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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="realset2.html#V3">trivial</a> non <a href="msualg_1.html#V2" target="_self">void</a> <a href="msualg_1.html#V6" target="_self">segmental</a>  <a href="msualg_1.html#L1" target="_self">ManySortedSign</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  <a href="msualg_1.html#V5" target="_self">non-empty</a>  <a href="msualg_1.html#L3" target="_self">MSAlgebra</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<a NAME="E1:35"/><i><font color="Green">E16</font></i>: 
 <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> <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="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E2:35"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st the <a href="msualg_1.html#U1" target="_self">OperSymbols</a> of <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Olive">b<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="msualg_1.html#D12" target="_self">Def12</a></i>;<br/>
<b>then reconsider </b><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> as   <a href="finseq_1.html#NM1">FinSequence</a> <b>by </b><i><a class="txt" href="msualg_1.html#E1:35"><i><font color="Green">E16</font></i></a>, <a class="ref" href="finseq_1.html#D2">FINSEQ_1:def 2</a></i>;<br/>
<a NAME="E4:35"/>
<font color="Maroon">c<sub>3</sub></font> is    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="partfun1.html#K4">PFuncs</a> <span class="p1">(<span class="default"><span class="p2">(<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> </span>)</span>,<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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_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="tarski.html#D3">TARSKI:def 3</a>,<a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a><br/>

<b>assume </b><a NAME="E1:35_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>5</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:35_1"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U1" target="_self">OperSymbols</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E4:35_1"/><i><font color="Green">E18</font></i>: 
<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>4</sub></font>
 <b>by </b><i><a class="txt" href="msualg_1.html#E1:35"><i><font color="Green">E16</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</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#E3:35_1"><i><font color="Green">E17</font></i></a></i>;<br/>
<a NAME="E6:35_1"/><i><font color="Green">E19</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="E7:35_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>6</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#E3:35_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/>
<a NAME="E8:35_1"/><b>then </b><i><font color="Green">E20</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>6</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/>
<a NAME="E9:35_1"/><i><font color="Green">E21</font></i>: 
 <a href="msualg_1.html#K3" target="_self">Args</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</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> <a href="finseq_2.html#K3">*</a> 
 
<b>by </b><i><a class="ref" href="msualg_1.html#T13" target="_self">Th13</a></i>;<br/>
<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>6</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>6</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="msualg_1.html#E3:35_1"><i><font color="Green">E17</font></i></a>, <a class="txt" href="msualg_1.html#E6:35_1"><i><font color="Green">E19</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#E8:35_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="msualg_1.html#D17" target="_self">Def17</a></i>
;<br/>
<a NAME="E11:35_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> is   <a href="funct_2.html#NM1">Function</a> of  <a href="msualg_1.html#K3" target="_self">Args</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font>, <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#E3:35_1"><i><font color="Green">E17</font></i></a>, <a class="txt" href="msualg_1.html#E4:35_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="pboole.html#D18">PBOOLE:def 18</a></i>;<br/>
<a NAME="E12:35_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> is   <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>
 
<b>by </b><i><a class="txt" href="msualg_1.html#E9:35_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="partfun1.html#T30">PARTFUN1:30</a></i>;<br/>
<b>hence </b><a NAME="E13:35_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="partfun1.html#K4">PFuncs</a> <span class="p1">(<span class="default"><span class="p2">(<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> </span>)</span>,<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="ref" href="partfun1.html#T119">PARTFUN1:119</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:35"/>
the <a href="msualg_1.html#U5" target="_self">Charact</a> of <font color="Maroon">c<sub>2</sub></font> is    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="partfun1.html#K4">PFuncs</a> <span class="p1">(<span class="default"><span class="p2">(<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> </span>)</span>,<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>
 ;<br/>


</div>
