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<span class="kw">let </span><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="struct_0.html#V11" title="STRUCT_0:attr.11">void</a>   <a href="msualg_1.html#L1" title="MSUALG_1:struct.1">ManySortedSign</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">A</font> being   <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">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/>  for <font color="Olive" title="b2">R</font> being   <a href="msualg_4.html#NM3" title="MSUALG_4:NM.3">ManySortedRelation</a> of <font color="Olive" title="b1">A</font> holds   <a href="msualg_6.html#K10" title="MSUALG_6:func.10">EqTh</a> <font color="Olive" title="b2">R</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="msualg_6.html#K9" title="MSUALG_6:func.9">EqCl</a> (<span class="p1">(<span class="default"><a href="msualg_6.html#K8" title="MSUALG_6:func.8">TRS</a> <font color="Olive" title="b2">R</font></span>)</span>,<font color="Olive" title="b1">A</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c2">A</font> be   <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a>   <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon" title="c1">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">R</font> being   <a href="msualg_4.html#NM3" title="MSUALG_4:NM.3">ManySortedRelation</a> of <font color="Maroon" title="c2">A</font> holds   <a href="msualg_6.html#K10" title="MSUALG_6:func.10">EqTh</a> <font color="Olive" title="b1">R</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="msualg_6.html#K9" title="MSUALG_6:func.9">EqCl</a> (<span class="p1">(<span class="default"><a href="msualg_6.html#K8" title="MSUALG_6:func.8">TRS</a> <font color="Olive" title="b1">R</font></span>)</span>,<font color="Maroon" title="c2">A</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c3">R</font> be   <a href="msualg_4.html#NM3" title="MSUALG_4:NM.3">ManySortedRelation</a> of <font color="Maroon" title="c2">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="msualg_6.html#K10" title="MSUALG_6:func.10">EqTh</a> <font color="Maroon" title="c3">R</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="msualg_6.html#K9" title="MSUALG_6:func.9">EqCl</a> (<span class="p1">(<span class="default"><a href="msualg_6.html#K8" title="MSUALG_6:func.8">TRS</a> <font color="Maroon" title="c3">R</font></span>)</span>,<font color="Maroon" title="c2">A</font>)</span><br/>





<a NAME="E1:84"/><span class="lab"><font color="Green" title="E56">A1</font></span>: 
 <a href="msualg_6.html#K8" title="MSUALG_6:func.8">TRS</a> <font color="Maroon" title="c3">R</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a>  <a href="msualg_6.html#K9" title="MSUALG_6:func.9">EqCl</a> (<span class="p1">(<span class="default"><a href="msualg_6.html#K8" title="MSUALG_6:func.8">TRS</a> <font color="Maroon" title="c3">R</font></span>)</span>,<font color="Maroon" title="c2">A</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_6.html#T43" target="_self" title="MSUALG_6:th.43">Th43</a></span>;<br/>
<a NAME="E2:84"/>
<font color="Maroon" title="c3">R</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a>  <a href="msualg_6.html#K8" title="MSUALG_6:func.8">TRS</a> <font color="Maroon" title="c3">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_6.html#D13" target="_self" title="MSUALG_6:def.13">Def13</a></span>;<br/>
<a NAME="E3:84"/><span class="kw">then </span>
<font color="Maroon" title="c3">R</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a>  <a href="msualg_6.html#K9" title="MSUALG_6:func.9">EqCl</a> (<span class="p1">(<span class="default"><a href="msualg_6.html#K8" title="MSUALG_6:func.8">TRS</a> <font color="Maroon" title="c3">R</font></span>)</span>,<font color="Maroon" title="c2">A</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:84"><span class="lab"><font color="Green" title="E56">A1</font></span></a>, <a class="ref" href="pboole.html#T13" title="PBOOLE:th.13">PBOOLE:13</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:84"/>
 <a href="msualg_6.html#K10" title="MSUALG_6:func.10">EqTh</a> <font color="Maroon" title="c3">R</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a>  <a href="msualg_6.html#K9" title="MSUALG_6:func.9">EqCl</a> (<span class="p1">(<span class="default"><a href="msualg_6.html#K8" title="MSUALG_6:func.8">TRS</a> <font color="Maroon" title="c3">R</font></span>)</span>,<font color="Maroon" title="c2">A</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_6.html#D15" target="_self" title="MSUALG_6:def.15">Def15</a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="pboole.html#D10" title="PBOOLE:def.10">PBOOLE:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="msualg_6.html#K9" title="MSUALG_6:func.9">EqCl</a> (<span class="p1">(<span class="default"><a href="msualg_6.html#K8" title="MSUALG_6:func.8">TRS</a> <font color="Maroon" title="c3">R</font></span>)</span>,<font color="Maroon" title="c2">A</font>) <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a>  <a href="msualg_6.html#K10" title="MSUALG_6:func.10">EqTh</a> <font color="Maroon" title="c3">R</font></span><br/>

<a NAME="E5:84"/>
<font color="Maroon" title="c3">R</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a>  <a href="msualg_6.html#K10" title="MSUALG_6:func.10">EqTh</a> <font color="Maroon" title="c3">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_6.html#D15" target="_self" title="MSUALG_6:def.15">Def15</a></span>;<br/>
<a NAME="E6:84"/><span class="kw">then </span>
 <a href="msualg_6.html#K8" title="MSUALG_6:func.8">TRS</a> <font color="Maroon" title="c3">R</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a>  <a href="msualg_6.html#K10" title="MSUALG_6:func.10">EqTh</a> <font color="Maroon" title="c3">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_6.html#D13" target="_self" title="MSUALG_6:def.13">Def13</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:84"/>
 <a href="msualg_6.html#K9" title="MSUALG_6:func.9">EqCl</a> (<span class="p1">(<span class="default"><a href="msualg_6.html#K8" title="MSUALG_6:func.8">TRS</a> <font color="Maroon" title="c3">R</font></span>)</span>,<font color="Maroon" title="c2">A</font>) <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a>  <a href="msualg_6.html#K10" title="MSUALG_6:func.10">EqTh</a> <font color="Maroon" title="c3">R</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_6.html#T44" target="_self" title="MSUALG_6:th.44">Th44</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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