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<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="100_1_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:100_1_1_1"/>
(  not <font color="Maroon" title="c1">M</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  or <font color="Maroon" title="c1">M</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="100_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:100_1_1_1_1"/><span class="lab"><font color="Green" title="E52">A1</font></span>: 
 not <font color="Maroon" title="c1">M</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">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"> <font color="Maroon" title="c1">M</font> <a href="algstr_4.html#K4" title="ALGSTR_4:func.4">./.</a> <font color="Maroon" title="c2">R</font> is  <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a> </span><br/><div class="add"><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="100_1_1_1_1_1"><span class="kw">now </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">  for <font color="Olive" title="b1">X</font>, <font color="Olive" title="b2">Y</font>, <font color="Olive" title="b3">Z</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="algstr_4.html#K4" title="ALGSTR_4:func.4">./.</a> <font color="Maroon" title="c2">R</font></span>)</span> holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">X</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b2">Y</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b3">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">X</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">Y</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b3">Z</font></span>)</span></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c3">X</font>, <font color="Maroon" title="c4">Y</font>, <font color="Maroon" title="c5">Z</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="algstr_4.html#K4" title="ALGSTR_4:func.4">./.</a> <font color="Maroon" title="c2">R</font></span>)</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"> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">Y</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">Y</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">Z</font></span>)</span></span><br/><span class="kw">consider </span><font color="Maroon" title="c6">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">M</font><span class="kw"> such that </span><br/><a NAME="E2:100_1_1_1_1_1"/><span class="lab"><font color="Green" title="E53">A2</font></span>: 
<font color="Maroon" title="c3">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c6">x</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:100_1_1_1_1"><span class="lab"><font color="Green" title="E52">A1</font></span></a>, <a class="ref" href="eqrel_1.html#T36" onmouseover="rs('eqrel_1/T36')" onmouseout="rh()">EQREL_1:36</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c7">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">M</font><span class="kw"> such that </span><br/><a NAME="E4:100_1_1_1_1_1"/><span class="lab"><font color="Green" title="E54">A3</font></span>: 
<font color="Maroon" title="c4">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c7">y</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:100_1_1_1_1"><span class="lab"><font color="Green" title="E52">A1</font></span></a>, <a class="ref" href="eqrel_1.html#T36" onmouseover="rs('eqrel_1/T36')" onmouseout="rh()">EQREL_1:36</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c8">z</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">M</font><span class="kw"> such that </span><br/><a NAME="E6:100_1_1_1_1_1"/><span class="lab"><font color="Green" title="E55">A4</font></span>: 
<font color="Maroon" title="c5">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c8">z</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:100_1_1_1_1"><span class="lab"><font color="Green" title="E52">A1</font></span></a>, <a class="ref" href="eqrel_1.html#T36" onmouseover="rs('eqrel_1/T36')" onmouseout="rh()">EQREL_1:36</a></span>;<br/><a NAME="E7:100_1_1_1_1_1"/><span class="lab"><font color="Green" title="E56">A5</font></span>: 
(  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<font color="Maroon" title="c2">R</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c7">y</font></span>)</span>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <font color="Maroon" title="c2">R</font> &amp;  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<font color="Maroon" title="c2">R</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c8">z</font></span>)</span>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <font color="Maroon" title="c2">R</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:100_1_1_1_1"><span class="lab"><font color="Green" title="E52">A1</font></span></a>, <a class="ref" href="eqrel_1.html#D3" onmouseover="rs('eqrel_1/D3')" onmouseout="rh()">EQREL_1:def 3</a></span>;<br/><span class="kw">thus </span><span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">Y</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">Z</font> = 
<span class="p1">(<span class="default"><a href="algstr_4.html#K3" title="ALGSTR_4:func.3">ClassOp</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<font color="Maroon" title="c2">R</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c7">y</font></span>)</span>)</span>)</span>,<font color="Maroon" title="c5">Z</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:100_1_1_1_1"><span class="lab"><font color="Green" title="E52">A1</font></span></a>, <a class="txt" href="#E2:100_1_1_1_1_1"><span class="lab"><font color="Green" title="E53">A2</font></span></a>, <a class="txt" href="#E4:100_1_1_1_1_1"><span class="lab"><font color="Green" title="E54">A3</font></span></a>, <a class="ref" href="algstr_4.html#D4" target="_self" onmouseover="rs('algstr_4/D4')" onmouseout="rh()">Def4</a></span>
<br/>.= 
 <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<font color="Maroon" title="c2">R</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c7">y</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c8">z</font></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:100_1_1_1_1"><span class="lab"><font color="Green" title="E52">A1</font></span></a>, <a class="txt" href="#E6:100_1_1_1_1_1"><span class="lab"><font color="Green" title="E55">A4</font></span></a>, <a class="txt" href="#E7:100_1_1_1_1_1"><span class="lab"><font color="Green" title="E56">A5</font></span></a>, <a class="ref" href="algstr_4.html#D4" target="_self" onmouseover="rs('algstr_4/D4')" onmouseout="rh()">Def4</a></span>
<br/>.= 
 <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<font color="Maroon" title="c2">R</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c8">z</font></span>)</span></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D3" onmouseover="rs('group_1/D3')" onmouseout="rh()">GROUP_1:def 3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="algstr_4.html#K3" title="ALGSTR_4:func.3">ClassOp</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c3">X</font>,<span class="p1">(<span class="default"><a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<font color="Maroon" title="c2">R</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c8">z</font></span>)</span>)</span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:100_1_1_1_1"><span class="lab"><font color="Green" title="E52">A1</font></span></a>, <a class="txt" href="#E2:100_1_1_1_1_1"><span class="lab"><font color="Green" title="E53">A2</font></span></a>, <a class="txt" href="#E7:100_1_1_1_1_1"><span class="lab"><font color="Green" title="E56">A5</font></span></a>, <a class="ref" href="algstr_4.html#D4" target="_self" onmouseover="rs('algstr_4/D4')" onmouseout="rh()">Def4</a></span>
<br/>.= 
<font color="Maroon" title="c3">X</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">Y</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">Z</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:100_1_1_1_1"><span class="lab"><font color="Green" title="E52">A1</font></span></a>, <a class="txt" href="#E4:100_1_1_1_1_1"><span class="lab"><font color="Green" title="E54">A3</font></span></a>, <a class="txt" href="#E6:100_1_1_1_1_1"><span class="lab"><font color="Green" title="E55">A4</font></span></a>, <a class="ref" href="algstr_4.html#D4" target="_self" onmouseover="rs('algstr_4/D4')" onmouseout="rh()">Def4</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">hence </span><a NAME="E3:100_1_1_1_1"/>
<font color="Maroon" title="c1">M</font> <a href="algstr_4.html#K4" title="ALGSTR_4:func.4">./.</a> <font color="Maroon" title="c2">R</font> is  <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D3" onmouseover="rs('group_1/D3')" onmouseout="rh()">GROUP_1:def 3</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="100_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:100_1_1_1_2"/>
<font color="Maroon" title="c1">M</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">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"> <font color="Maroon" title="c1">M</font> <a href="algstr_4.html#K4" title="ALGSTR_4:func.4">./.</a> <font color="Maroon" title="c2">R</font> is  <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a> </span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:100_1_1_1_2"/>
<font color="Maroon" title="c1">M</font> <a href="algstr_4.html#K4" title="ALGSTR_4:func.4">./.</a> <font color="Maroon" title="c2">R</font> is  <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</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"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

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