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<b>let </b><font color="Maroon" title="c1">S</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> non <a href="msualg_1.html#V2" title="MSUALG_1:attr.2">void</a>  <a href="msualg_1.html#L1" title="MSUALG_1:struct.1">ManySortedSign</a> ;<br/><b>let </b><font color="Maroon" title="c2">o</font> be   <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">S</font>;<br/><b>let </b><font color="Maroon" title="c3">i</font> be    <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> ;<br/>





<b>assume </b><a NAME="E1:19"/>
<font color="Maroon" title="c3">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c2">o</font></span>)</span>
 ;<br/>

<b>let </b><font color="Maroon" title="c4">A</font> be  <a href="msualg_6.html#V1" title="MSUALG_6:attr.1">feasible</a>  <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon" title="c1">S</font>;<br/><b>let </b><font color="Maroon" title="c5">a</font> be   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>;<br/>



<b>assume </b><a NAME="E2:19"/><i><font color="Green" title="E15">A1</font></i>: 
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c4">A</font>
 ;<br/>

<a NAME="E3:19"/><b>then </b><i><font color="Green" title="E16">A2</font></i>: 
 <a href="msualg_1.html#K4" title="MSUALG_1:func.4">Result</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c4">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="msualg_6.html#D1" target="_self" title="MSUALG_6:def.1">Def1</a></i>;<br/>
<a NAME="E4:19"/><i><font color="Green" title="E17">A3</font></i>: 
 <a href="msualg_1.html#K4" title="MSUALG_1:func.4">Result</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c4">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c2">o</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pralg_2.html#T10" title="PRALG_2:th.10">PRALG_2:10</a></i>;<br/>
<a NAME="E5:19"/><i><font color="Green" title="E18">A4</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="msualg_6.html#K4" title="MSUALG_6:func.4">transl</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c4">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c4">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c2">o</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c3">i</font></span>)</span>
 
<b>by </b><i><a class="ref" href="msualg_6.html#D4" target="_self" title="MSUALG_6:def.4">Def4</a></i>;<br/>
<a NAME="E6:19"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="msualg_6.html#K4" title="MSUALG_6:func.4">transl</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c4">A</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c4">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c2">o</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">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><br/>

<b>assume </b><a NAME="E1:19_1"/>
<font color="Maroon" title="c6">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> <span class="p1">(<span class="default"><a href="msualg_6.html#K4" title="MSUALG_6:func.4">transl</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c4">A</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c7">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:19_1"/><i><font color="Green" title="E19">A5</font></i>: 
( <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="msualg_6.html#K4" title="MSUALG_6:func.4">transl</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c4">A</font></span>)</span> &amp; <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msualg_6.html#K4" title="MSUALG_6:func.4">transl</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c4">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">y</font> )
 <b>by </b><i><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="c8">y</font> = <font color="Maroon" title="c7">y</font> as   <a href="msualg_6.html#NM1" title="MSUALG_6:NM.1">Element</a> of <font color="Maroon" title="c4">A</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c2">o</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c3">i</font></span>)</span> <b>by </b><i><a class="txt" href="msualg_6.html#E3:19_1"><i><font color="Green" title="E19">A5</font></i></a>, <a class="ref" href="msualg_6.html#D4" target="_self" title="MSUALG_6:def.4">Def4</a></i>;<br/>
<b>set </b><font color="Maroon" title="c9">b</font> = <font color="Maroon" title="c5">a</font> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c8">y</font>;<br/>
<a NAME="E5:19_1"/>
<font color="Maroon" title="c5">a</font> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c4">A</font>
 
<b>by </b><i><a class="txt" href="msualg_6.html#E2:19"><i><font color="Green" title="E15">A1</font></i></a>, <a class="ref" href="msualg_6.html#T7" target="_self" title="MSUALG_6:th.7">Th7</a></i>;<br/>
<a NAME="E6:19_1"/><b>then </b>
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c4">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c8">y</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c4">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c8">y</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msualg_1.html#K4" title="MSUALG_1:func.4">Result</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c4">A</font> )
 
<b>by </b><i><a class="txt" href="msualg_6.html#E3:19"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="msualg_6.html#E5:19"><i><font color="Green" title="E18">A4</font></i></a>, <a class="txt" href="msualg_6.html#E3:19_1"><i><font color="Green" title="E19">A5</font></i></a>, <a class="ref" href="msualg_6.html#D4" target="_self" title="MSUALG_6:def.4">Def4</a>, <a class="ref" href="funct_2.html#T7" title="FUNCT_2:th.7">FUNCT_2:7</a></i>;<br/>
<b>hence </b><a NAME="E7:19_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c4">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c2">o</font></span>)</span>
 <b>by </b><i><a class="ref" href="pralg_2.html#T10" title="PRALG_2:th.10">PRALG_2:10</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:19"/>
 <a href="msualg_6.html#K4" title="MSUALG_6:func.4">transl</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c4">A</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c4">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c2">o</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c3">i</font></span>)</span>,the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c4">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c2">o</font></span>)</span>
 <b>by </b><i><a class="txt" href="msualg_6.html#E3:19"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="msualg_6.html#E4:19"><i><font color="Green" title="E17">A3</font></i></a>, <a class="txt" href="msualg_6.html#E5:19"><i><font color="Green" title="E18">A4</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11" title="RELSET_1:th.11">RELSET_1:11</a></i>;<br/>


</div>
