sub></font>
<b>by </b><i><a class="txt" href="equation.html#E26:34_3_1_1_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="equation.html#E1:34_3_1_1_2_6"><i><font color="Green">E41</font></i></a>, <a class="txt" href="equation.html#E3:34_3_1_1_2_6"><i><font color="Green">E42</font></i></a>, <a class="ref" href="pralg_3.html#T22">PRALG_3:22</a></i>
<br/>.= 
<font color="Maroon">c<sub>28</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>29</sub></font>
<b>by </b><i><a class="txt" href="equation.html#E4:34_3_1_1_2_6"><i><font color="Green">E43</font></i></a>, <a class="txt" href="equation.html#E7:34_3_1_1_2_6"><i><font color="Green">E46</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>28</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>26</sub></font>
<b>by </b><i><a class="txt" href="equation.html#E4:34_3_1_1_2_6"><i><font color="Green">E43</font></i></a>, <a class="txt" href="equation.html#E5:34_3_1_1_2_6"><i><font color="Green">E44</font></i></a>, <a class="txt" href="equation.html#E6:34_3_1_1_2_6"><i><font color="Green">E45</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_6.html#K10">commute</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>26</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>27</sub></font>
<b>by </b><i><a class="txt" href="equation.html#E26:34_3_1_1_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="equation.html#E1:34_3_1_1_2_6"><i><font color="Green">E41</font></i></a>, <a class="txt" href="equation.html#E3:34_3_1_1_2_6"><i><font color="Green">E42</font></i></a>, <a class="ref" href="pralg_3.html#T22">PRALG_3:22</a></i>
;<br/>


</div><b>end;</b></div><a NAME="E28:34_3_1_1_2"/><i><font color="Green">E41</font></i>: 
<span class="p1">(<span class="default"><a href="funct_6.html#K10">commute</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>25</sub></font> is    <a href="subset_1.html#M1">Element</a> of  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">c<sub>12</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">c<sub>25</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="equation.html#E1:34_3_1_1_2"><i><font color="Green">E29</font></i></a>, <a class="ref" href="pralg_3.html#T21">PRALG_3:21</a></i>;<br/><a NAME="E29:34_3_1_1_2"/><i><font color="Green">E42</font></i>: 
<span class="p1">(<span class="default"><a href="funct_6.html#K10">commute</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>26</sub></font> is    <a href="subset_1.html#M1">Element</a> of  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">c<sub>12</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">c<sub>26</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="equation.html#E1:34_3_1_1_2"><i><font color="Green">E29</font></i></a>, <a class="ref" href="pralg_3.html#T21">PRALG_3:21</a></i>;<br/><b>thus </b><font color="Maroon">c<sub>16</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>23</sub></font> = 
<span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">c<sub>12</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">c<sub>26</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_6.html#K10">commute</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>26</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="equation.html#E10:34_3"><i><font color="Green">E27</font></i></a>, <a class="txt" href="equation.html#E1:34_3_1_1_2"><i><font color="Green">E29</font></i></a>, <a class="ref" href="pralg_3.html#T23">PRALG_3:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_6.html#K10">commute</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>26</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="equation.html#E23:34_3_1_1_2"><i><font color="Green">E36</font></i></a>, <a class="txt" href="equation.html#E29:34_3_1_1_2"><i><font color="Green">E42</font></i></a>, <a class="ref" href="equation.html#T21" target="_self">Th21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_6.html#K10">commute</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>25</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="equation.html#E25:34_3_1_1_2"><i><font color="Green">E38</font></i></a>, <a class="txt" href="equation.html#E26:34_3_1_1_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="equation.html#E27:34_3_1_1_2"><i><font color="Green">E40</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">c<sub>12</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">c<sub>25</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_6.html#K10">commute</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>25</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="equation.html#E23:34_3_1_1_2"><i><font color="Green">E36</font></i></a>, <a class="txt" href="equation.html#E28:34_3_1_1_2"><i><font color="Green">E41</font></i></a>, <a class="ref" href="equation.html#T21" target="_self">Th21</a></i>
<br/>.= 
<font color="Maroon">c<sub>16</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font>
<b>by </b><i><a class="txt" href="equation.html#E10:34_3"><i><font color="Green">E27</font></i></a>, <a class="txt" href="equation.html#E1:34_3_1_1_2"><i><font color="Green">E29</font></i></a>, <a class="ref" href="pralg_3.html#T23">PRALG_3:23</a></i>
;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E14:34_3"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="equation.html#E8:34"><i><font color="Green">E21</font></i></a></i>;<br/>
<a NAME="E15:34_3"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
 
;<br/>
<b>hence </b><a NAME="E16:34_3"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E12:34"/><b>then </b><i><font color="Green">E22</font></i>: 
<span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="msualg_2.html#K10">|</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#G3">MSAlgebra</a>(# <font color="Maroon">c<sub>11</sub></font>,<span class="p1">(<span class="default"><a href="msualg_2.html#K4">Opers</a> <span class="p2">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<font color="Maroon">c<sub>11</sub></font></span>)</span> #)
 
<b>by </b><i><a class="ref" href="msualg_2.html#D16">MSUALG_2:def 16</a></i>;<br/>
<a NAME="E13:34"/>
<span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="msualg_2.html#K10">|</a> <font color="Maroon">c<sub>11</sub></font> is <a href="msualg_1.html#V5">non-empty</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_4"><b>proof </b></a><div class="add">

<a NAME="E1:34_4"/>
<font color="Maroon">c<sub>11</sub></font> is <a href="relat_1.html#V2">non-empty</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>12</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="pboole.html#D16">PBOOLE:def 16</a><br/>

<b>assume </b><a NAME="E1:34_4_1"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>12</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="E2:34_4_1"/><b>then </b>
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</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>then consider </b><font color="Maroon">c<sub>13</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:34_4_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ] means ex <font color="Olive">b<sub>1</sub></font> being <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="msafree2.html#V3">finitely-generated</a> );<br/>
<b>consider </b><font color="Maroon">c<sub>14</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:34_4_1"/><i><font color="Green">E25</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font> iff ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="equation.html#K2" target="_self">SuperAlgebraSet</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1">XBOOLE_0:sch 1</a>();<br/></i>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] means ex <font color="Olive">b<sub>1</sub></font> being   <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">c<sub>2</sub></font>ex <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> );<br/>
<a NAME="E7:34_4_1"/><i><font color="Green">E26</font></i>: 
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> <font color="Maroon">c<sub>6</sub></font> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_4_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>15</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:34_4_1_1"/>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>16</sub></font> being    <a href="subset_1.html#M1">Element</a> of  <a href="msualg_2.html#K14">MSSub</a> <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E3:34_4_1_1"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font>
 <b>and </b><br/><a NAME="E4:34_4_1_1"/><i><font color="Green">E28</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="txt" href="equation.html#E1:34"><i><font color="Green">E18</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>17</sub></font> being  <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E6:34_4_1_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="txt" href="equation.html#E4:34_4_1_1"><i><font color="Green">E28</font></i></a></i>;<br/>
<a NAME="E7:34_4_1_1"/>
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>17</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</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="equation.html#E1:34_4_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="pboole.html#D16">PBOOLE:def 16</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>18</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E9:34_4_1_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>17</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>18</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>17</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>18</sub></font>
;<br/>

<b>thus </b><a NAME="E10:34_4_1_1"/>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="equation.html#E3:34_4_1_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="equation.html#E6:34_4_1_1"><i><font color="Green">E29</font></i></a></i>;<br/>

<b>thus </b><a NAME="E11:34_4_1_1"/>
( <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>17</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> )
 <b>by </b><i><a class="txt" href="equation.html#E9:34_4_1_1"><i><font color="Green">E30</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>15</sub></font> being    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">c<sub>6</sub></font><b> such that </b><br/><a NAME="E9:34_4_1"/><i><font color="Green">E27</font></i>: 
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> <font color="Maroon">c<sub>6</sub></font> holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>15</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="pboole.html#S3">PBOOLE:sch 3</a>(<a class="txt" href="equation.html#E7:34_4_1"><i><font color="Green">E26</font></i></a>);<br/></i>
<b>set </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>15</sub></font> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>;<br/>
<a NAME="E10:34_4_1"/><i><font color="Green">E28</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/>
<a NAME="E11:34_4_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_4_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>17</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:34_4_1_2"/>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>18</sub></font> being  <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E3:34_4_1_2"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>18</sub></font> is <a href="msafree2.html#V3">finitely-generated</a> )
 <b>by </b><i><a class="txt" href="equation.html#E6:34_4_1"><i><font color="Green">E25</font></i></a></i>;<br/>
<a NAME="E4:34_4_1_2"/>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a>  <a href="msualg_2.html#K14">MSSub</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="equation.html#E3:34_4_1_2"><i><font color="Green">E30</font></i></a>, <a class="ref" href="msualg_2.html#D20">MSUALG_2:def 20</a></i>;<br/>
<b>hence </b><a NAME="E5:34_4_1_2"/>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="equation.html#E1:34"><i><font color="Green">E18</font></i></a>, <a class="txt" href="equation.html#E3:34_4_1_2"><i><font color="Green">E30</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>12</sub></font> as   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="equation.html#E1:34_4_1"><i><font color="Green">E23</font></i></a></i>;<br/>
<i><font color="Green">E30</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>
<br/>.= 
<font color="Maroon">c<sub>6</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
<br/>.= 
<font color="Maroon">c<sub>6</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>14</sub></font>
<b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>
<br/>.= 
<font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="txt" href="equation.html#E11:34_4_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>
<br/>.= 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>17</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
;<br/>
<a NAME="E14:34_4_1"/>
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="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>17</sub></font></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>17</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="34_4_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>18</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:34_4_1_4"/>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>17</sub></font></span>)</span>
 ;<br/>

<a NAME="E2:34_4_1_4"/><b>then </b><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>19</sub></font> being    <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E4:34_4_1_4"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font> &amp; <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>19</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font> )
 <b>by </b><i><a class="ref" href="pralg_2.html#D16">PRALG_2:def 16</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>20</sub></font> being    <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>21</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:34_4_1_4"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>20</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> )
 <b>by </b><i><a class="txt" href="equation.html#E9:34_4_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="equation.html#E2:34_4_1_4"><i><font color="Green">E31</font></i></a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_4_1_4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:34_4_1_4_1"/>
( <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font> or  not <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_4_1_4_1_1"><b>suppose </b></a><a NAME="E1:34_4_1_4_1_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font>
 ;<br/><div class="add"><a NAME="E2:34_4_1_4_1_1"/><b>then </b>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/><a NAME="E3:34_4_1_4_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font>
 <b>by </b><i><a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>;<br/><a NAME="E4:34_4_1_4_1_1"/><b>then </b><i><font color="Green">E35</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="equation.html#E1:34_4_1_4_1_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/><a NAME="E5:34_4_1_4_1_1"/>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a>  <a href="equation.html#K2" target="_self">SuperAlgebraSet</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="equation.html#E6:34_4_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="equation.html#E1:34_4_1_4_1_1"><i><font color="Green">E34</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>22</sub></font> being  <a href="msualg_1.html#V4">strict</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E7:34_4_1_4_1_1"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> is    <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">c<sub>22</sub></font> )
 <b>by </b><i><a class="ref" href="equation.html#D2" target="_self">Def2</a></i>;<br/><a NAME="E8:34_4_1_4_1_1"/>
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>7</sub></font> is    <a href="pboole.html#M4">ManySortedSubset</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>22</sub></font>
 <b>by </b><i><a class="txt" href="equation.html#E7:34_4_1_4_1_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="msualg_2.html#D10">MSUALG_2:def 10</a></i>;<br/><a NAME="E9:34_4_1_4_1_1"/><b>then </b>
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>7</sub></font> <a href="pboole.html#R2">c=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>22</sub></font>
 <b>by </b><i><a class="ref" href="pboole.html#D23">PBOOLE:def 23</a></i>;<br/><a NAME="E10:34_4_1_4_1_1"/><b>then </b>
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="tarski.html#R1">c=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>22</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="equation.html#E1:34_4_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="pboole.html#D5">PBOOLE:def 5</a></i>;<br/><a NAME="E11:34_4_1_4_1_1"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>22</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="equation.html#E4:34_4_1"><i><font color="Green">E24</font></i></a></i>;<br/><b>hence </b><a NAME="E12:34_4_1_4_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font>
 <b>by </b><i><a class="txt" href="equation.html#E2:34"><i><font color="Green">E19</font></i></a>, <a class="txt" href="equation.html#E2:34_4_1_4"><i><font color="Green">E31</font></i></a>, <a class="txt" href="equation.html#E4:34_4_1_4"><i><font color="Green">E32</font></i></a>, <a class="txt" href="equation.html#E4:34_4_1_4_1_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="equation.html#E7:34_4_1_4_1_1"><i><font color="Green">E36</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_4_1_4_1_2"><b>suppose </b></a><a NAME="E1:34_4_1_4_1_2"/>
not <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font>
 ;<br/><div class="add"><a NAME="E2:34_4_1_4_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font>
 <b>by </b><i><a class="txt" href="equation.html#E10:34_4_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="equation.html#E6:34_4_1_4"><i><font color="Green">E33</font></i></a>, <a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>;<br/><b>hence </b><a NAME="E3:34_4_1_4_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font>
 <b>by </b><i><a class="txt" href="equation.html#E2:34"><i><font color="Green">E19</font></i></a>, <a class="txt" href="equation.html#E2:34_4_1_4"><i><font color="Green">E31</font></i></a>, <a class="txt" href="equation.html#E4:34_4_1_4"><i><font color="Green">E32</font></i></a>, <a class="txt" href="equation.html#E6:34_4_1_4"><i><font color="Green">E33</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E15:34_4_1"/><b>then </b><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>17</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="equation.html#E13:34_4_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/>
<a NAME="E16:34_4_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>15</sub></font> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_4_1_5"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon">c<sub>17</sub></font>
;<br/>

<b>reconsider </b><font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>15</sub></font> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> as    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K11">SORTS</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font> <b>by </b><i><a class="txt" href="equation.html#E15:34_4_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="pralg_2.html#D17">PRALG_2:def 17</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>18</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>7</sub></font>
;<br/>

<b>thus </b><a NAME="E2:34_4_1_5"/>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font>
 ;<br/>

<b>thus </b><a NAME="E3:34_4_1_5"/>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>19</sub></font> be  <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>

<b>assume </b><a NAME="E4:34_4_1_5"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>7</sub></font> is    <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">c<sub>19</sub></font>
 ;<br/>

<a NAME="E5:34_4_1_5"/>
<font color="Maroon">c<sub>7</sub></font> is    <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="ref" href="msualg_2.html#T6">MSUALG_2:6</a></i>;<br/>
<a NAME="E6:34_4_1_5"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="equation.html#K2" target="_self">SuperAlgebraSet</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="ref" href="equation.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E7:34_4_1_5"/><b>then </b><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="txt" href="equation.html#E6:34_4_1"><i><font color="Green">E25</font></i></a></i>;<br/>
<a NAME="E8:34_4_1_5"/>
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a>  <a href="equation.html#K2" target="_self">SuperAlgebraSet</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="equation.html#E4:34_4_1_5"><i><font color="Green">E32</font></i></a>, <a class="ref" href="equation.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E9:34_4_1_5"/><b>then </b><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="txt" href="equation.html#E6:34_4_1"><i><font color="Green">E25</font></i></a></i>;<br/>
<b>thus </b><font color="Maroon">c<sub>18</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">c<sub>13</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="equation.html#E10:34_4_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="equation.html#E7:34_4_1_5"><i><font color="Green">E33</font></i></a>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
<br/>.= 
<font color="Maroon">c<sub>13</sub></font>
<b>by </b><i><a class="txt" href="equation.html#E7:34_4_1_5"><i><font color="Green">E33</font></i></a>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font>
<b>by </b><i><a class="txt" href="equation.html#E9:34_4_1_5"><i><font color="Green">E34</font></i></a>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
<br/>.= 
<font color="Maroon">c<sub>18</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font>
<b>by </b><i><a class="txt" href="equation.html#E10:34_4_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="equation.html#E9:34_4_1_5"><i><font color="Green">E34</font></i></a>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E17:34_4_1"/>
not <font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i><a class="txt" href="equation.html#E8:34"><i><font color="Green">E21</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:34_4"/>
the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="msualg_2.html#K10">|</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> is <a href="relat_1.html#V2">non-empty</a>
 <b>by </b><i><a class="txt" href="equation.html#E12:34"><i><font color="Green">E22</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="msualg_1.html#D8">MSUALG_1:def 8</a><br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>12</sub></font> = <span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="msualg_2.html#K10">|</a> <font color="Maroon">c<sub>11</sub></font> as  <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of  <a href="pralg_2.html#K15">product</a> <font color="Maroon">c<sub>4</sub></font> ;<br/>
<b>take </b>
<font color="Maroon">c<sub>12</sub></font>
;<br/>

<div><i><font color="Green">E23</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="34_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>13</sub></font> be   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>14</sub></font> be    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K11">SORTS</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>15</sub></font> be  <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><a NAME="E1:34_5"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a>  <a href="msualg_2.html#K14">MSSub</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="msualg_2.html#D20">MSUALG_2:def 20</a></i>;<br/><a NAME="E2:34_5"/><i><font color="Green">E25</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><a NAME="E3:34_5"/><b>then </b><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>13</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="equation.html#E1:34"><i><font color="Green">E18</font></i></a>, <a class="txt" href="equation.html#E1:34_5"><i><font color="Green">E24</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>16</sub></font> being    <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E5:34_5"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font>
 <b>and </b><br/><a NAME="E6:34_5"/><i><font color="Green">E28</font></i>: 
<span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>16</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="equation.html#E2:34_5"><i><font color="Green">E25</font></i></a>, <a class="ref" href="pralg_2.html#D16">PRALG_2:def 16</a></i>;<br/><a NAME="E7:34_5"/>
<font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="equation.html#E2:34"><i><font color="Green">E19</font></i></a>, <a class="txt" href="equation.html#E2:34_5"><i><font color="Green">E25</font></i></a>, <a class="txt" href="equation.html#E3:34_5"><i><font color="Green">E26</font></i></a></i>;<br/><a NAME="E8:34_5"/><b>then </b>
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>16</sub></font> is    <a href="pboole.html#M4">ManySortedSubset</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="equation.html#E5:34_5"><i><font color="Green">E27</font></i></a>, <a class="ref" href="msualg_2.html#D10">MSUALG_2:def 10</a></i>;<br/><a NAME="E9:34_5"/><b>then </b>
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>16</sub></font> <a href="pboole.html#R2">c=</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#D23">PBOOLE:def 23</a></i>;<br/><a NAME="E10:34_5"/><b>then </b><i><font color="Green">E29</font></i>: 
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>16</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="tarski.html#R1">c=</a> 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>13</sub></font>
 <b>by </b><i><a class="ref" href="pboole.html#D5">PBOOLE:def 5</a></i>;<br/><a NAME="E11:34_5"/>
<span class="p1">(<span class="default"><a href="pralg_2.html#K11">SORTS</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>13</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="pralg_2.html#D17">PRALG_2:def 17</a></i>;<br/><a NAME="E12:34_5"/><b>then </b>
<font color="Maroon">c<sub>14</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="equation.html#E3:34_5"><i><font color="Green">E26</font></i></a>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/><b>hence </b><a NAME="E13:34_5"/>
<font color="Maroon">c<sub>14</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> 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>13</sub></font>
 <b>by </b><i><a class="txt" href="equation.html#E6:34_5"><i><font color="Green">E28</font></i></a>, <a class="txt" href="equation.html#E10:34_5"><i><font color="Green">E29</font></i></a></i>;<br/></div><b>end;</b></div>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] means ex <font color="Olive">b<sub>1</sub></font> being  <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">c<sub>1</sub></font>ex <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K11">SORTS</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>ex <font color="Olive">b<sub>3</sub></font> being <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>3</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>2</sub></font> &amp; ( for <font color="Olive">b<sub>4</sub></font> being <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>3</sub></font> is    <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Olive">b<sub>4</sub></font> holds <br/><font color="Olive">b<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>4</sub></font> ) &amp; <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>3</sub></font> );<br/>
<div><i><font color="Green">E24</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="34_6"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>13</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:34_6"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>13</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/><b>let </b><font color="Maroon">c<sub>14</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E2:34_6"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>15</sub></font> being   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>16</sub></font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K11">SORTS</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font>, <font color="Maroon">c<sub>17</sub></font> being  <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E4:34_6"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font>
 <b>and </b><br/><a NAME="E5:34_6"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font>
 <b>and </b><br/><a NAME="E6:34_6"/><i><font color="Green">E28</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Maroon">c<sub>17</sub></font> is    <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>16</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="equation.html#E8:34"><i><font color="Green">E21</font></i></a>, <a class="txt" href="equation.html#E12:34"><i><font color="Green">E22</font></i></a>, <a class="txt" href="equation.html#E1:34_6"><i><font color="Green">E25</font></i></a></i>;<br/><b>take </b><font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>16</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font>;<br/><b>thus </b><a NAME="E7:34_6"/>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> 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>13</sub></font>
 <b>by </b><i><a class="txt" href="equation.html#E15:34"><i><font color="Green">E23</font></i></a>, <a class="txt" href="equation.html#E4:34_6"><i><font color="Green">E26</font></i></a></i>;<br/><b>thus </b><a NAME="E8:34_6"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>13</sub></font>]
 <b>by </b><i><a class="txt" href="equation.html#E4:34_6"><i><font color="Green">E26</font></i></a>, <a class="txt" href="equation.html#E5:34_6"><i><font color="Green">E27</font></i></a>, <a class="txt" href="equation.html#E6:34_6"><i><font color="Green">E28</font></i></a></i>;<br/></div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>13</sub></font> being   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E18:34"/><i><font color="Green">E25</font></i>: 
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> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being  <a href="funct_2.html#NM1">Function</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>,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="Olive">b<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>3</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>] ) )
 <b>from </b><i><a class="ref" href="mssubfam.html#S1">MSSUBFAM:sch 1</a>(<a class="txt" href="equation.html#E16:34"><i><font color="Green">E24</font></i></a>);<br/></i>
<b>take </b>
<font color="Maroon">c<sub>13</sub></font>
;<br/>

<b>thus </b><a NAME="E19:34"/>
<font color="Maroon">c<sub>13</sub></font> <a href="msualg_3.html#R1">is_homomorphism</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>2</sub></font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="msualg_3.html#D10">MSUALG_3:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>14</sub></font> be   <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">c<sub>1</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="msualg_3.html#D9">MSUALG_3:def 9</a><br/>

<b>assume </b><a NAME="E1:34_7"/>
 <a href="msualg_1.html#K3">Args</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>let </b><font color="Maroon">c<sub>15</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>14</sub></font>,<font color="Maroon">c<sub>12</sub></font>;<br/>

<b>set </b><font color="Maroon">c<sub>16</sub></font> =  <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">c<sub>14</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>17</sub></font> =  <a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>14</sub></font>;<br/>
<b>consider </b><font color="Maroon">c<sub>18</sub></font> being   <a href="funct_2.html#NM1">Function</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">c<sub>14</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> <span class="p1">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">c<sub>14</sub></font></span>)</span><b> such that </b><br/><a NAME="E3:34_7"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="msualg_3.html#K1">.</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E4:34_7"/><i><font color="Green">E27</font></i>: 
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> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>18</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>1</sub></font>, <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">c<sub>14</sub></font>]
 <b>by </b><i><a class="txt" href="equation.html#E18:34"><i><font color="Green">E25</font></i></a></i>;<br/>
<a NAME="E5:34_7"/>
<span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="msualg_9.html#T19">MSUALG_9:19</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>19</sub></font> being   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>20</sub></font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K11">SORTS</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font>, <font color="Maroon">c<sub>21</sub></font> being  <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E7:34_7"/>
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">c<sub>14</sub></font>
 <b>and </b><br/><a NAME="E8:34_7"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>15</sub></font>
 <b>and </b><br/><a NAME="E9:34_7"/><i><font color="Green">E29</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Maroon">c<sub>21</sub></font> is    <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>20</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 <b>and </b><br/><a NAME="E10:34_7"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>18</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#K5">Den</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font>
 <b>by </b><i><a class="txt" href="equation.html#E4:34_7"><i><font color="Green">E27</font></i></a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_7_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:34_7_1"/>
(  <a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or  <a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_7_1_1"><b>suppose </b></a><a NAME="E1:34_7_1_1"/><i><font color="Green">E31</font></i>: 
 <a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:34_7_1_1"/>
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a>  <a href="msualg_2.html#K14">MSSub</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="msualg_2.html#D20">MSUALG_2:def 20</a></i>;<br/><a NAME="E3:34_7_1_1"/><b>then </b>
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="equation.html#E1:34"><i><font color="Green">E18</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>22</sub></font> = <font color="Maroon">c<sub>21</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>6</sub></font> ;<br/><b>reconsider </b><font color="Maroon">c<sub>23</sub></font> = <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">c<sub>14</sub></font>,<span class="p2">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font> as   <a href="funct_1.html#NM1">Function</a> <b>by </b><i><a class="ref" href="equation.html#T22" target="_self">Th22</a></i>;<br/><a NAME="E6:34_7_1_1"/><i><font color="Green">E32</font></i>: 
 <a href="msualg_1.html#K3">Args</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="equation.html#E1:34_7_1_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="pralg_2.html#T11">PRALG_2:11</a></i>;<br/><a NAME="E7:34_7_1_1"/><b>then </b><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E8:34_7_1_1"/>
 <a href="msualg_1.html#K3">Args</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="equation.html#E1:34_7_1_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="pralg_2.html#T11">PRALG_2:11</a></i>;<br/><a NAME="E9:34_7_1_1"/><b>then </b><i><font color="Green">E34</font></i>: 
 <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a>  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>21</sub></font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E10:34_7_1_1"/><i><font color="Green">E35</font></i>: 
<span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">c<sub>14</sub></font>,<span class="p2">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a>  <a href="pralg_3.html#K2">const</a> <font color="Maroon">c<sub>14</sub></font>,<span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="pralg_3.html#D1">PRALG_3:def 1</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="msualg_3.html#K1">.</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> = 
<font color="Maroon">c<sub>23</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font>
<b>by </b><i><a class="txt" href="equation.html#E3:34_7"><i><font color="Green">E26</font></i></a>, <a class="txt" href="equation.html#E8:34_7"><i><font color="Green">E28</font></i></a>, <a class="txt" href="equation.html#E10:34_7"><i><font color="Green">E30</font></i></a>, <a class="ref" href="equation.html#T21" target="_self">Th21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pralg_3.html#K2">const</a> <font color="Maroon">c<sub>14</sub></font>,<span class="p2">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font>
<b>by </b><i><a class="txt" href="equation.html#E6:34_7_1_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="equation.html#E10:34_7_1_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>
<br/>.= 
 <a href="pralg_3.html#K2">const</a> <font color="Maroon">c<sub>14</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">c<sub>22</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="equation.html#E1:34_7_1_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="pralg_3.html#T10">PRALG_3:10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">c<sub>14</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">c<sub>22</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i><a class="ref" href="pralg_3.html#D1">PRALG_3:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i><a class="txt" href="equation.html#E2:34"><i><font color="Green">E19</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i><a class="txt" href="equation.html#E9:34_7_1_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="equation.html#T21" target="_self">Th21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="msualg_3.html#K6">#</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="equation.html#E1:34_7_1_1"><i><font color="Green">E31</font></i></a>, <a class="txt" href="equation.html#E7:34_7_1_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="pralg_3.html#T12">PRALG_3:12</a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_7_1_2"><b>suppose </b></a><a NAME="E1:34_7_1_2"/><i><font color="Green">E31</font></i>: 
 <a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">c<sub>22</sub></font> =  <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>14</sub></font></span>)</span> as  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="ref" href="finseq_1.html#T26">FINSEQ_1:26</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] means ex <font color="Olive">b<sub>1</sub></font> being  <a href="funct_1.html#NM1">Function</a>ex <font color="Olive">b<sub>2</sub></font> being <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>3</sub></font> being <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>2</sub></font> is    <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Olive">b<sub>3</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>3</sub></font> ) &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="funct_1.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>14</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> );<br/><a NAME="E3:34_7_1_2"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>22</sub></font>ex <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="msualg_2.html#K14">MSSub</a> <font color="Maroon">c<sub>2</sub></font> st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_7_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>23</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>22</sub></font>;<br/>

<a NAME="E1:34_7_1_2_1"/>
<span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>23</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>
 
<b>by </b><i><a class="ref" href="partfun1.html#T27">PARTFUN1:27</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>24</sub></font> being   <a href="funct_2.html#NM1">Function</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>12</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>14</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>23</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> <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>14</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span><b> such that </b><br/><a NAME="E3:34_7_1_2_1"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</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>14</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> &amp; ( 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> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>12</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>14</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>24</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>23</sub></font>] ) )
 <b>by </b><i><a class="txt" href="equation.html#E18:34"><i><font color="Green">E25</font></i></a></i>;<br/>
<a NAME="E4:34_7_1_2_1"/>
<font color="Maroon">c<sub>15</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>12</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>14</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="msualg_6.html#T2">MSUALG_6:2</a></i>;<br/>
<a NAME="E5:34_7_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>15</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>12</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>14</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>25</sub></font> being   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>26</sub></font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K11">SORTS</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>25</sub></font>, <font color="Maroon">c<sub>27</sub></font> being  <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E7:34_7_1_2_1"/>
<font color="Maroon">c<sub>25</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>14</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>23</sub></font>
 <b>and </b><br/><a NAME="E8:34_7_1_2_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>23</sub></font>
 <b>and </b><br/><a NAME="E9:34_7_1_2_1"/><i><font color="Green">E35</font></i>: 
( ( for <font color="Olive">b<sub>1</sub></font> being <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Maroon">c<sub>27</sub></font> is    <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>26</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>26</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> ) &amp; <font color="Maroon">c<sub>24</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>26</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>27</sub></font> )
 <b>by </b><i><a class="txt" href="equation.html#E3:34_7_1_2_1"><i><font color="Green">E33</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>28</sub></font> = <font color="Maroon">c<sub>27</sub></font> as    <a href="subset_1.html#M1">Element</a> of  <a href="msualg_2.html#K14">MSSub</a> <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="msualg_2.html#D20">MSUALG_2:def 20</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>28</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>26</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>27</sub></font>
;<br/>

<b>thus </b><a NAME="E11:34_7_1_2_1"/>
<font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>23</sub></font>
 <b>by </b><i><a class="txt" href="equation.html#E8:34_7_1_2_1"><i><font color="Green">E34</font></i></a></i>;<br/>

<b>thus </b><a NAME="E12:34_7_1_2_1"/>
for <font color="Olive">b<sub>1</sub></font> being <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Maroon">c<sub>27</sub></font> is    <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>26</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>26</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="equation.html#E9:34_7_1_2_1"><i><font color="Green">E35</font></i></a></i>;<br/>

<b>thus </b><a NAME="E13:34_7_1_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="funct_1.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>14</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>26</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>27</sub></font>
 <b>by </b><i><a class="txt" href="equation.html#E3:34_7_1_2_1"><i><font color="Green">E33</font></i></a>, <a class="txt" href="equation.html#E8:34_7_1_2_1"><i><font color="Green">E34</font></i></a>, <a class="txt" href="equation.html#E9:34_7_1_2_1"><i><font color="Green">E35</font></i></a></i>;<br/>

<b>thus </b><a NAME="E14:34_7_1_2_1"/>
<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>27</sub></font>
 ;<br/>


</div><b>end;</b></div><b>consider </b><font color="Maroon">c<sub>23</sub></font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>22</sub></font>, <a href="msualg_2.html#K14">MSSub</a> <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E5:34_7_1_2"/><i><font color="Green">E33</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>22</sub></font> holds <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>23</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>(<a class="txt" href="equation.html#E3:34_7_1_2"><i><font color="Green">E32</font></i></a>);<br/></i><a NAME="E6:34_7_1_2"/>
<font color="Maroon">c<sub>23</sub></font> is    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">c<sub>22</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_7_1_2_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:34_7_1_2_2"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a><br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">c<sub>24</sub></font> = <font color="Maroon">c<sub>23</sub></font> as    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">c<sub>22</sub></font> ;<br/><a NAME="E8:34_7_1_2"/>
<font color="Maroon">c<sub>24</sub></font> is    <a href="pralg_2.html#M2">MSAlgebra-Family</a> of <font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_7_1_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>25</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="pralg_2.html#D12">PRALG_2:def 12</a><br/>

<b>assume </b><a NAME="E1:34_7_1_2_3"/>
<font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>22</sub></font>
 ;<br/>

<a NAME="E2:34_7_1_2_3"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="funct_1.html#NM1">Function</a>ex <font color="Olive">b<sub>2</sub></font> being <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>25</sub></font> &amp; ( for <font color="Olive">b<sub>3</sub></font> being <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>2</sub></font> is    <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Olive">b<sub>3</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>3</sub></font> ) &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="funct_1.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>14</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>25</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Maroon">c<sub>24</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="equation.html#E5:34_7_1_2"><i><font color="Green">E33</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:34_7_1_2_3"/>
<font color="Maroon">c<sub>24</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>25</sub></font> is  <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">c<sub>1</sub></font>
 ;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">c<sub>25</sub></font> = <font color="Maroon">c<sub>24</sub></font> as    <a href="pralg_2.html#M2">MSAlgebra-Family</a> of <font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>1</sub></font> ;<br/><a NAME="E10:34_7_1_2"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>22</sub></font>ex <font color="Olive">b<sub>2</sub></font> being <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>25</sub></font> <a href="pralg_2.html#K6">.</a> <font color="Olive">b<sub>1</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_7_1_2_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>26</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>22</sub></font>;<br/>

<a NAME="E1:34_7_1_2_4"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="funct_1.html#NM1">Function</a>ex <font color="Olive">b<sub>2</sub></font> being <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>26</sub></font> &amp; ( for <font color="Olive">b<sub>3</sub></font> being <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>2</sub></font> is    <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Olive">b<sub>3</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>3</sub></font> ) &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="funct_1.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>14</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>26</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Maroon">c<sub>25</sub></font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="equation.html#E5:34_7_1_2"><i><font color="Green">E33</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:34_7_1_2_4"/>
ex <font color="Olive">b<sub>1</sub></font> being <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>25</sub></font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">c<sub>26</sub></font>
 ;<br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon">c<sub>26</sub></font> being  <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E12:34_7_1_2"/><i><font color="Green">E34</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>22</sub></font> holds <font color="Maroon">c<sub>25</sub></font> <a href="pralg_2.html#K6">.</a> <font color="Olive">b<sub>1</sub></font> is    <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">c<sub>26</sub></font>
 <b>by </b><i><a class="ref" href="equation.html#T27" target="_self">Th27</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>27</sub></font> being  <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E14:34_7_1_2"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>21</sub></font> is    <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">c<sub>27</sub></font>
 <b>and </b><br/><a NAME="E15:34_7_1_2"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>26</sub></font> is    <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">c<sub>27</sub></font>
 <b>by </b><i><a class="ref" href="equation.html#T28" target="_self">Th28</a></i>;<br/><a NAME="E16:34_7_1_2"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a>  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">c<sub>14</sub></font>,<span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="equation.html#T20" target="_self">Th20</a></i>;<br/><a NAME="E17:34_7_1_2"/>
<font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R2">in</a>  <a href="msualg_2.html#K14">MSSub</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="msualg_2.html#D20">MSUALG_2:def 20</a></i>;<br/><a NAME="E18:34_7_1_2"/><b>then </b>
<font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="equation.html#E1:34"><i><font color="Green">E18</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>28</sub></font> = <font color="Maroon">c<sub>27</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>6</sub></font> ;<br/><b>reconsider </b><font color="Maroon">c<sub>29</sub></font> = <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">c<sub>14</sub></font>,<span class="p2">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font> as   <a href="funct_1.html#NM1">Function</a> <b>by </b><i><a class="ref" href="equation.html#T22" target="_self">Th22</a></i>;<br/><a NAME="E21:34_7_1_2"/>
<span class="p1">(<span class="default"><a href="funct_6.html#K10">commute</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font> is    <a href="subset_1.html#M1">Element</a> of  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">c<sub>14</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">c<sub>28</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="equation.html#E1:34_7_1_2"><i><font color="Green">E31</font></i></a>, <a class="txt" href="equation.html#E16:34_7_1_2"><i><font color="Green">E37</font></i></a>, <a class="ref" href="pralg_3.html#T21">PRALG_3:21</a></i>;<br/><a NAME="E22:34_7_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="funct_6.html#K10">commute</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a>  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">c<sub>14</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">c<sub>28</sub></font></span>)</span>
 ;<br/><a NAME="E23:34_7_1_2"/><b>then </b><i><font color="Green">E38</font></i>: 
<span class="p1">(<span class="default"><a href="funct_6.html#K10">commute</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a>  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>27</sub></font>
 <b>by </b><i><a class="txt" href="equation.html#E2:34"><i><font color="Green">E19</font></i></a></i>;<br/><a NAME="E24:34_7_1_2"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>6</sub></font>,<span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <span class="p5">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pralg_2.html#K6">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>6</sub></font>, B is    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> : verum </span>}</span> </span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="equation.html#E16:34_7_1_2"><i><font color="Green">E37</font></i></a>, <a class="ref" href="pralg_3.html#T15">PRALG_3:15</a></i>;<br/><a NAME="E25:34_7_1_2"/><b>then </b><i><font color="Green">E40</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_6.html#K10">commute</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font>
 <b>by </b><i><a class="ref" href="equation.html#T3" target="_self">Th3</a></i>;<br/><a NAME="E26:34_7_1_2"/><i><font color="Green">E41</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="msualg_3.html#K6">#</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font>
 <b>by </b><i><a class="ref" href="msualg_6.html#T2">MSUALG_6:2</a></i>;<br/><a NAME="E27:34_7_1_2"/><i><font color="Green">E42</font></i>: 
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="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_6.html#K10">commute</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="msualg_3.html#K6">#</a> <font color="Maroon">c<sub>15</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"><span class="p2">(<span class="default"><a href="funct_6.html#K10">commute</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</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="34_7_1_2_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>30</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:34_7_1_2_5"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_6.html#K10">commute</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>31</sub></font> = <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>30</sub></font> as   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="equation.html#E25:34_7_1_2"><i><font color="Green">E40</font></i></a>, <a class="ref" href="partfun1.html#T27">PARTFUN1:27</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>32</sub></font> being   <a href="funct_1.html#NM1">Function</a>, <font color="Maroon">c<sub>33</sub></font> being  <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E4:34_7_1_2_5"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>30</sub></font>
 <b>and </b><br/><a NAME="E5:34_7_1_2_5"/><i><font color="Green">E45</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Maroon">c<sub>33</sub></font> is    <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>32</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>32</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 <b>and </b><br/><a NAME="E6:34_7_1_2_5"/><i><font color="Green">E46</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>31</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>32</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>33</sub></font>
 <b>and </b><br/><a NAME="E7:34_7_1_2_5"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>25</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>33</sub></font>
 <b>by </b><i><a class="txt" href="equation.html#E5:34_7_1_2"><i><font color="Green">E33</font></i></a>, <a class="txt" href="equation.html#E25:34_7_1_2"><i><font color="Green">E40</font></i></a>, <a class="txt" href="equation.html#E1:34_7_1_2_5"><i><font color="Green">E43</font></i></a></i>;<br/>
<a NAME="E8:34_7_1_2_5"/>
<font color="Maroon">c<sub>33</sub></font> is    <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">c<sub>26</sub></font>
 
<b>by </b><i><a class="txt" href="equation.html#E12:34_7_1_2"><i><font color="Green">E34</font></i></a>, <a class="txt" href="equation.html#E25:34_7_1_2"><i><font color="Green">E40</font></i></a>, <a class="txt" href="equation.html#E1:34_7_1_2_5"><i><font color="Green">E43</font></i></a>, <a class="txt" href="equation.html#E7:34_7_1_2_5"><i><font color="Green">E47</font></i></a></i>;<br/>
<a NAME="E9:34_7_1_2_5"/><b>then </b><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>33</sub></font> is    <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">c<sub>27</sub></font>
 
<b>by </b><i><a class="txt" href="equation.html#E15:34_7_1_2"><i><font color="Green">E36</font></i></a>, <a class="ref" href="msualg_2.html#T7">MSUALG_2:7</a></i>;<br/>
<a NAME="E10:34_7_1_2_5"/>
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font>
 
<b>by </b><i><a class="txt" href="equation.html#E24:34_7_1_2"><i><font color="Green">E39</font></i></a>, <a class="ref" href="funct_2.html#T169">FUNCT_2:169</a></i>;<br/>
<b>hence </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="msualg_3.html#K6">#</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>30</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>13</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>14</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>30</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>30</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="equation.html#E25:34_7_1_2"><i><font color="Green">E40</font></i></a>, <a class="txt" href="equation.html#E1:34_7_1_2_5"><i><font color="Green">E43</font></i></a>, <a class="ref" href="msualg_3.html#D8">MSUALG_3:def 8</a></i>
<br/>.= 
<font color="Maroon">c<sub>32</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>25</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>30</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="equation.html#E25:34_7_1_2"><i><font color="Green">E40</font></i></a>, <a class="txt" href="equation.html#E1:34_7_1_2_5"><i><font color="Green">E43</font></i></a>, <a class="txt" href="equation.html#E4:34_7_1_2_5"><i><font color="Green">E44</font></i></a>, <a class="txt" href="equation.html#E6:34_7_1_2_5"><i><font color="Green">E46</font></i></a>, <a class="txt" href="equation.html#E7:34_7_1_2_5"><i><font color="Green">E47</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<font color="Maroon">c<sub>32</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font>
<b>by </b><i><a class="txt" href="equation.html#E5:34_7_1_2_5"><i><font color="Green">E45</font></i></a>, <a class="txt" href="equation.html#E7:34_7_1_2_5"><i><font color="Green">E47</font></i></a>, <a class="txt" href="equation.html#E9:34_7_1_2_5"><i><font color="Green">E48</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_6.html#K10">commute</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>30</sub></font>
<b>by </b><i><a class="txt" href="equation.html#E16:34_7_1_2"><i><font color="Green">E37</font></i></a>, <a class="txt" href="equation.html#E25:34_7_1_2"><i><font color="Green">E40</font></i></a>, <a class="txt" href="equation.html#E1:34_7_1_2_5"><i><font color="Green">E43</font></i></a>, <a class="txt" href="equation.html#E4:34_7_1_2_5"><i><font color="Green">E44</font></i></a>, <a class="ref" href="pralg_3.html#T22">PRALG_3:22</a></i>
;<br/><br/>


</div><b>end;</b></div><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="msualg_3.html#K1">.</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> = 
<font color="Maroon">c<sub>20</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>27</sub></font>
<b>by </b><i><a class="txt" href="equation.html#E3:34_7"><i><font color="Green">E26</font></i></a>, <a class="txt" href="equation.html#E9:34_7"><i><font color="Green">E29</font></i></a>, <a class="txt" href="equation.html#E10:34_7"><i><font color="Green">E30</font></i></a>, <a class="txt" href="equation.html#E14:34_7_1_2"><i><font color="Green">E35</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>29</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>27</sub></font>
<b>by </b><i><a class="txt" href="equation.html#E8:34_7"><i><font color="Green">E28</font></i></a>, <a class="ref" href="equation.html#T21" target="_self">Th21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">c<sub>14</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">c<sub>28</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_6.html#K10">commute</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="equation.html#E1:34_7_1_2"><i><font color="Green">E31</font></i></a>, <a class="txt" href="equation.html#E16:34_7_1_2"><i><font color="Green">E37</font></i></a>, <a class="ref" href="pralg_3.html#T23">PRALG_3:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>27</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_6.html#K10">commute</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="equation.html#E2:34"><i><font color="Green">E19</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_6.html#K10">commute</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="equation.html#E23:34_7_1_2"><i><font color="Green">E38</font></i></a>, <a class="ref" href="equation.html#T21" target="_self">Th21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="msualg_3.html#K6">#</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="equation.html#E25:34_7_1_2"><i><font color="Green">E40</font></i></a>, <a class="txt" href="equation.html#E26:34_7_1_2"><i><font color="Green">E41</font></i></a>, <a class="txt" href="equation.html#E27:34_7_1_2"><i><font color="Green">E42</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>
;<br/></div><b>end;</b></div></div><b>end;</b></div>

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

<b>let </b><font color="Maroon">c<sub>14</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="E20:34"/>
<font color="Maroon">c<sub>14</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/>

<b>then reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>14</sub></font> as   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E22:34"/><i><font color="Green">E26</font></i>: 
<span class="p1">(<span class="default"><a href="pralg_2.html#K11">SORTS</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>15</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="pralg_2.html#D17">PRALG_2:def 17</a></i>;<br/>
<a NAME="E23:34"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> 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>15</sub></font>
 
;<br/>
<b>hence </b><a NAME="E24:34"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> 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>14</sub></font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

<b>let </b><font color="Maroon">c<sub>16</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E25:34"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> 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>14</sub></font>
 ;<br/>

<a NAME="E26:34"/><i><font color="Green">E28</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>17</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E28:34"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E29:34"/>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>15</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="equation.html#E8:34"><i><font color="Green">E21</font></i></a>, <a class="txt" href="equation.html#E12:34"><i><font color="Green">E22</font></i></a>, <a class="txt" href="equation.html#E22:34"><i><font color="Green">E26</font></i></a>, <a class="txt" href="equation.html#E28:34"><i><font color="Green">E29</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>18</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E31:34"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>15</sub></font></span>)</span> &amp; ( 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="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>15</sub></font></span>)</span> holds <br/><font color="Maroon">c<sub>18</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>by </b><i><a class="ref" href="card_3.html#D5">CARD_3:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>19</sub></font> being  <a href="relat_1.html#V2">V2</a> <a href="pre_circ.html#V1">locally-finite</a> <a href="msualg_2.html#NM1">MSSubset</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>20</sub></font> = <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>15</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="pre_circ.html#K2">--&gt;</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>16</sub></font></span><a href="tarski.html#K1">}</a></span>;<br/>
<a NAME="E33:34"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>15</sub></font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E34:34"/><i><font color="Green">E32</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>15</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="pre_circ.html#K2">--&gt;</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>16</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>15</sub></font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/>
<i><font color="Green">E33</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>15</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="pre_circ.html#K2">--&gt;</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>16</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>15</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="pre_circ.html#K2">--&gt;</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>16</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>15</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="pre_circ.html#K2">--&gt;</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>16</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>15</sub></font></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="ref" href="zfmisc_1.html#T46">ZFMISC_1:46</a></i>
;<br/>
<a NAME="E36:34"/>
<font color="Maroon">c<sub>19</sub></font> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>15</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="pre_circ.html#K2">--&gt;</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>16</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span> is    <a href="pboole.html#M1">ManySortedSet</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_9"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:34_9"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>15</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="pre_circ.html#K2">--&gt;</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>16</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="equation.html#E35:34"><i><font color="Green">E33</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a><br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>21</sub></font> = <font color="Maroon">c<sub>19</sub></font> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>15</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="pre_circ.html#K2">--&gt;</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>16</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span> as    <a href="pboole.html#M1">ManySortedSet</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<i><font color="Green">E34</font></i>: <font color="Maroon">c<sub>21</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font> = 
<span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>15</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="pre_circ.html#K2">--&gt;</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>16</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font>
<b>by </b><i><a class="txt" href="equation.html#E33:34"><i><font color="Green">E31</font></i></a>, <a class="txt" href="equation.html#E34:34"><i><font color="Green">E32</font></i></a>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>16</sub></font></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="txt" href="equation.html#E33:34"><i><font color="Green">E31</font></i></a>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
;<br/>
<div><i><font color="Green">E35</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="34_11"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>22</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:34_11"/>
( <font color="Maroon">c<sub>22</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>22</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>15</sub></font> )
 ;<br/><a NAME="E2:34_11"/><b>then </b>
not <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>15</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="pre_circ.html#K2">--&gt;</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>16</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="equation.html#E34:34"><i><font color="Green">E32</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:34_11"/>
<font color="Maroon">c<sub>21</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font>
 <b>by </b><i><a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>;<br/></div><b>end;</b></div>
<a NAME="E40:34"/>
<font color="Maroon">c<sub>21</sub></font> is    <a href="pboole.html#M4">ManySortedSubset</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_12"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>22</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="pboole.html#D5">PBOOLE:def 5</a>,<a class="ref" href="pboole.html#D23">PBOOLE:def 23</a><br/>

<b>assume </b><a NAME="E1:34_12"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>22</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/>

<b>let </b><font color="Maroon">c<sub>23</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:34_12"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>21</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_12_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:34_12_1"/>
( <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>15</sub></font> or <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_12_1_1"><b>suppose </b></a><a NAME="E1:34_12_1_1"/>
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>15</sub></font>
 ;<br/><div class="add"><a NAME="E2:34_12_1_1"/><b>then </b><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>19</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font>
 <b>by </b><i><a class="txt" href="equation.html#E39:34"><i><font color="Green">E35</font></i></a>, <a class="txt" href="equation.html#E1:34_12"><i><font color="Green">E36</font></i></a>, <a class="txt" href="equation.html#E2:34_12"><i><font color="Green">E37</font></i></a></i>;<br/><a NAME="E3:34_12_1_1"/>
<font color="Maroon">c<sub>19</sub></font> <a href="pboole.html#R2">c=</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#D23">PBOOLE:def 23</a></i>;<br/><a NAME="E4:34_12_1_1"/><b>then </b>
<font color="Maroon">c<sub>19</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font> <a href="tarski.html#R1">c=</a> 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>22</sub></font>
 <b>by </b><i><a class="txt" href="equation.html#E1:34_12"><i><font color="Green">E36</font></i></a>, <a class="ref" href="pboole.html#D5">PBOOLE:def 5</a></i>;<br/><b>hence </b><a NAME="E5:34_12_1_1"/>
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a> 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>22</sub></font>
 <b>by </b><i><a class="txt" href="equation.html#E2:34_12_1_1"><i><font color="Green">E38</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_12_1_2"><b>suppose </b></a><a NAME="E1:34_12_1_2"/>
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:34_12_1_2"/>
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a> 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>22</sub></font>
 <b>by </b><i><a class="txt" href="equation.html#E25:34"><i><font color="Green">E27</font></i></a>, <a class="txt" href="equation.html#E38:34"><i><font color="Green">E34</font></i></a>, <a class="txt" href="equation.html#E2:34_12"><i><font color="Green">E37</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>22</sub></font> = <font color="Maroon">c<sub>21</sub></font> as   <a href="msualg_2.html#NM1">MSSubset</a> of <font color="Maroon">c<sub>2</sub></font> ;<br/>
<a NAME="E42:34"/>
<font color="Maroon">c<sub>22</sub></font> is <a href="relat_1.html#V2">non-empty</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_13"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>23</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="pboole.html#D16">PBOOLE:def 16</a><br/>

<b>assume </b><a NAME="E1:34_13"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>23</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/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_13_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:34_13_1"/>
( <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>15</sub></font> or <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_13_1_1"><b>suppose </b></a><a NAME="E1:34_13_1_1"/>
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>15</sub></font>
 ;<br/><div class="add"><a NAME="E2:34_13_1_1"/><b>then </b>
<font color="Maroon">c<sub>22</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>23</sub></font>
 <b>by </b><i><a class="txt" href="equation.html#E39:34"><i><font color="Green">E35</font></i></a>, <a class="txt" href="equation.html#E1:34_13"><i><font color="Green">E36</font></i></a></i>;<br/><b>hence </b><a NAME="E3:34_13_1_1"/>
not <font color="Maroon">c<sub>22</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>23</sub></font> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i><a class="txt" href="equation.html#E1:34_13"><i><font color="Green">E36</font></i></a>, <a class="ref" href="pboole.html#D16">PBOOLE:def 16</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_13_1_2"><b>suppose </b></a><a NAME="E1:34_13_1_2"/>
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:34_13_1_2"/>
not <font color="Maroon">c<sub>22</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>23</sub></font> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i><a class="txt" href="equation.html#E38:34"><i><font color="Green">E34</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>23</sub></font> = <font color="Maroon">c<sub>22</sub></font> as  <a href="relat_1.html#V2">V2</a> <a href="msualg_2.html#NM1">MSSubset</a> of <font color="Maroon">c<sub>2</sub></font> ;<br/>
<a NAME="E44:34"/>
<font color="Maroon">c<sub>23</sub></font> is <a href="pre_circ.html#V1">locally-finite</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_14"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>24</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="pre_circ.html#D3">PRE_CIRC:def 3</a><br/>

<b>assume </b><a NAME="E1:34_14"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>24</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/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_14_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:34_14_1"/>
( <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>15</sub></font> or <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_14_1_1"><b>suppose </b></a><a NAME="E1:34_14_1_1"/>
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>15</sub></font>
 ;<br/><div class="add"><a NAME="E2:34_14_1_1"/><b>then </b>
<font color="Maroon">c<sub>23</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font>
 <b>by </b><i><a class="txt" href="equation.html#E39:34"><i><font color="Green">E35</font></i></a>, <a class="txt" href="equation.html#E1:34_14"><i><font color="Green">E36</font></i></a></i>;<br/><b>hence </b><a NAME="E3:34_14_1_1"/>
<font color="Maroon">c<sub>23</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="txt" href="equation.html#E1:34_14"><i><font color="Green">E36</font></i></a>, <a class="ref" href="pre_circ.html#D3">PRE_CIRC:def 3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_14_1_2"><b>suppose </b></a><a NAME="E1:34_14_1_2"/>
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:34_14_1_2"/>
<font color="Maroon">c<sub>23</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="txt" href="equation.html#E38:34"><i><font color="Green">E34</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>24</sub></font> = <font color="Maroon">c<sub>23</sub></font> as  <a href="relat_1.html#V2">V2</a> <a href="pre_circ.html#V1">locally-finite</a> <a href="msualg_2.html#NM1">MSSubset</a> of <font color="Maroon">c<sub>2</sub></font> ;<br/>
<a NAME="E46:34"/>
<font color="Maroon">c<sub>24</sub></font> is   <a href="msualg_2.html#NM1">MSSubset</a> of <span class="p1">(<span class="default"><a href="msualg_2.html#K12">GenMSAlg</a> <font color="Maroon">c<sub>24</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="msualg_2.html#D18">MSUALG_2:def 18</a></i>;<br/>
<a NAME="E47:34"/><b>then </b>
<font color="Maroon">c<sub>24</sub></font> <a href="pboole.html#R2">c=</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_2.html#K12">GenMSAlg</a> <font color="Maroon">c<sub>24</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="pboole.html#D23">PBOOLE:def 23</a></i>;<br/>
<a NAME="E48:34"/><b>then </b><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>24</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font> <a href="tarski.html#R1">c=</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_2.html#K12">GenMSAlg</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font>
 
<b>by </b><i><a class="ref" href="pboole.html#D5">PBOOLE:def 5</a></i>;<br/>
<a NAME="E49:34"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>24</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font>
 
<b>by </b><i><a class="txt" href="equation.html#E38:34"><i><font color="Green">E34</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E50:34"/><b>then </b><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_2.html#K12">GenMSAlg</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font>
 
<b>by </b><i><a class="txt" href="equation.html#E48:34"><i><font color="Green">E36</font></i></a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="hidden.html#M1">set</a> ] means ex <font color="Olive">b<sub>1</sub></font> being   <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font> );<br/>
<b>consider </b><font color="Maroon">c<sub>25</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E52:34"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>25</sub></font> iff ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1">XBOOLE_0:sch 1</a>();<br/></i>
<a NAME="E53:34"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>25</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_15"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>26</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:34_15"/>
<font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>25</sub></font>
 ;<br/>

<b>hence </b><a NAME="E2:34_15"/>
<font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="equation.html#E52:34"><i><font color="Green">E39</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>set </b><font color="Maroon">c<sub>26</sub></font> = <font color="Maroon">c<sub>18</sub></font> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>25</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">c<sub>16</sub></font></span>)</span>;<br/>
<a NAME="E54:34"/><i><font color="Green">E41</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>25</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>25</sub></font>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/>
<i><font color="Green">E42</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>25</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">c<sub>16</sub></font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>25</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">c<sub>16</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>
<br/>.= 
<font color="Maroon">c<sub>6</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>25</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">c<sub>16</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="equation.html#E31:34"><i><font color="Green">E30</font></i></a>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
<br/>.= 
<font color="Maroon">c<sub>6</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>25</sub></font>
<b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>
<br/>.= 
<font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="txt" href="equation.html#E53:34"><i><font color="Green">E40</font></i></a>, <a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>
;<br/>
<a NAME="E56:34"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>6</sub></font> st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>25</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">c<sub>16</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_17"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>27</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>6</sub></font>;<br/>

<b>assume </b><a NAME="E1:34_17"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">c<sub>27</sub></font>]
 ;<br/>

<a NAME="E2:34_17"/><b>then </b><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>25</sub></font>
 
<b>by </b><i><a class="txt" href="equation.html#E52:34"><i><font color="Green">E39</font></i></a></i>;<br/>
<b>hence </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>25</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">c<sub>16</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>27</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>25</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>27</sub></font>
<b>by </b><i><a class="txt" href="equation.html#E54:34"><i><font color="Green">E41</font></i></a>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
<br/>.= 
<font color="Maroon">c<sub>16</sub></font>
<b>by </b><i><a class="txt" href="equation.html#E2:34_17"><i><font color="Green">E44</font></i></a>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
;<br/><br/>


</div><b>end;</b></div>
<a NAME="E57:34"/><i><font color="Green">E44</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E58:34"/>
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="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>15</sub></font></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>25</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">c<sub>16</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>15</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="34_18"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>27</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:34_18"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>15</sub></font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>28</sub></font> being    <a href="subset_1.html#M1">Element</a> of  <a href="msualg_2.html#K14">MSSub</a> <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E3:34_18"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>28</sub></font>
 <b>and </b><br/><a NAME="E4:34_18"/><i><font color="Green">E47</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>28</sub></font>
 <b>by </b><i><a class="txt" href="equation.html#E1:34"><i><font color="Green">E18</font></i></a>, <a class="txt" href="equation.html#E57:34"><i><font color="Green">E44</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>29</sub></font> being  <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E6:34_18"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>28</sub></font>
 <b>by </b><i><a class="txt" href="equation.html#E4:34_18"><i><font color="Green">E47</font></i></a></i>;<br/>
<a NAME="E7:34_18"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="equation.html#E1:34"><i><font color="Green">E18</font></i></a>, <a class="txt" href="equation.html#E6:34_18"><i><font color="Green">E48</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>30</sub></font> being    <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E9:34_18"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>29</sub></font>
 <b>and </b><br/><a NAME="E10:34_18"/><i><font color="Green">E51</font></i>: 
<span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>30</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="ref" href="pralg_2.html#D16">PRALG_2:def 16</a></i>;<br/>
<a NAME="E11:34_18"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>29</sub></font>
 
<b>by </b><i><a class="txt" href="equation.html#E2:34"><i><font color="Green">E19</font></i></a>, <a class="txt" href="equation.html#E7:34_18"><i><font color="Green">E49</font></i></a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_18_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:34_18_1"/>
( <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">c<sub>27</sub></font>] or  not <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">c<sub>27</sub></font>] )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_18_1_1"><b>suppose </b></a><a NAME="E1:34_18_1_1"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">c<sub>27</sub></font>]
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>31</sub></font> being    <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E3:34_18_1_1"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>27</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>31</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font> )
 ;<br/><b>thus </b><a NAME="E4:34_18_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>25</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">c<sub>16</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>27</sub></font>
 <b>by </b><i><a class="txt" href="equation.html#E56:34"><i><font color="Green">E43</font></i></a>, <a class="txt" href="equation.html#E3:34_18"><i><font color="Green">E46</font></i></a>, <a class="txt" href="equation.html#E6:34_18"><i><font color="Green">E48</font></i></a>, <a class="txt" href="equation.html#E7:34_18"><i><font color="Green">E49</font></i></a>, <a class="txt" href="equation.html#E9:34_18"><i><font color="Green">E50</font></i></a>, <a class="txt" href="equation.html#E10:34_18"><i><font color="Green">E51</font></i></a>, <a class="txt" href="equation.html#E11:34_18"><i><font color="Green">E52</font></i></a>, <a class="txt" href="equation.html#E3:34_18_1_1"><i><font color="Green">E53</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_18_1_2"><b>suppose </b></a><a NAME="E1:34_18_1_2"/>
 not <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">c<sub>27</sub></font>]
 ;<br/><div class="add"><a NAME="E2:34_18_1_2"/><b>then </b>
not <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>25</sub></font>
 <b>by </b><i><a class="txt" href="equation.html#E52:34"><i><font color="Green">E39</font></i></a></i>;<br/><a NAME="E3:34_18_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>25</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">c<sub>16</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>27</sub></font>
 <b>by </b><i><a class="txt" href="equation.html#E54:34"><i><font color="Green">E41</font></i></a>, <a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>;<br/><b>hence </b><a NAME="E4:34_18_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>25</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">c<sub>16</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>27</sub></font>
 <b>by </b><i><a class="txt" href="equation.html#E31:34"><i><font color="Green">E30</font></i></a>, <a class="txt" href="equation.html#E1:34_18"><i><font color="Green">E45</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>27</sub></font> = <font color="Maroon">c<sub>18</sub></font> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>25</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> as    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K11">SORTS</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font> <b>by </b><i><a class="txt" href="equation.html#E22:34"><i><font color="Green">E26</font></i></a>, <a class="txt" href="equation.html#E55:34"><i><font color="Green">E42</font></i></a>, <a class="txt" href="equation.html#E57:34"><i><font color="Green">E44</font></i></a>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>28</sub></font> being   <a href="funct_2.html#NM1">Function</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font>,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>15</sub></font><b> such that </b><br/><a NAME="E61:34"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>28</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font>
 <b>and </b><br/><a NAME="E62:34"/><i><font color="Green">E46</font></i>: 
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> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>28</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>15</sub></font>]
 <b>by </b><i><a class="txt" href="equation.html#E18:34"><i><font color="Green">E25</font></i></a></i>;<br/>
<a NAME="E63:34"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>27</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_19"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon">c<sub>15</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>27</sub></font>
;<br/>

<b>take </b><font color="Maroon">c<sub>29</sub></font> =  <a href="msualg_2.html#K12">GenMSAlg</a> <font color="Maroon">c<sub>24</sub></font>;<br/>

<b>thus </b><a NAME="E1:34_19"/>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font>
 ;<br/>

<b>thus </b><a NAME="E2:34_19"/>
<font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>27</sub></font>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>30</sub></font> be  <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>

<b>assume </b><a NAME="E3:34_19"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>29</sub></font> is    <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">c<sub>30</sub></font>
 ;<br/>

<a NAME="E4:34_19"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> )
 
<b>by </b><i><a class="txt" href="equation.html#E3:34"><i><font color="Green">E20</font></i></a></i>;<br/>
<a NAME="E5:34_19"/>
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>29</sub></font> is    <a href="pboole.html#M4">ManySortedSubset</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>30</sub></font>
 
<b>by </b><i><a class="txt" href="equation.html#E3:34_19"><i><font color="Green">E47</font></i></a>, <a class="ref" href="msualg_2.html#D10">MSUALG_2:def 10</a></i>;<br/>
<a NAME="E6:34_19"/><b>then </b>
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>29</sub></font> <a href="pboole.html#R2">c=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>30</sub></font>
 
<b>by </b><i><a class="ref" href="pboole.html#D23">PBOOLE:def 23</a></i>;<br/>
<a NAME="E7:34_19"/><b>then </b><i><font color="Green">E49</font></i>: 
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>29</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font> <a href="tarski.html#R1">c=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>30</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font>
 
<b>by </b><i><a class="ref" href="pboole.html#D5">PBOOLE:def 5</a></i>;<br/>
<b>thus </b><font color="Maroon">c<sub>27</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>29</sub></font> = 
<font color="Maroon">c<sub>16</sub></font>
<b>by </b><i><a class="txt" href="equation.html#E48:34"><i><font color="Green">E36</font></i></a>, <a class="txt" href="equation.html#E49:34"><i><font color="Green">E37</font></i></a>, <a class="txt" href="equation.html#E56:34"><i><font color="Green">E43</font></i></a>, <a class="txt" href="equation.html#E4:34_19"><i><font color="Green">E48</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>27</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>30</sub></font>
<b>by </b><i><a class="txt" href="equation.html#E50:34"><i><font color="Green">E38</font></i></a>, <a class="txt" href="equation.html#E56:34"><i><font color="Green">E43</font></i></a>, <a class="txt" href="equation.html#E4:34_19"><i><font color="Green">E48</font></i></a>, <a class="txt" href="equation.html#E7:34_19"><i><font color="Green">E49</font></i></a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E64:34"/><b>then </b><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font>
 
<b>by </b><i><a class="txt" href="equation.html#E8:34"><i><font color="Green">E21</font></i></a></i>;<br/>
<a NAME="E65:34"/><b>then </b>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>28</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>27</sub></font>,<font color="Maroon">c<sub>27</sub></font>,<font color="Maroon">c<sub>15</sub></font>]
 
<b>by </b><i><a class="txt" href="equation.html#E12:34"><i><font color="Green">E22</font></i></a>, <a class="txt" href="equation.html#E62:34"><i><font color="Green">E46</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>29</sub></font> being   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>30</sub></font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K11">SORTS</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font>, <font color="Maroon">c<sub>31</sub></font> being  <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E67:34"/>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>29</sub></font>
 <b>and </b><br/><a NAME="E68:34"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>27</sub></font>
 <b>and </b><br/><a NAME="E69:34"/><i><font color="Green">E49</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Maroon">c<sub>31</sub></font> is    <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>30</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>30</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 <b>and </b><br/><a NAME="E70:34"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>28</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>30</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>31</sub></font>
 ;<br/>
<b>consider </b><font color="Maroon">c<sub>32</sub></font> being  <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E72:34"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>31</sub></font> is    <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">c<sub>32</sub></font>
 <b>and </b><br/><a NAME="E73:34"/><i><font color="Green">E52</font></i>: 
 <a href="msualg_2.html#K12">GenMSAlg</a> <font color="Maroon">c<sub>24</sub></font> is    <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">c<sub>32</sub></font>
 <b>by </b><i><a class="ref" href="equation.html#T28" target="_self">Th28</a></i>;<br/>
<a NAME="E74:34"/>
the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_2.html#K12">GenMSAlg</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> is    <a href="pboole.html#M4">ManySortedSubset</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>32</sub></font>
 
<b>by </b><i><a class="txt" href="equation.html#E73:34"><i><font color="Green">E52</font></i></a>, <a class="ref" href="msualg_2.html#D10">MSUALG_2:def 10</a></i>;<br/>
<a NAME="E75:34"/><b>then </b>
the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_2.html#K12">GenMSAlg</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="pboole.html#R2">c=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>32</sub></font>
 
<b>by </b><i><a class="ref" href="pboole.html#D23">PBOOLE:def 23</a></i>;<br/>
<a NAME="E76:34"/><b>then </b><i><font color="Green">E53</font></i>: 
the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_2.html#K12">GenMSAlg</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font> <a href="tarski.html#R1">c=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>32</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font>
 
<b>by </b><i><a class="ref" href="pboole.html#D5">PBOOLE:def 5</a></i>;<br/>
<a NAME="E77:34"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="equation.html#E3:34"><i><font color="Green">E20</font></i></a></i>;<br/>
<font color="Maroon">c<sub>28</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>27</sub></font> = 
<font color="Maroon">c<sub>27</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>32</sub></font>
<b>by </b><i><a class="txt" href="equation.html#E68:34"><i><font color="Green">E48</font></i></a>, <a class="txt" href="equation.html#E69:34"><i><font color="Green">E49</font></i></a>, <a class="txt" href="equation.html#E70:34"><i><font color="Green">E50</font></i></a>, <a class="txt" href="equation.html#E72:34"><i><font color="Green">E51</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>16</sub></font>
<b>by </b><i><a class="txt" href="equation.html#E50:34"><i><font color="Green">E38</font></i></a>, <a class="txt" href="equation.html#E56:34"><i><font color="Green">E43</font></i></a>, <a class="txt" href="equation.html#E76:34"><i><font color="Green">E53</font></i></a>, <a class="txt" href="equation.html#E77:34"><i><font color="Green">E54</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E79:34"/>
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="equation.html#E12:34"><i><font color="Green">E22</font></i></a>, <a class="txt" href="equation.html#E26:34"><i><font color="Green">E28</font></i></a>, <a class="txt" href="equation.html#E61:34"><i><font color="Green">E45</font></i></a>, <a class="txt" href="equation.html#E64:34"><i><font color="Green">E47</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


</div>
