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<b>let </b><font color="Maroon" title="c1">S</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">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">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">S</font>;<br/><b>let </b><font color="Maroon" title="c3">F</font> be   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon" title="c2">A</font>,<span class="p1">(<span class="default"><a href="msafree2.html#K8" title="MSAFREE2:func.8">Trivial_Algebra</a> <font color="Maroon" title="c1">S</font></span>)</span>;<br/><b>let </b><font color="Maroon" title="c4">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="c5">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c2">A</font>;<br/>









<b>set </b><font color="Maroon" title="c6">I</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font>;<br/>
<b>set </b><font color="Maroon" title="c7">SA</font> = the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">A</font>;<br/>
<b>set </b><font color="Maroon" title="c8">T</font> =  <a href="msafree2.html#K8" title="MSAFREE2:func.8">Trivial_Algebra</a> <font color="Maroon" title="c1">S</font>;<br/>
<b>set </b><font color="Maroon" title="c9">ST</font> = the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree2.html#K8" title="MSAFREE2:func.8">Trivial_Algebra</a> <font color="Maroon" title="c1">S</font></span>)</span>;<br/>
<b>consider </b><font color="Maroon" title="c10">XX</font> being    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font><b> such that </b><br/><a NAME="E2:36"/><i><font color="Green" title="E14">A1</font></i>: 
<span class="p1"><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">{</a><span class="default"><font color="Maroon" title="c10">XX</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="msualg_9.html#T6" target="_self" title="MSUALG_9:th.6">Th6</a></i>;<br/>
<a NAME="E3:36"/><i><font color="Green" title="E15">A2</font></i>: 
the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree2.html#K8" title="MSAFREE2:func.8">Trivial_Algebra</a> <font color="Maroon" title="c1">S</font></span>)</span> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <span class="p1"><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">{</a><span class="default"><font color="Maroon" title="c10">XX</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span>
 
<b>by </b><i><a class="txt" href="msualg_9.html#E2:36"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="msafree2.html#D12" title="MSAFREE2:def.12">MSAFREE2:def 12</a></i>;<br/>
<b>set </b><font color="Maroon" title="c11">r</font> =  <a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c4">o</font>;<br/>
<b>consider </b><font color="Maroon" title="c12">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:36"/><i><font color="Green" title="E16">A3</font></i>: 
<font color="Maroon" title="c12">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">S</font>
 <b>and </b><br/><a NAME="E6:36"/><i><font color="Green" title="E17">A4</font></i>: 
 <a href="msualg_1.html#K4" title="MSUALG_1:func.4">Result</a> <font color="Maroon" title="c4">o</font>,<span class="p1">(<span class="default"><a href="msafree2.html#K8" title="MSAFREE2:func.8">Trivial_Algebra</a> <font color="Maroon" title="c1">S</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 <span class="p1">(<span class="default"><a href="msafree2.html#K8" title="MSAFREE2:func.8">Trivial_Algebra</a> <font color="Maroon" title="c1">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font>
 <b>by </b><i><a class="ref" href="pboole.html#T150" title="PBOOLE:th.150">PBOOLE:150</a></i>;<br/>
<a NAME="E7:36"/><i><font color="Green" title="E18">A5</font></i>: 
the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree2.html#K8" title="MSAFREE2:func.8">Trivial_Algebra</a> <font color="Maroon" title="c1">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="txt" href="msualg_9.html#E2:36"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="msualg_9.html#E3:36"><i><font color="Green" title="E15">A2</font></i></a>, <a class="txt" href="msualg_9.html#E5:36"><i><font color="Green" title="E16">A3</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c13">d</font> = <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of 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"><a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c4">o</font></span>)</span> <b>by </b><i><a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>;<br/>
<a NAME="E9:36"/><i><font color="Green" title="E19">A6</font></i>: 
the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree2.html#K8" title="MSAFREE2:func.8">Trivial_Algebra</a> <font color="Maroon" title="c1">S</font></span>)</span> <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="c4">o</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="txt" href="msualg_9.html#E2:36"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="msualg_9.html#E3:36"><i><font color="Green" title="E15">A2</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c4">o</font></span>)</span></span>)</span> <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#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c4">o</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">d</font>

<br/>.= 
0
<b>by </b><i><a class="txt" href="msualg_9.html#E9:36"><i><font color="Green" title="E19">A6</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>
;<br/>

<b>thus </b><a NAME="E11:36"/>
<span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c4">o</font>,<span class="p2">(<span class="default"><a href="msafree2.html#K8" title="MSAFREE2:func.8">Trivial_Algebra</a> <font color="Maroon" title="c1">S</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="msualg_3.html#K6" title="MSUALG_3:func.6">#</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="txt" href="msualg_9.html#E6:36"><i><font color="Green" title="E17">A4</font></i></a>, <a class="txt" href="msualg_9.html#E7:36"><i><font color="Green" title="E18">A5</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>


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