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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">d</font> be    <a href="subset_1.html#M1">Element</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">k</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">l</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>









<a NAME="E1:32"/>
( <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">l</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 implies (  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="finsop_1.html#K1" target="_self">|-&gt;</a> <font color="Maroon">d</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="finsop_1.html#K1" target="_self">|-&gt;</a> <font color="Maroon">d</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 ) )
 
<b>by </b><i><a class="ref" href="finseq_2.html#T69">FINSEQ_2:69</a></i>;<br/>
<a NAME="E2:32"/><b>then </b><i><font color="Green">E35</font></i>: 
( <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">l</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 implies (  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="finsop_1.html#K1" target="_self">|-&gt;</a> <font color="Maroon">d</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="finsop_1.html#K1" target="_self">|-&gt;</a> <font color="Maroon">d</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 ) )
 
<b>by </b><i><a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/>
<a NAME="E3:32"/>
<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">l</font></span>)</span> <a href="finsop_1.html#K1" target="_self">|-&gt;</a> <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="finsop_1.html#K1" target="_self">|-&gt;</a> <font color="Maroon">d</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="finsop_1.html#K1" target="_self">|-&gt;</a> <font color="Maroon">d</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T143">FINSEQ_2:143</a></i>;<br/>
<b>hence </b><a NAME="E4:32"/>
( <font color="Maroon">g</font> is <a href="binop_1.html#V2">associative</a> &amp; ( <font color="Maroon">g</font> <a href="setwiseo.html#NR1" target="_self">has_a_unity</a>  or ( <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">l</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 ) ) implies <font color="Maroon">g</font> <a href="finsop_1.html#K2" target="_self">"**"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">l</font></span>)</span> <a href="finsop_1.html#K1" target="_self">|-&gt;</a> <font color="Maroon">d</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">g</font> <a href="finsop_1.html#K2" target="_self">"**"</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="finsop_1.html#K1" target="_self">|-&gt;</a> <font color="Maroon">d</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finsop_1.html#K2" target="_self">"**"</a> <span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="finsop_1.html#K1" target="_self">|-&gt;</a> <font color="Maroon">d</font></span>)</span></span>)</span> )
 <b>by </b><i>, </i>;<br/>


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