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<b>let </b><font color="Maroon">D</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">e</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font>;<br/><b>let </b><font color="Maroon">F</font> be   <a href="binop_1.html#NM2">BinOp</a> of <font color="Maroon">D</font>;<br/><b>let </b><font color="Maroon">G</font> be   <a href="binop_1.html#NM2">BinOp</a> of <font color="Maroon">D</font>;<br/><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">T1</font> be    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">i</font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font>;<br/><b>let </b><font color="Maroon">T2</font> be    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">i</font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font>;<br/>













<a NAME="E1:43"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">T1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">T2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> )
 
<b>by </b><i><a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>;<br/>
<b>hence </b><a NAME="E2:43"/>
( <font color="Maroon">F</font> is <a href="binop_1.html#V1">commutative</a> &amp; <font color="Maroon">F</font> is <a href="binop_1.html#V2">associative</a> &amp; <font color="Maroon">F</font> <a href="setwiseo.html#NR1" target="_self">has_a_unity</a>  &amp; <font color="Maroon">e</font> <a href="hidden.html#R1">=</a>  <a href="binop_1.html#K3">the_unity_wrt</a> <font color="Maroon">F</font> &amp; <font color="Maroon">G</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">e</font>,<font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font> &amp; ( for <font color="Olive">d1</font>, <font color="Olive">d2</font>, <font color="Olive">d3</font>, <font color="Olive">d4</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> holds  <font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="binop_1.html#K2">.</a> <font color="Olive">d1</font>,<font color="Olive">d2</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="binop_1.html#K2">.</a> <font color="Olive">d3</font>,<font color="Olive">d4</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <font color="Olive">d1</font>,<font color="Olive">d3</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <font color="Olive">d2</font>,<font color="Olive">d4</font></span>)</span> ) implies <font color="Maroon">G</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finsop_1.html#K2">"**"</a> <font color="Maroon">T1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finsop_1.html#K2">"**"</a> <font color="Maroon">T2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="finsop_1.html#K2">"**"</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="finseqop.html#K1">.:</a> <font color="Maroon">T1</font>,<font color="Maroon">T2</font></span>)</span> )
 <b>by </b><i/>;<br/>


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