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<b>let </b><font color="Maroon" title="c1">MS</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="realset2.html#V3" title="REALSET2:attr.3">trivial</a> non <a href="msualg_1.html#V2" title="MSUALG_1:attr.2">void</a> <a href="msualg_1.html#V6" title="MSUALG_1:attr.6">segmental</a>  <a href="msualg_1.html#L1" title="MSUALG_1:struct.1">ManySortedSign</a> ;<br/><b>let </b><font color="Maroon" title="c2">A</font> be  <a href="msualg_1.html#V5" title="MSUALG_1:attr.5">non-empty</a>  <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon" title="c1">MS</font>;<br/>



<a NAME="E1:35"/><i><font color="Green" title="E16">A1</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> the <a href="msualg_1.html#U5" title="MSUALG_1:sel.5">Charact</a> of <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <font color="Maroon" title="c1">MS</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E2:35"/>
 ex <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <font color="Maroon" title="c1">MS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Olive" title="b1">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon" title="c3">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E2:35_1"/><i><font color="Green" title="E17">A2</font></i>: 
the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <font color="Maroon" title="c1">MS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">n</font>
 <b>by </b><i><a class="ref" href="msualg_1.html#D12" target="_self" title="MSUALG_1:def.12">Def12</a></i>;<br/>
<a NAME="E3:35_1"/>
<font color="Maroon" title="c3">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<b>hence </b><a NAME="E4:35_1"/>
 ex <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <font color="Maroon" title="c1">MS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Olive" title="b1">n</font>
 <b>by </b><i><a class="txt" href="msualg_1.html#E2:35_1"><i><font color="Green" title="E17">A2</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c3">f</font> = the <a href="msualg_1.html#U5" title="MSUALG_1:sel.5">Charact</a> of <font color="Maroon" title="c2">A</font> as   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> <b>by </b><i><a class="txt" href="msualg_1.html#E1:35"><i><font color="Green" title="E16">A1</font></i></a>, <a class="ref" href="finseq_1.html#D2" title="FINSEQ_1:def.2">FINSEQ_1:def 2</a></i>;<br/>
<a NAME="E4:35"/>
<font color="Maroon" title="c3">f</font> is    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K11" title="MSUALG_1:func.11">the_sort_of</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> </span>)</span>,<span class="p1">(<span class="default"><a href="msualg_1.html#K11" title="MSUALG_1:func.11">the_sort_of</a> <font color="Maroon" title="c2">A</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a>,<a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a><br/>

<b>assume </b><a NAME="E1:35_2"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">f</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c5">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:35_2"/><i><font color="Green" title="E17">A3</font></i>: 
<font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <font color="Maroon" title="c1">MS</font>
 <b>and </b><br/><a NAME="E4:35_2"/><i><font color="Green" title="E18">A4</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font>
 <b>by </b><i><a class="txt" href="msualg_1.html#E1:35"><i><font color="Green" title="E16">A1</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">i</font> = <font color="Maroon" title="c5">i</font> as   <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">MS</font> <b>by </b><i><a class="txt" href="msualg_1.html#E3:35_2"><i><font color="Green" title="E17">A3</font></i></a></i>;<br/>
<a NAME="E6:35_2"/><i><font color="Green" title="E19">A5</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">ResultSort</a> of <font color="Maroon" title="c1">MS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <font color="Maroon" title="c1">MS</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E7:35_2"/>
the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">ResultSort</a> of <font color="Maroon" title="c1">MS</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">MS</font>
 
<b>by </b><i><a class="txt" href="msualg_1.html#E3:35_2"><i><font color="Green" title="E17">A3</font></i></a>, <a class="ref" href="funct_2.html#T7" title="FUNCT_2:th.7">FUNCT_2:7</a></i>;<br/>
<a NAME="E8:35_2"/><b>then </b><i><font color="Green" title="E20">A6</font></i>: 
the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">ResultSort</a> of <font color="Maroon" title="c1">MS</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font></span>)</span> is   <a href="pboole.html#NM3" title="PBOOLE:NM.3">Component</a> of the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">A</font>
 
<b>by </b><i><a class="ref" href="pboole.html#T151" title="PBOOLE:th.151">PBOOLE:151</a></i>;<br/>
<a NAME="E9:35_2"/><i><font color="Green" title="E21">A7</font></i>: 
 <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K11" title="MSUALG_1:func.11">the_sort_of</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> 
 
<b>by </b><i><a class="ref" href="msualg_1.html#T13" target="_self" title="MSUALG_1:th.13">Th13</a></i>;<br/>
<span class="p1">(<span class="default">the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">A</font> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">ResultSort</a> of <font color="Maroon" title="c1">MS</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> = 
the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">ResultSort</a> of <font color="Maroon" title="c1">MS</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font></span>)</span>
<b>by </b><i><a class="txt" href="msualg_1.html#E3:35_2"><i><font color="Green" title="E17">A3</font></i></a>, <a class="txt" href="msualg_1.html#E6:35_2"><i><font color="Green" title="E19">A5</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
 <a href="msualg_1.html#K11" title="MSUALG_1:func.11">the_sort_of</a> <font color="Maroon" title="c2">A</font>
<b>by </b><i><a class="txt" href="msualg_1.html#E8:35_2"><i><font color="Green" title="E20">A6</font></i></a>, <a class="ref" href="msualg_1.html#D17" target="_self" title="MSUALG_1:def.17">Def17</a></i>
;<br/>
<a NAME="E11:35_2"/><b>then </b>
<font color="Maroon" title="c4">x</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c2">A</font>, <a href="msualg_1.html#K11" title="MSUALG_1:func.11">the_sort_of</a> <font color="Maroon" title="c2">A</font>
 
<b>by </b><i><a class="txt" href="msualg_1.html#E3:35_2"><i><font color="Green" title="E17">A3</font></i></a>, <a class="txt" href="msualg_1.html#E4:35_2"><i><font color="Green" title="E18">A4</font></i></a>, <a class="ref" href="pboole.html#D18" title="PBOOLE:def.18">PBOOLE:def 18</a></i>;<br/>
<a NAME="E12:35_2"/><b>then </b>
<font color="Maroon" title="c4">x</font> is   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1">(<span class="default"><a href="msualg_1.html#K11" title="MSUALG_1:func.11">the_sort_of</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> , <a href="msualg_1.html#K11" title="MSUALG_1:func.11">the_sort_of</a> <font color="Maroon" title="c2">A</font>
 
<b>by </b><i><a class="txt" href="msualg_1.html#E9:35_2"><i><font color="Green" title="E21">A7</font></i></a>, <a class="ref" href="partfun1.html#T30" title="PARTFUN1:th.30">PARTFUN1:30</a></i>;<br/>
<b>hence </b><a NAME="E13:35_2"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K11" title="MSUALG_1:func.11">the_sort_of</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> </span>)</span>,<span class="p1">(<span class="default"><a href="msualg_1.html#K11" title="MSUALG_1:func.11">the_sort_of</a> <font color="Maroon" title="c2">A</font></span>)</span>
 <b>by </b><i><a class="ref" href="partfun1.html#T119" title="PARTFUN1:th.119">PARTFUN1:119</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:35"/>
the <a href="msualg_1.html#U5" title="MSUALG_1:sel.5">Charact</a> of <font color="Maroon" title="c2">A</font> is    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K11" title="MSUALG_1:func.11">the_sort_of</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> </span>)</span>,<span class="p1">(<span class="default"><a href="msualg_1.html#K11" title="MSUALG_1:func.11">the_sort_of</a> <font color="Maroon" title="c2">A</font></span>)</span>
 ;<br/>


</div>
