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

<b>reconsider </b><font color="Maroon">c<sub>2</sub></font> = the <a href="msualg_1.html#U1" target="_self">OperSymbols</a> of <span class="p1">(<span class="default"><a href="msualg_1.html#K7" target="_self">MSSign</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="ref" href="msualg_1.html#D5" target="_self">Def5</a></i>;<br/>
<a NAME="E2:27_1_1"/><i><font color="Green">E11</font></i>: 
( the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="msualg_1.html#K7" target="_self">MSSign</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> &amp; the <a href="msualg_1.html#U1" target="_self">OperSymbols</a> of <span class="p1">(<span class="default"><a href="msualg_1.html#K7" target="_self">MSSign</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K6" target="_self">signature</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp; the <a href="msualg_1.html#U2" target="_self">Arity</a> of <span class="p1">(<span class="default"><a href="msualg_1.html#K7" target="_self">MSSign</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pboole.html#K8">*--&gt;</a> 0</span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K6" target="_self">signature</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp; the <a href="msualg_1.html#U3" target="_self">ResultSort</a> of <span class="p1">(<span class="default"><a href="msualg_1.html#K7" target="_self">MSSign</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#R1">=</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#K6" target="_self">signature</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="pboole.html#K9">--&gt;</a> 0 )
 
<b>by </b><i><a class="ref" href="msualg_1.html#D13" target="_self">Def13</a></i>;<br/>
<a NAME="E3:27_1_1"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K6" target="_self">signature</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="unialg_1.html#D11">UNIALG_1:def 11</a></i>;<br/>
<a NAME="E4:27_1_1"/><b>then </b><i><font color="Green">E12</font></i>: 
 <a href="finseq_1.html#K4">dom</a> the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="msualg_1.html#E2:27_1_1"><i><font color="Green">E11</font></i></a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>3</sub></font> = the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">c<sub>1</sub></font> as    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E6:27_1_1"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="funcop_1.html#V1">Function-yielding</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funcop_1.html#D6">FUNCOP_1:def 6</a><br/>

<b>assume </b><a NAME="E1:27_1_1_1"/><i><font color="Green">E13</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>
 ;<br/>

<a NAME="E2:27_1_1_1"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">c<sub>1</sub></font>
 
;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="msualg_1.html#E1:27_1_1_1"><i><font color="Green">E13</font></i></a></i>;<br/>
<a NAME="E4:27_1_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> is   <a href="funct_1.html#NM1">Function</a>
 
<b>by </b><i><a class="txt" href="msualg_1.html#E1:27_1_1_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="unialg_1.html#T5">UNIALG_1:5</a></i>;<br/>
<b>hence </b><a NAME="E5:27_1_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> is   <a href="funct_1.html#NM1">Function</a>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="pboole.html#NM2">ManySortedFunction</a> of <font color="Maroon">c<sub>2</sub></font> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K8" target="_self">MSSorts</a> <font color="Maroon">c<sub>1</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" target="_self">Arity</a> of <span class="p1">(<span class="default"><a href="msualg_1.html#K7" target="_self">MSSign</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>, <font color="Maroon">c<sub>6</sub></font> = <span class="p1">(<span class="default"><a href="msualg_1.html#K8" target="_self">MSSorts</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U3" target="_self">ResultSort</a> of <span class="p1">(<span class="default"><a href="msualg_1.html#K7" target="_self">MSSign</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> as    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">c<sub>2</sub></font> ;<br/>
<a NAME="E9:27_1_1"/>
<font color="Maroon">c<sub>4</sub></font> is    <a href="pboole.html#M3">ManySortedFunction</a> of <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="pboole.html#D18">PBOOLE:def 18</a><br/>

<b>assume </b><a NAME="E1:27_1_1_2"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>7</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="msualg_1.html#E2:27_1_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="msualg_1.html#E1:27_1_1_2"><i><font color="Green">E13</font></i></a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>10</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> as  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 the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="finseq_2.html#K3">*</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#E4:27_1_1"><i><font color="Green">E12</font></i></a>, <a class="txt" href="msualg_1.html#E1:27_1_1_2"><i><font color="Green">E13</font></i></a>, <a class="ref" href="funct_1.html#D15">FUNCT_1:def 15</a>, <a class="ref" href="unialg_1.html#T5">UNIALG_1:5</a>, <a class="ref" href="unialg_1.html#D4">UNIALG_1:def 4</a>, <a class="ref" href="unialg_1.html#D5">UNIALG_1:def 5</a></i>;<br/>
<a NAME="E5:27_1_1_2"/>
 <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">0</span><a href="tarski.html#K1">}</a></span> <a href="pboole.html#K9">--&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</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/>
<b>then reconsider </b><font color="Maroon">c<sub>12</sub></font> = <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> <a href="pboole.html#K9">--&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> as    <a href="pboole.html#M1">ManySortedSet</a> of <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E7:27_1_1_2"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>9</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="finseq_1.html#K4">dom</a> <span class="p3">(<span class="default"><a href="msualg_1.html#K6" target="_self">signature</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="pboole.html#K9">--&gt;</a> 0</span>)</span>
 
<b>by </b><i><a class="txt" href="msualg_1.html#E2:27_1_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="msualg_1.html#E1:27_1_1_2"><i><font color="Green">E13</font></i></a>, <a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/>
<a NAME="E8:27_1_1_2"/><i><font color="Green">E15</font></i>: 
0 <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</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/>
<i><font color="Green">E16</font></i>: <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="msualg_1.html#K8" target="_self">MSSorts</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="pboole.html#K6">#</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p3">(<span class="default"><a href="pboole.html#K8">*--&gt;</a> 0</span>)</span></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K6" target="_self">signature</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="txt" href="msualg_1.html#E2:27_1_1"><i><font color="Green">E11</font></i></a>, <a class="ref" href="relat_1.html#T55">RELAT_1:55</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="msualg_1.html#K8" target="_self">MSSorts</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="pboole.html#K6">#</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="pboole.html#K8">*--&gt;</a> 0</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#K6" target="_self">signature</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="msualg_1.html#E2:27_1_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="msualg_1.html#E1:27_1_1_2"><i><font color="Green">E13</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="pboole.html#K6">#</a> </span>)</span> <a href="pboole.html#K7">*</a> <span class="p2">(<span class="default"><a href="pboole.html#K8">*--&gt;</a> 0</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#K6" target="_self">signature</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="msualg_1.html#E2:27_1_1"><i><font color="Green">E11</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="pboole.html#K6">#</a> </span>)</span> <a href="pboole.html#K7">*</a> <span class="p2">(<span class="default"><a href="pboole.html#K8">*--&gt;</a> 0</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="unialg_1.html#K2">arity</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="msualg_1.html#E2:27_1_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="msualg_1.html#E1:27_1_1_2"><i><font color="Green">E13</font></i></a>, <a class="ref" href="unialg_1.html#D11">UNIALG_1:def 11</a></i>
<br/>.= 
 <a href="funct_2.html#K1">Funcs</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><a href="unialg_1.html#K2">arity</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span>,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="pboole.html#T153">PBOOLE:153</a></i>
<br/>.= 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="msualg_1.html#T8" target="_self">Th8</a></i>
;<br/>
<i><font color="Green">E17</font></i>: <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> = 
<span class="p1">(<span class="default"><a href="msualg_1.html#K8" target="_self">MSSorts</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K4">dom</a> <span class="p4">(<span class="default"><a href="msualg_1.html#K6" target="_self">signature</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="pboole.html#K9">--&gt;</a> 0</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="msualg_1.html#E2:27_1_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="msualg_1.html#E7:27_1_1_2"><i><font color="Green">E14</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="msualg_1.html#K8" target="_self">MSSorts</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> 0
<b>by </b><i><a class="txt" href="msualg_1.html#E2:27_1_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="msualg_1.html#E1:27_1_1_2"><i><font color="Green">E13</font></i></a>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> <a href="pboole.html#K9">--&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> 0

<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="txt" href="msualg_1.html#E8:27_1_1_2"><i><font color="Green">E15</font></i></a>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
;<br/>
<a NAME="E11:27_1_1_2"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>11</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/>
<b>hence </b><a NAME="E12:27_1_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>6</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#E9:27_1_1_2"><i><font color="Green">E16</font></i></a>, <a class="txt" href="msualg_1.html#E10:27_1_1_2"><i><font color="Green">E17</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E10:27_1_1"/>
the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">c<sub>1</sub></font> is    <a href="pboole.html#M3">ManySortedFunction</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K8" target="_self">MSSorts</a> <font color="Maroon">c<sub>1</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" target="_self">Arity</a> of <span class="p1">(<span class="default"><a href="msualg_1.html#K7" target="_self">MSSign</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_1.html#K8" target="_self">MSSorts</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U3" target="_self">ResultSort</a> of <span class="p1">(<span class="default"><a href="msualg_1.html#K7" target="_self">MSSign</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 ;<br/>


</div>
