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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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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#L3" title="MSUALG_1:struct.3">MSAlgebra</a> over <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#V2" title="MSUALG_4:attr.2">MSEquivalence-like</a>  <a href="msualg_4.html#NM3" title="MSUALG_4:NM.3">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> over <font color="Maroon" title="c1">S</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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#V2" title="MSUALG_4:attr.2">MSEquivalence-like</a>  <a href="msualg_4.html#NM3" title="MSUALG_4:NM.3">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#V2" title="MSUALG_4:attr.2">MSEquivalence-like</a>  <a href="msualg_4.html#NM3" title="MSUALG_4:NM.3">ManySortedRelation</a> of <font color="Maroon" title="c2">A</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="E1:42"/><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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="E2:42"/><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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="E3:42"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">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#T12" onmouseover="rs('orders_1/T12')" onmouseout="rh()">ORDERS_1:12</a></span>;<br/>
<a NAME="E4:42"/><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" onmouseover="rs('relat_2/D16')" onmouseout="rh()">RELAT_2:def 16</a></span>;<br/>
<a NAME="E5:42"/><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="#E1:42"><span class="lab"><font color="Green" title="E16">A1</font></span></a></span>;<br/>
<span class="kw">thus </span><a NAME="E6:42"/>
 <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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="42_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">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="E1:42_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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="E2:42_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#T19" onmouseover="rs('eqrel_1/T19')" onmouseout="rh()">EQREL_1:19</a></span>;<br/>
<a NAME="E3:42_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="#E2:42"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="txt" href="#E4:42"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="txt" href="#E1:42_1"><span class="lab"><font color="Green" title="E20">A5</font></span></a>, <a class="ref" href="relat_2.html#D8" onmouseover="rs('relat_2/D8')" onmouseout="rh()">RELAT_2:def 8</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:42_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="#E5:42"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="ref" href="eqrel_1.html#T19" onmouseover="rs('eqrel_1/T19')" onmouseout="rh()">EQREL_1:19</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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

<span class="kw">assume </span><a NAME="E7:42"/>
<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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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> over <font color="Maroon" title="c1">S</font> ;<br/>
<a NAME="E9:42"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">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#T12" onmouseover="rs('orders_1/T12')" onmouseout="rh()">ORDERS_1:12</a></span>;<br/>
<a NAME="E10:42"/><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" onmouseover="rs('relat_2/D11')" onmouseout="rh()">RELAT_2:def 11</a></span>;<br/>
<a NAME="E11:42"/><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="#E2:42"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="ref" href="relat_2.html#D3" onmouseover="rs('relat_2/D3')" onmouseout="rh()">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">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="E12:42"/><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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="E13:42"/>
<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="#E12:42"><span class="lab"><font color="Green" title="E21">A7</font></span></a>, <a class="ref" href="eqrel_1.html#T19" onmouseover="rs('eqrel_1/T19')" onmouseout="rh()">EQREL_1:19</a></span>;<br/>
<a NAME="E14:42"/><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="#E4:42"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="txt" href="#E12:42"><span class="lab"><font color="Green" title="E21">A7</font></span></a>, <a class="txt" href="#E11:42"><span class="lab"><font color="Green" title="E20">A6</font></span></a>, <a class="ref" href="relat_2.html#D8" onmouseover="rs('relat_2/D8')" onmouseout="rh()">RELAT_2:def 8</a></span>;<br/>
<span class="kw">hence </span><a NAME="E15:42"/>
<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="#E5:42"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="ref" href="eqrel_1.html#T19" onmouseover="rs('eqrel_1/T19')" onmouseout="rh()">EQREL_1:19</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
