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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  <a href="relat_1.html#V2">V3</a>  <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="pralg_1.html#M1" target="_self">ManySortedOperation</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>





<b>thus </b><a NAME="E1:39"/>
( <font color="Maroon">c<sub>3</sub></font> is <a href="pralg_1.html#V4" target="_self">equal-arity</a> implies for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds  <a href="unialg_1.html#K2">arity</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="pralg_1.html#K14" target="_self">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="unialg_1.html#K2">arity</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="pralg_1.html#K14" target="_self">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:39_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="pralg_1.html#V4" target="_self">equal-arity</a>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<a NAME="E2:39_1"/><i><font color="Green">E23</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E3:39_1"/>
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="pralg_1.html#K14" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="unialg_1.html#K2">arity</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="pralg_1.html#K14" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="pralg_1.html#K14" target="_self">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="unialg_1.html#K2">arity</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="pralg_1.html#K14" target="_self">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="unialg_2.html#T2">UNIALG_2:2</a></i>;<br/>
<b>hence </b><a NAME="E4:39_1"/>
 <a href="unialg_1.html#K2">arity</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="pralg_1.html#K14" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="unialg_1.html#K2">arity</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="pralg_1.html#K14" target="_self">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="pralg_1.html#E1:39_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="pralg_1.html#E2:39_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="pralg_1.html#D16" target="_self">Def16</a></i>;<br/>


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

<b>assume </b><a NAME="E2:39"/><i><font color="Green">E22</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds  <a href="unialg_1.html#K2">arity</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="pralg_1.html#K14" target="_self">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="unialg_1.html#K2">arity</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="pralg_1.html#K14" target="_self">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="pralg_1.html#D16" target="_self">PRALG_1:def 16</a><br/>

<b>assume </b><a NAME="E3:39"/><i><font color="Green">E23</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/>

<b>let </b><font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font> be   <a href="funct_1.html#NM1">Function</a>;<br/>

<b>assume </b><a NAME="E4:39"/><i><font color="Green">E24</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> )
 ;<br/>

<b>let </b><font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>11</sub></font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E5:39"/><i><font color="Green">E25</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">c<sub>10</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">c<sub>11</sub></font> )
 ;<br/>

<b>let </b><font color="Maroon">c<sub>12</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="unialg_1.html#V1">homogeneous</a> <a href="unialg_1.html#V2">quasi_total</a> <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">c<sub>10</sub></font> <a href="finseq_2.html#K3">*</a> ,<font color="Maroon">c<sub>10</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>13</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="unialg_1.html#V1">homogeneous</a> <a href="unialg_1.html#V2">quasi_total</a> <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">c<sub>11</sub></font> <a href="finseq_2.html#K3">*</a> ,<font color="Maroon">c<sub>11</sub></font>;<br/>



<b>assume </b><a NAME="E6:39"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> )
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>5</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="pralg_1.html#E3:39"><i><font color="Green">E23</font></i></a>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E8:39"/>
 <a href="unialg_1.html#K2">arity</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="pralg_1.html#K14" target="_self">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="unialg_1.html#K2">arity</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="pralg_1.html#K14" target="_self">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="pralg_1.html#E2:39"><i><font color="Green">E22</font></i></a></i>;<br/>
<a NAME="E9:39"/><b>then </b>
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="unialg_1.html#K2">arity</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="pralg_1.html#K14" target="_self">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="unialg_1.html#K2">arity</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="pralg_1.html#K14" target="_self">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="pralg_1.html#E4:39"><i><font color="Green">E24</font></i></a>, <a class="ref" href="unialg_2.html#T2">UNIALG_2:2</a></i>;<br/>
<a NAME="E10:39"/><b>then </b><i><font color="Green">E27</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="unialg_1.html#K2">arity</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="pralg_1.html#K14" target="_self">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="unialg_1.html#K2">arity</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="pralg_1.html#K14" target="_self">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="pralg_1.html#E5:39"><i><font color="Green">E25</font></i></a>, <a class="ref" href="pralg_1.html#T1" target="_self">Th1</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>16</sub></font> being    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">c<sub>8</sub></font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">c<sub>10</sub></font>;<br/>
<i><font color="Green">E28</font></i>:  <a href="unialg_1.html#K2">arity</a> <font color="Maroon">c<sub>12</sub></font> = 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>16</sub></font>
<b>by </b><i><a class="txt" href="pralg_1.html#E5:39"><i><font color="Green">E25</font></i></a>, <a class="txt" href="pralg_1.html#E6:39"><i><font color="Green">E26</font></i></a>, <a class="ref" href="unialg_1.html#D10">UNIALG_1:def 10</a></i>
<br/>.= 
<font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>
;<br/>
<b>consider </b><font color="Maroon">c<sub>17</sub></font> being    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">c<sub>9</sub></font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">c<sub>11</sub></font>;<br/>
 <a href="unialg_1.html#K2">arity</a> <font color="Maroon">c<sub>13</sub></font> = 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>17</sub></font>
<b>by </b><i><a class="txt" href="pralg_1.html#E5:39"><i><font color="Green">E25</font></i></a>, <a class="txt" href="pralg_1.html#E6:39"><i><font color="Green">E26</font></i></a>, <a class="ref" href="unialg_1.html#D10">UNIALG_1:def 10</a></i>
<br/>.= 
<font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>
;<br/>
<b>hence </b><a NAME="E15:39"/>
 <a href="unialg_1.html#K2">arity</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="unialg_1.html#K2">arity</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="pralg_1.html#E10:39"><i><font color="Green">E27</font></i></a>, <a class="txt" href="pralg_1.html#E12:39"><i><font color="Green">E28</font></i></a></i>;<br/>


</div>
