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

<a NAME="E1:37_1_1"/><i><font color="Green">E16</font></i>: 
 <a href="msualg_1.html#K12" target="_self">the_charact_of</a> <font color="Maroon">c<sub>2</sub></font> is <a href="unialg_1.html#V5">quasi_total</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="nat_1.html#NM1">Nat</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="unialg_1.html#D5">UNIALG_1:def 5</a><br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="partfun1.html#NM1">PartFunc</a> of <span class="p1">(<span class="default"><a href="msualg_1.html#K11" target="_self">the_sort_of</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_2.html#K3">*</a> , <a href="msualg_1.html#K11" target="_self">the_sort_of</a> <font color="Maroon">c<sub>2</sub></font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:37_1_1_1"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K12" target="_self">the_charact_of</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E2:37_1_1_1"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K12" target="_self">the_charact_of</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="msualg_1.html#K11" target="_self">the_sort_of</a> <font color="Maroon">c<sub>2</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="unialg_1.html#D2">UNIALG_1:def 2</a><br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="msualg_1.html#K11" target="_self">the_sort_of</a> <font color="Maroon">c<sub>2</sub></font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E3:37_1_1_1"/><i><font color="Green">E19</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font>
 <b>and </b><br/><a NAME="E4:37_1_1_1"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<i><font color="Green">E21</font></i>:  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K12" target="_self">the_charact_of</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> = 
 <a href="relat_1.html#K1">dom</a> the <a href="msualg_1.html#U5" target="_self">Charact</a> of <font color="Maroon">c<sub>2</sub></font>

<br/>.= 
the <a href="msualg_1.html#U1" target="_self">OperSymbols</a> of <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/>
<b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="msualg_1.html#E1:37_1_1_1"><i><font color="Green">E17</font></i></a></i>;<br/>
<a NAME="E7:37_1_1_1"/><i><font color="Green">E22</font></i>: 
 <a href="relat_1.html#K1">dom</a> the <a href="msualg_1.html#U3" target="_self">ResultSort</a> of <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U1" target="_self">OperSymbols</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E8:37_1_1_1"/><i><font color="Green">E23</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="msualg_1.html#U4" target="_self">Sorts</a> of <font color="Maroon">c<sub>2</sub></font> <a href="pboole.html#K6">#</a> </span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U2" target="_self">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>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K3" target="_self">Args</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 
;<br/>
<a NAME="E9:37_1_1_1"/>
the <a href="msualg_1.html#U3" target="_self">ResultSort</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</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="txt" href="msualg_1.html#E1:37_1_1_1"><i><font color="Green">E17</font></i></a>, <a class="txt" href="msualg_1.html#E5:37_1_1_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/>
<a NAME="E10:37_1_1_1"/><b>then </b><i><font color="Green">E24</font></i>: 
the <a href="msualg_1.html#U4" target="_self">Sorts</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U3" target="_self">ResultSort</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> is   <a href="pboole.html#NM3">Component</a> of the <a href="msualg_1.html#U4" target="_self">Sorts</a> of <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="pboole.html#T151">PBOOLE:151</a></i>;<br/>
<i><font color="Green">E25</font></i>: <span class="p1">(<span class="default">the <a href="msualg_1.html#U4" target="_self">Sorts</a> of <font color="Maroon">c<sub>2</sub></font> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U3" target="_self">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>7</sub></font> = 
the <a href="msualg_1.html#U4" target="_self">Sorts</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U3" target="_self">ResultSort</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="msualg_1.html#E1:37_1_1_1"><i><font color="Green">E17</font></i></a>, <a class="txt" href="msualg_1.html#E5:37_1_1_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="msualg_1.html#E7:37_1_1_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
 <a href="msualg_1.html#K11" target="_self">the_sort_of</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="msualg_1.html#E10:37_1_1_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="msualg_1.html#D17" target="_self">Def17</a></i>
;<br/>
<a NAME="E12:37_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U5" target="_self">Charact</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="msualg_1.html#E2:37_1_1_1"><i><font color="Green">E18</font></i></a></i>;<br/>
<a NAME="E13:37_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> is   <a href="funct_2.html#NM1">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="msualg_1.html#U4" target="_self">Sorts</a> of <font color="Maroon">c<sub>2</sub></font> <a href="pboole.html#K6">#</a> </span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U2" target="_self">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>7</sub></font>,<span class="p1">(<span class="default">the <a href="msualg_1.html#U4" target="_self">Sorts</a> of <font color="Maroon">c<sub>2</sub></font> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U3" target="_self">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>7</sub></font>
 
<b>by </b><i><a class="txt" href="msualg_1.html#E1:37_1_1_1"><i><font color="Green">E17</font></i></a>, <a class="txt" href="msualg_1.html#E5:37_1_1_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="pboole.html#D18">PBOOLE:def 18</a></i>;<br/>
<b>then </b><i><font color="Green">E26</font></i>:  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="msualg_1.html#U4" target="_self">Sorts</a> of <font color="Maroon">c<sub>2</sub></font> <a href="pboole.html#K6">#</a> </span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U2" target="_self">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>7</sub></font>
<b>by </b><i><a class="txt" href="msualg_1.html#E11:37_1_1_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1" target="_self">the_arity_of</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K11" target="_self">the_sort_of</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="msualg_1.html#E8:37_1_1_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="msualg_1.html#T11" target="_self">Th11</a></i>
;<br/>
<a NAME="E15:37_1_1_1"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" target="_self">the_arity_of</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="msualg_1.html#E3:37_1_1_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="msualg_1.html#E4:37_1_1_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>;<br/>
<a NAME="E16:37_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> is    <a href="subset_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="msualg_1.html#E14:37_1_1_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="finseq_2.html#T110">FINSEQ_2:110</a></i>;<br/>
<b>hence </b><a NAME="E17:37_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="msualg_1.html#E4:37_1_1_1"><i><font color="Green">E20</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:37_1_1"/><i><font color="Green">E17</font></i>: 
 <a href="msualg_1.html#K12" target="_self">the_charact_of</a> <font color="Maroon">c<sub>2</sub></font> is <a href="unialg_1.html#V4">homogeneous</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="nat_1.html#NM1">Nat</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="unialg_1.html#D4">UNIALG_1:def 4</a><br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="partfun1.html#NM1">PartFunc</a> of <span class="p1">(<span class="default"><a href="msualg_1.html#K11" target="_self">the_sort_of</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_2.html#K3">*</a> , <a href="msualg_1.html#K11" target="_self">the_sort_of</a> <font color="Maroon">c<sub>2</sub></font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:37_1_1_2"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K12" target="_self">the_charact_of</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E2:37_1_1_2"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K12" target="_self">the_charact_of</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="msualg_1.html#K11" target="_self">the_sort_of</a> <font color="Maroon">c<sub>2</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="unialg_1.html#D1">UNIALG_1:def 1</a><br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="msualg_1.html#K11" target="_self">the_sort_of</a> <font color="Maroon">c<sub>2</sub></font>;<br/>



<b>assume </b><a NAME="E3:37_1_1_2"/><i><font color="Green">E20</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></font> )
 ;<br/>

<i><font color="Green">E21</font></i>:  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K12" target="_self">the_charact_of</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> = 
 <a href="relat_1.html#K1">dom</a> the <a href="msualg_1.html#U5" target="_self">Charact</a> of <font color="Maroon">c<sub>2</sub></font>

<br/>.= 
the <a href="msualg_1.html#U1" target="_self">OperSymbols</a> of <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/>
<b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="msualg_1.html#E1:37_1_1_2"><i><font color="Green">E18</font></i></a></i>;<br/>
<a NAME="E6:37_1_1_2"/><i><font color="Green">E22</font></i>: 
 <a href="relat_1.html#K1">dom</a> the <a href="msualg_1.html#U3" target="_self">ResultSort</a> of <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U1" target="_self">OperSymbols</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E7:37_1_1_2"/><i><font color="Green">E23</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="msualg_1.html#U4" target="_self">Sorts</a> of <font color="Maroon">c<sub>2</sub></font> <a href="pboole.html#K6">#</a> </span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U2" target="_self">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>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K3" target="_self">Args</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 
;<br/>
<a NAME="E8:37_1_1_2"/>
the <a href="msualg_1.html#U3" target="_self">ResultSort</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</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="txt" href="msualg_1.html#E1:37_1_1_2"><i><font color="Green">E18</font></i></a>, <a class="txt" href="msualg_1.html#E4:37_1_1_2"><i><font color="Green">E21</font></i></a>, <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/>
<a NAME="E9:37_1_1_2"/><b>then </b><i><font color="Green">E24</font></i>: 
the <a href="msualg_1.html#U4" target="_self">Sorts</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U3" target="_self">ResultSort</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> is   <a href="pboole.html#NM3">Component</a> of the <a href="msualg_1.html#U4" target="_self">Sorts</a> of <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="pboole.html#T151">PBOOLE:151</a></i>;<br/>
<i><font color="Green">E25</font></i>: <span class="p1">(<span class="default">the <a href="msualg_1.html#U4" target="_self">Sorts</a> of <font color="Maroon">c<sub>2</sub></font> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U3" target="_self">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>7</sub></font> = 
the <a href="msualg_1.html#U4" target="_self">Sorts</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U3" target="_self">ResultSort</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="msualg_1.html#E1:37_1_1_2"><i><font color="Green">E18</font></i></a>, <a class="txt" href="msualg_1.html#E4:37_1_1_2"><i><font color="Green">E21</font></i></a>, <a class="txt" href="msualg_1.html#E6:37_1_1_2"><i><font color="Green">E22</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
 <a href="msualg_1.html#K11" target="_self">the_sort_of</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="msualg_1.html#E9:37_1_1_2"><i><font color="Green">E24</font></i></a>, <a class="ref" href="msualg_1.html#D17" target="_self">Def17</a></i>
;<br/>
<a NAME="E11:37_1_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U5" target="_self">Charact</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="msualg_1.html#E2:37_1_1_2"><i><font color="Green">E19</font></i></a></i>;<br/>
<a NAME="E12:37_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> is   <a href="funct_2.html#NM1">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="msualg_1.html#U4" target="_self">Sorts</a> of <font color="Maroon">c<sub>2</sub></font> <a href="pboole.html#K6">#</a> </span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U2" target="_self">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>7</sub></font>,<span class="p1">(<span class="default">the <a href="msualg_1.html#U4" target="_self">Sorts</a> of <font color="Maroon">c<sub>2</sub></font> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U3" target="_self">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>7</sub></font>
 
<b>by </b><i><a class="txt" href="msualg_1.html#E1:37_1_1_2"><i><font color="Green">E18</font></i></a>, <a class="txt" href="msualg_1.html#E4:37_1_1_2"><i><font color="Green">E21</font></i></a>, <a class="ref" href="pboole.html#D18">PBOOLE:def 18</a></i>;<br/>
<b>then </b><i><font color="Green">E26</font></i>:  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="msualg_1.html#U4" target="_self">Sorts</a> of <font color="Maroon">c<sub>2</sub></font> <a href="pboole.html#K6">#</a> </span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U2" target="_self">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>7</sub></font>
<b>by </b><i><a class="txt" href="msualg_1.html#E10:37_1_1_2"><i><font color="Green">E25</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1" target="_self">the_arity_of</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K11" target="_self">the_sort_of</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="msualg_1.html#E7:37_1_1_2"><i><font color="Green">E23</font></i></a>, <a class="ref" href="msualg_1.html#T11" target="_self">Th11</a></i>
;<br/>
<b>hence </b> <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font> = 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" target="_self">the_arity_of</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="msualg_1.html#E3:37_1_1_2"><i><font color="Green">E20</font></i></a>, <a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="txt" href="msualg_1.html#E3:37_1_1_2"><i><font color="Green">E20</font></i></a>, <a class="txt" href="msualg_1.html#E13:37_1_1_2"><i><font color="Green">E26</font></i></a>, <a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>
;<br/><br/>


</div><b>end;</b></div>
<a NAME="E3:37_1_1"/>
the <a href="msualg_1.html#U1" target="_self">OperSymbols</a> of <font color="Maroon">c<sub>1</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="msualg_1.html#D5" target="_self">Def5</a></i>;<br/>
<a NAME="E4:37_1_1"/><b>then </b>
the <a href="msualg_1.html#U5" target="_self">Charact</a> of <font color="Maroon">c<sub>2</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#D3">PBOOLE:def 3</a>, <a class="ref" href="relat_1.html#T60">RELAT_1:60</a></i>;<br/>
<a NAME="E5:37_1_1"/><b>then </b><i><font color="Green">E18</font></i>: 
 <a href="msualg_1.html#K12" target="_self">the_charact_of</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
;<br/>
<a NAME="E6:37_1_1"/>
 <a href="msualg_1.html#K12" target="_self">the_charact_of</a> <font color="Maroon">c<sub>2</sub></font> is <a href="relat_1.html#V2">non-empty</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_1.html#D15">FUNCT_1:def 15</a><br/>

<b>assume </b><a NAME="E1:37_1_1_3"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K12" target="_self">the_charact_of</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/>

<b>set </b><font color="Maroon">c<sub>4</sub></font> = <span class="p1">(<span class="default"><a href="msualg_1.html#K12" target="_self">the_charact_of</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font>;<br/>
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K12" target="_self">the_charact_of</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> = 
 <a href="relat_1.html#K1">dom</a> the <a href="msualg_1.html#U5" target="_self">Charact</a> of <font color="Maroon">c<sub>2</sub></font>

<br/>.= 
the <a href="msualg_1.html#U1" target="_self">OperSymbols</a> of <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/>
<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="msualg_1.html#E1:37_1_1_3"><i><font color="Green">E19</font></i></a></i>;<br/>
<span class="p1">(<span class="default"><a href="msualg_1.html#K12" target="_self">the_charact_of</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> = 
the <a href="msualg_1.html#U5" target="_self">Charact</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>

<br/>.= 
 <a href="msualg_1.html#K5" target="_self">Den</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>

;<br/>
<b>hence </b><a NAME="E5:37_1_1_3"/>
not <span class="p1">(<span class="default"><a href="msualg_1.html#K12" target="_self">the_charact_of</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relat_1.html#T60">RELAT_1:60</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:37_1_1"/>
 <a href="unialg_1.html#G1">UAStr</a>(# <span class="p1">(<span class="default"><a href="msualg_1.html#K11" target="_self">the_sort_of</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_1.html#K12" target="_self">the_charact_of</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> #) is  <a href="unialg_1.html#V3">strict</a> <a href="unialg_1.html#V8">non-empty</a> <a href="unialg_1.html#NM2">Universal_Algebra</a>
 <b>by </b><i><a class="txt" href="msualg_1.html#E1:37_1_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="msualg_1.html#E2:37_1_1"><i><font color="Green">E17</font></i></a>, <a class="txt" href="msualg_1.html#E5:37_1_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="unialg_1.html#D7">UNIALG_1:def 7</a>, <a class="ref" href="unialg_1.html#D8">UNIALG_1:def 8</a>, <a class="ref" href="unialg_1.html#D9">UNIALG_1:def 9</a></i>;<br/>


</div>
