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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="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">A</font> being   <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">C1</font>, <font color="Olive" title="b3">C2</font> being  <a href="msualg_4.html#V3" title="MSUALG_4:attr.3">MSEquivalence-like</a>  <a href="msualg_4.html#NM4" title="MSUALG_4:NM.4">ManySortedRelation</a> of <font color="Olive" title="b1">A</font>  st <font color="Olive" title="b2">C1</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Olive" title="b3">C2</font> holds <br/>for <font color="Olive" title="b4">i</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font><br/> for <font color="Olive" title="b5">x</font>, <font color="Olive" title="b6">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Olive" title="b1">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">i</font>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b5">x</font>,<font color="Olive" title="b6">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">C1</font> <a href="msualg_4.html#K2" title="MSUALG_4:func.2">.</a> <font color="Olive" title="b4">i</font> holds <br/>(  <a href="msualg_4.html#K3" title="MSUALG_4:func.3">Class</a> <font color="Olive" title="b2">C1</font>,<font color="Olive" title="b5">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="msualg_4.html#K3" title="MSUALG_4:func.3">Class</a> <font color="Olive" title="b3">C2</font>,<font color="Olive" title="b6">y</font> &amp; ( <font color="Olive" title="b1">A</font> is  <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a>  implies  <a href="msualg_4.html#K3" title="MSUALG_4:func.3">Class</a> <font color="Olive" title="b2">C1</font>,<font color="Olive" title="b6">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="msualg_4.html#K3" title="MSUALG_4:func.3">Class</a> <font color="Olive" title="b3">C2</font>,<font color="Olive" title="b5">x</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">A</font> be    <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="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">C1</font>, <font color="Olive" title="b2">C2</font> being  <a href="msualg_4.html#V3" title="MSUALG_4:attr.3">MSEquivalence-like</a>  <a href="msualg_4.html#NM4" title="MSUALG_4:NM.4">ManySortedRelation</a> of <font color="Maroon" title="c2">A</font>  st <font color="Olive" title="b1">C1</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Olive" title="b2">C2</font> holds <br/>for <font color="Olive" title="b3">i</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font><br/> for <font color="Olive" title="b4">x</font>, <font color="Olive" title="b5">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">i</font>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b4">x</font>,<font color="Olive" title="b5">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C1</font> <a href="msualg_4.html#K2" title="MSUALG_4:func.2">.</a> <font color="Olive" title="b3">i</font> holds <br/>(  <a href="msualg_4.html#K3" title="MSUALG_4:func.3">Class</a> <font color="Olive" title="b1">C1</font>,<font color="Olive" title="b4">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="msualg_4.html#K3" title="MSUALG_4:func.3">Class</a> <font color="Olive" title="b2">C2</font>,<font color="Olive" title="b5">y</font> &amp; ( <font color="Maroon" title="c2">A</font> is  <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a>  implies  <a href="msualg_4.html#K3" title="MSUALG_4:func.3">Class</a> <font color="Olive" title="b1">C1</font>,<font color="Olive" title="b5">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="msualg_4.html#K3" title="MSUALG_4:func.3">Class</a> <font color="Olive" title="b2">C2</font>,<font color="Olive" title="b4">x</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">C1</font>, <font color="Maroon" title="c4">C2</font> be   <a href="msualg_4.html#V3" title="MSUALG_4:attr.3">MSEquivalence-like</a>  <a href="msualg_4.html#NM4" title="MSUALG_4:NM.4">ManySortedRelation</a> of <font color="Maroon" title="c2">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">C1</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c4">C2</font> implies for <font color="Olive" title="b1">i</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font><br/> for <font color="Olive" title="b2">x</font>, <font color="Olive" title="b3">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">x</font>,<font color="Olive" title="b3">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C1</font> <a href="msualg_4.html#K2" title="MSUALG_4:func.2">.</a> <font color="Olive" title="b1">i</font> holds <br/>(  <a href="msualg_4.html#K3" title="MSUALG_4:func.3">Class</a> <font color="Maroon" title="c3">C1</font>,<font color="Olive" title="b2">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="msualg_4.html#K3" title="MSUALG_4:func.3">Class</a> <font color="Maroon" title="c4">C2</font>,<font color="Olive" title="b3">y</font> &amp; ( <font color="Maroon" title="c2">A</font> is  <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a>  implies  <a href="msualg_4.html#K3" title="MSUALG_4:func.3">Class</a> <font color="Maroon" title="c3">C1</font>,<font color="Olive" title="b3">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="msualg_4.html#K3" title="MSUALG_4:func.3">Class</a> <font color="Maroon" title="c4">C2</font>,<font color="Olive" title="b2">x</font> ) ) )</span><br/>





<span class="kw">assume </span><a NAME="E1:43"/><span class="lab"><font color="Green" title="E16">A1</font></span>: 
<font color="Maroon" title="c3">C1</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c4">C2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">i</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font><br/> for <font color="Olive" title="b2">x</font>, <font color="Olive" title="b3">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">x</font>,<font color="Olive" title="b3">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C1</font> <a href="msualg_4.html#K2" title="MSUALG_4:func.2">.</a> <font color="Olive" title="b1">i</font> holds <br/>(  <a href="msualg_4.html#K3" title="MSUALG_4:func.3">Class</a> <font color="Maroon" title="c3">C1</font>,<font color="Olive" title="b2">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="msualg_4.html#K3" title="MSUALG_4:func.3">Class</a> <font color="Maroon" title="c4">C2</font>,<font color="Olive" title="b3">y</font> &amp; ( <font color="Maroon" title="c2">A</font> is  <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a>  implies  <a href="msualg_4.html#K3" title="MSUALG_4:func.3">Class</a> <font color="Maroon" title="c3">C1</font>,<font color="Olive" title="b3">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="msualg_4.html#K3" title="MSUALG_4:func.3">Class</a> <font color="Maroon" title="c4">C2</font>,<font color="Olive" title="b2">x</font> ) )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c5">i</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C1</font> <a href="msualg_4.html#K2" title="MSUALG_4:func.2">.</a> <font color="Maroon" title="c5">i</font> holds <br/>(  <a href="msualg_4.html#K3" title="MSUALG_4:func.3">Class</a> <font color="Maroon" title="c3">C1</font>,<font color="Olive" title="b1">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="msualg_4.html#K3" title="MSUALG_4:func.3">Class</a> <font color="Maroon" title="c4">C2</font>,<font color="Olive" title="b2">y</font> &amp; ( <font color="Maroon" title="c2">A</font> is  <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a>  implies  <a href="msualg_4.html#K3" title="MSUALG_4:func.3">Class</a> <font color="Maroon" title="c3">C1</font>,<font color="Olive" title="b2">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="msualg_4.html#K3" title="MSUALG_4:func.3">Class</a> <font color="Maroon" title="c4">C2</font>,<font color="Olive" title="b1">x</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c7">y</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C1</font> <a href="msualg_4.html#K2" title="MSUALG_4:func.2">.</a> <font color="Maroon" title="c5">i</font> implies (  <a href="msualg_4.html#K3" title="MSUALG_4:func.3">Class</a> <font color="Maroon" title="c3">C1</font>,<font color="Maroon" title="c6">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="msualg_4.html#K3" title="MSUALG_4:func.3">Class</a> <font color="Maroon" title="c4">C2</font>,<font color="Maroon" title="c7">y</font> &amp; ( <font color="Maroon" title="c2">A</font> is  <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a>  implies  <a href="msualg_4.html#K3" title="MSUALG_4:func.3">Class</a> <font color="Maroon" title="c3">C1</font>,<font color="Maroon" title="c7">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="msualg_4.html#K3" title="MSUALG_4:func.3">Class</a> <font color="Maroon" title="c4">C2</font>,<font color="Maroon" title="c6">x</font> ) ) )</span><br/>



<span class="kw">assume </span><a NAME="E2:43"/><span class="lab"><font color="Green" title="E17">A2</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C1</font> <a href="msualg_4.html#K2" title="MSUALG_4:func.2">.</a> <font color="Maroon" title="c5">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="msualg_4.html#K3" title="MSUALG_4:func.3">Class</a> <font color="Maroon" title="c3">C1</font>,<font color="Maroon" title="c6">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="msualg_4.html#K3" title="MSUALG_4:func.3">Class</a> <font color="Maroon" title="c4">C2</font>,<font color="Maroon" title="c7">y</font> &amp; ( <font color="Maroon" title="c2">A</font> is  <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a>  implies  <a href="msualg_4.html#K3" title="MSUALG_4:func.3">Class</a> <font color="Maroon" title="c3">C1</font>,<font color="Maroon" title="c7">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="msualg_4.html#K3" title="MSUALG_4:func.3">Class</a> <font color="Maroon" title="c4">C2</font>,<font color="Maroon" title="c6">x</font> ) )</span><br/>

<a NAME="E3:43"/>
 <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">C1</font> <a href="msualg_4.html#K2" title="MSUALG_4:func.2">.</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="orders_1.html#T97" title="ORDERS_1:th.97">ORDERS_1:97</a></span>;<br/>
<a NAME="E4:43"/><span class="kw">then </span><span class="lab"><font color="Green" title="E18">A3</font></span>: 
<font color="Maroon" title="c3">C1</font> <a href="msualg_4.html#K2" title="MSUALG_4:func.2">.</a> <font color="Maroon" title="c5">i</font> <a href="relat_2.html#R8" title="RELAT_2:pred.8">is_transitive_in</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_2.html#D16" title="RELAT_2:def.16">RELAT_2:def 16</a></span>;<br/>
<a NAME="E5:43"/><span class="lab"><font color="Green" title="E19">A4</font></span>: 
<font color="Maroon" title="c3">C1</font> <a href="msualg_4.html#K2" title="MSUALG_4:func.2">.</a> <font color="Maroon" title="c5">i</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Maroon" title="c4">C2</font> <a href="msualg_4.html#K2" title="MSUALG_4:func.2">.</a> <font color="Maroon" title="c5">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43"><span class="lab"><font color="Green" title="E16">A1</font></span></a>, <a class="ref" href="pboole.html#D5" title="PBOOLE:def.5">PBOOLE:def 5</a></span>;<br/>
<span class="kw">thus </span><a NAME="E6:43"/>
 <a href="msualg_4.html#K3" title="MSUALG_4:func.3">Class</a> <font color="Maroon" title="c3">C1</font>,<font color="Maroon" title="c6">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="msualg_4.html#K3" title="MSUALG_4:func.3">Class</a> <font color="Maroon" title="c4">C2</font>,<font color="Maroon" title="c7">y</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">A</font> is  <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a>  implies  <a href="msualg_4.html#K3" title="MSUALG_4:func.3">Class</a> <font color="Maroon" title="c3">C1</font>,<font color="Maroon" title="c7">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="msualg_4.html#K3" title="MSUALG_4:func.3">Class</a> <font color="Maroon" title="c4">C2</font>,<font color="Maroon" title="c6">x</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">q</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msualg_4.html#K3" title="MSUALG_4:func.3">Class</a> <font color="Maroon" title="c3">C1</font>,<font color="Maroon" title="c6">x</font> or <font color="Maroon" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msualg_4.html#K3" title="MSUALG_4:func.3">Class</a> <font color="Maroon" title="c4">C2</font>,<font color="Maroon" title="c7">y</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:43_1"/><span class="lab"><font color="Green" title="E20">A5</font></span>: 
<font color="Maroon" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msualg_4.html#K3" title="MSUALG_4:func.3">Class</a> <font color="Maroon" title="c3">C1</font>,<font color="Maroon" title="c6">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msualg_4.html#K3" title="MSUALG_4:func.3">Class</a> <font color="Maroon" title="c4">C2</font>,<font color="Maroon" title="c7">y</font></span><br/>

<a NAME="E2:43_1"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">q</font>,<font color="Maroon" title="c6">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C1</font> <a href="msualg_4.html#K2" title="MSUALG_4:func.2">.</a> <font color="Maroon" title="c5">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="eqrel_1.html#T27" title="EQREL_1:th.27">EQREL_1:27</a></span>;<br/>
<a NAME="E3:43_1"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">q</font>,<font color="Maroon" title="c7">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C1</font> <a href="msualg_4.html#K2" title="MSUALG_4:func.2">.</a> <font color="Maroon" title="c5">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="txt" href="#E4:43"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="txt" href="#E1:43_1"><span class="lab"><font color="Green" title="E20">A5</font></span></a>, <a class="ref" href="relat_2.html#D8" title="RELAT_2:def.8">RELAT_2:def 8</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:43_1"/>
<font color="Maroon" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msualg_4.html#K3" title="MSUALG_4:func.3">Class</a> <font color="Maroon" title="c4">C2</font>,<font color="Maroon" title="c7">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:43"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="ref" href="eqrel_1.html#T27" title="EQREL_1:th.27">EQREL_1:27</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E7:43"/>
<font color="Maroon" title="c2">A</font> is  <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="msualg_4.html#K3" title="MSUALG_4:func.3">Class</a> <font color="Maroon" title="c3">C1</font>,<font color="Maroon" title="c7">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="msualg_4.html#K3" title="MSUALG_4:func.3">Class</a> <font color="Maroon" title="c4">C2</font>,<font color="Maroon" title="c6">x</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">B</font> = <font color="Maroon" title="c2">A</font> as   <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/>
<a NAME="E9:43"/>
 <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">C1</font> <a href="msualg_4.html#K2" title="MSUALG_4:func.2">.</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="orders_1.html#T97" title="ORDERS_1:th.97">ORDERS_1:97</a></span>;<br/>
<a NAME="E10:43"/><span class="kw">then </span>
<font color="Maroon" title="c3">C1</font> <a href="msualg_4.html#K2" title="MSUALG_4:func.2">.</a> <font color="Maroon" title="c5">i</font> <a href="relat_2.html#R3" title="RELAT_2:pred.3">is_symmetric_in</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c8">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_2.html#D11" title="RELAT_2:def.11">RELAT_2:def 11</a></span>;<br/>
<a NAME="E11:43"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A6</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c6">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C1</font> <a href="msualg_4.html#K2" title="MSUALG_4:func.2">.</a> <font color="Maroon" title="c5">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="ref" href="relat_2.html#D3" title="RELAT_2:def.3">RELAT_2:def 3</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c9">q</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c9">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msualg_4.html#K3" title="MSUALG_4:func.3">Class</a> <font color="Maroon" title="c3">C1</font>,<font color="Maroon" title="c7">y</font> or <font color="Maroon" title="c9">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msualg_4.html#K3" title="MSUALG_4:func.3">Class</a> <font color="Maroon" title="c4">C2</font>,<font color="Maroon" title="c6">x</font> )</span><br/>

<span class="kw">assume </span><a NAME="E12:43"/><span class="lab"><font color="Green" title="E21">A7</font></span>: 
<font color="Maroon" title="c9">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msualg_4.html#K3" title="MSUALG_4:func.3">Class</a> <font color="Maroon" title="c3">C1</font>,<font color="Maroon" title="c7">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msualg_4.html#K3" title="MSUALG_4:func.3">Class</a> <font color="Maroon" title="c4">C2</font>,<font color="Maroon" title="c6">x</font></span><br/>

<a NAME="E13:43"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">q</font>,<font color="Maroon" title="c7">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C1</font> <a href="msualg_4.html#K2" title="MSUALG_4:func.2">.</a> <font color="Maroon" title="c5">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:43"><span class="lab"><font color="Green" title="E21">A7</font></span></a>, <a class="ref" href="eqrel_1.html#T27" title="EQREL_1:th.27">EQREL_1:27</a></span>;<br/>
<a NAME="E14:43"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">q</font>,<font color="Maroon" title="c6">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C1</font> <a href="msualg_4.html#K2" title="MSUALG_4:func.2">.</a> <font color="Maroon" title="c5">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:43"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="txt" href="#E12:43"><span class="lab"><font color="Green" title="E21">A7</font></span></a>, <a class="txt" href="#E11:43"><span class="lab"><font color="Green" title="E20">A6</font></span></a>, <a class="ref" href="relat_2.html#D8" title="RELAT_2:def.8">RELAT_2:def 8</a></span>;<br/>
<span class="kw">hence </span><a NAME="E15:43"/>
<font color="Maroon" title="c9">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msualg_4.html#K3" title="MSUALG_4:func.3">Class</a> <font color="Maroon" title="c4">C2</font>,<font color="Maroon" title="c6">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:43"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="ref" href="eqrel_1.html#T27" title="EQREL_1:th.27">EQREL_1:27</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
