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<b>let </b><font color="Maroon" title="c1">D</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">d1</font>, <font color="Maroon" title="c3">d2</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</font>;<br/><b>let </b><font color="Maroon" title="c4">F</font>, <font color="Maroon" title="c5">G</font> be   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c1">D</font>;<br/><b>let </b><font color="Maroon" title="c6">u</font> be   <a href="binop_1.html#NM1" title="BINOP_1:NM.1">UnOp</a> of <font color="Maroon" title="c1">D</font>;<br/>











<b>assume </b><b>that </b><br/><a NAME="E1:81"/><i><font color="Green" title="E40">A1</font></i>: 
( <font color="Maroon" title="c4">F</font> <a href="setwiseo.html#NR1" title="SETWISEO:NR.1">has_a_unity</a>  &amp; <font color="Maroon" title="c4">F</font> is <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a> &amp; <font color="Maroon" title="c4">F</font> <a href="finseqop.html#NR2" title="FINSEQOP:NR.2">has_an_inverseOp</a>  )
 <b>and </b><br/><a NAME="E2:81"/><i><font color="Green" title="E41">A2</font></i>: 
<font color="Maroon" title="c6">u</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="finseqop.html#K6" title="FINSEQOP:func.6">the_inverseOp_wrt</a> <font color="Maroon" title="c4">F</font>
 <b>and </b><br/><a NAME="E3:81"/><i><font color="Green" title="E42">A3</font></i>: 
<font color="Maroon" title="c5">G</font> <a href="binop_1.html#R6" title="BINOP_1:pred.6">is_distributive_wrt</a> <font color="Maroon" title="c4">F</font>
 ;<br/>

<b>set </b><font color="Maroon" title="c7">e</font> =  <a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> <font color="Maroon" title="c4">F</font>;<br/>
<font color="Maroon" title="c4">F</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c2">d1</font>,<font color="Maroon" title="c3">d2</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">u</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">d1</font></span>)</span>,<font color="Maroon" title="c3">d2</font></span>)</span> = 
<font color="Maroon" title="c5">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c2">d1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">u</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">d1</font></span>)</span></span>)</span>,<font color="Maroon" title="c3">d2</font>
<b>by </b><i><a class="txt" href="finseqop.html#E3:81"><i><font color="Green" title="E42">A3</font></i></a>, <a class="ref" href="binop_1.html#T23" title="BINOP_1:th.23">BINOP_1:23</a></i>
<br/>.= 
<font color="Maroon" title="c5">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> <font color="Maroon" title="c4">F</font></span>)</span>,<font color="Maroon" title="c3">d2</font>
<b>by </b><i><a class="txt" href="finseqop.html#E1:81"><i><font color="Green" title="E40">A1</font></i></a>, <a class="txt" href="finseqop.html#E2:81"><i><font color="Green" title="E41">A2</font></i></a>, <a class="ref" href="finseqop.html#T63" target="_self" title="FINSEQOP:th.63">Th63</a></i>
<br/>.= 
 <a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> <font color="Maroon" title="c4">F</font>
<b>by </b><i><a class="txt" href="finseqop.html#E1:81"><i><font color="Green" title="E40">A1</font></i></a>, <a class="txt" href="finseqop.html#E3:81"><i><font color="Green" title="E42">A3</font></i></a>, <a class="ref" href="finseqop.html#T70" target="_self" title="FINSEQOP:th.70">Th70</a></i>
;<br/>
<b>hence </b><a NAME="E5:81"/>
<font color="Maroon" title="c6">u</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c2">d1</font>,<font color="Maroon" title="c3">d2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">u</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">d1</font></span>)</span>,<font color="Maroon" title="c3">d2</font>
 <b>by </b><i><a class="txt" href="finseqop.html#E1:81"><i><font color="Green" title="E40">A1</font></i></a>, <a class="txt" href="finseqop.html#E2:81"><i><font color="Green" title="E41">A2</font></i></a>, <a class="ref" href="finseqop.html#T64" target="_self" title="FINSEQOP:th.64">Th64</a></i>;<br/>

<font color="Maroon" title="c4">F</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c2">d1</font>,<font color="Maroon" title="c3">d2</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c2">d1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">u</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">d2</font></span>)</span></span>)</span> = 
<font color="Maroon" title="c5">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c2">d1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c3">d2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">u</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">d2</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="finseqop.html#E3:81"><i><font color="Green" title="E42">A3</font></i></a>, <a class="ref" href="binop_1.html#T23" title="BINOP_1:th.23">BINOP_1:23</a></i>
<br/>.= 
<font color="Maroon" title="c5">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c2">d1</font>,<span class="p1">(<span class="default"><a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> <font color="Maroon" title="c4">F</font></span>)</span>
<b>by </b><i><a class="txt" href="finseqop.html#E1:81"><i><font color="Green" title="E40">A1</font></i></a>, <a class="txt" href="finseqop.html#E2:81"><i><font color="Green" title="E41">A2</font></i></a>, <a class="ref" href="finseqop.html#T63" target="_self" title="FINSEQOP:th.63">Th63</a></i>
<br/>.= 
 <a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> <font color="Maroon" title="c4">F</font>
<b>by </b><i><a class="txt" href="finseqop.html#E1:81"><i><font color="Green" title="E40">A1</font></i></a>, <a class="txt" href="finseqop.html#E3:81"><i><font color="Green" title="E42">A3</font></i></a>, <a class="ref" href="finseqop.html#T70" target="_self" title="FINSEQOP:th.70">Th70</a></i>
;<br/>
<b>hence </b><a NAME="E7:81"/>
<font color="Maroon" title="c6">u</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c2">d1</font>,<font color="Maroon" title="c3">d2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c2">d1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">u</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">d2</font></span>)</span>
 <b>by </b><i><a class="txt" href="finseqop.html#E1:81"><i><font color="Green" title="E40">A1</font></i></a>, <a class="txt" href="finseqop.html#E2:81"><i><font color="Green" title="E41">A2</font></i></a>, <a class="ref" href="finseqop.html#T64" target="_self" title="FINSEQOP:th.64">Th64</a></i>;<br/>


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