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<b>let </b><font color="Maroon">DK</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> , <font color="Maroon">DX</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">f</font> be   <a href="binop_1.html#NM2">BinOp</a> of <font color="Maroon">DK</font>;<br/>



<b>assume </b><a NAME="E1:25"/><i><font color="Green">E42</font></i>: 
( <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>  )
 ;<br/>

<b>let </b><font color="Maroon">Y1</font> be  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">DX</font>, <font color="Maroon">Y2</font> be  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">DX</font>;<br/>

<b>assume </b><a NAME="E2:25"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">Y1</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">Y2</font>
 ;<br/>

<b>let </b><font color="Maroon">F</font> be   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">DX</font>,<font color="Maroon">DK</font>;<br/>

<b>assume </b><a NAME="E3:25"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">Y1</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> &amp; <font color="Maroon">Y2</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> )
 ;<br/>

<b>let </b><font color="Maroon">Z</font> be  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">DX</font>;<br/>

<b>assume </b><a NAME="E4:25"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">Z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Y2</font>
 ;<br/>

<a NAME="E5:25"/><b>then </b><i><font color="Green">E46</font></i>: 
<font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/>
<b>consider </b><font color="Maroon">p1</font> being    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">DX</font><b> such that </b><br/><a NAME="E7:25"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">p1</font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>and </b><br/><a NAME="E8:25"/><i><font color="Green">E48</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y1</font>
 <b>and </b><br/><a NAME="E9:25"/><i><font color="Green">E49</font></i>: 
 <a href="bhsp_5.html#K5" target="_self">setopfunc</a> <font color="Maroon">Y1</font>,<font color="Maroon">DX</font>,<font color="Maroon">DK</font>,<font color="Maroon">F</font>,<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finsop_1.html#K2">"**"</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K4" target="_self">Func_Seq</a> <font color="Maroon">F</font>,<font color="Maroon">p1</font></span>)</span>
 <b>by </b><i>, , </i>;<br/>
<b>consider </b><font color="Maroon">p2</font> being    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">DX</font><b> such that </b><br/><a NAME="E11:25"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">p2</font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>and </b><br/><a NAME="E12:25"/><i><font color="Green">E52</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">p2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y2</font>
 <b>and </b><br/><a NAME="E13:25"/><i><font color="Green">E53</font></i>: 
 <a href="bhsp_5.html#K5" target="_self">setopfunc</a> <font color="Maroon">Y2</font>,<font color="Maroon">DX</font>,<font color="Maroon">DK</font>,<font color="Maroon">F</font>,<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finsop_1.html#K2">"**"</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K4" target="_self">Func_Seq</a> <font color="Maroon">F</font>,<font color="Maroon">p2</font></span>)</span>
 <b>by </b><i>, , </i>;<br/>
<b>set </b><font color="Maroon">q</font> = <font color="Maroon">p1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p2</font>;<br/>
<a NAME="E14:25"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">p1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p2</font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i>, , , , , <a class="ref" href="finseq_3.html#T98">FINSEQ_3:98</a></i>;<br/>
<a NAME="E15:25"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">Z</font>
 
<b>by </b><i>, , , <a class="ref" href="finseq_1.html#T44">FINSEQ_1:44</a></i>;<br/>
<a NAME="E16:25"/><b>then </b><i><font color="Green">E56</font></i>: 
 <a href="bhsp_5.html#K5" target="_self">setopfunc</a> <font color="Maroon">Z</font>,<font color="Maroon">DX</font>,<font color="Maroon">DK</font>,<font color="Maroon">F</font>,<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finsop_1.html#K2">"**"</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K4" target="_self">Func_Seq</a> <font color="Maroon">F</font>,<span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p2</font></span>)</span></span>)</span>
 
<b>by </b><i>, , , </i>;<br/>
<a NAME="E17:25"/><i><font color="Green">E57</font></i>: 
 ex <font color="Olive">fp1</font>, <font color="Olive">fp2</font> being  <a href="finseq_1.html#NM1">FinSequence</a> st <br/>( <font color="Olive">fp1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">p1</font> &amp; <font color="Olive">fp2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">p2</font> &amp; <font color="Maroon">F</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">fp1</font> <a href="finseq_1.html#K7">^</a> <font color="Olive">fp2</font> )
 
<b>by </b><i>, , , , <a class="ref" href="hilbasis.html#T1">HILBASIS:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">fp1</font> = <font color="Maroon">F</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">p1</font> as   <a href="finseq_1.html#NM1">FinSequence</a> ;<br/>
<b>reconsider </b><font color="Maroon">fp2</font> = <font color="Maroon">F</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">p2</font> as   <a href="finseq_1.html#NM1">FinSequence</a> <b>by </b><i/>;<br/>
<a NAME="E20:25"/>
 <a href="bhsp_5.html#K4" target="_self">Func_Seq</a> <font color="Maroon">F</font>,<span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K4" target="_self">Func_Seq</a> <font color="Maroon">F</font>,<font color="Maroon">p1</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K4" target="_self">Func_Seq</a> <font color="Maroon">F</font>,<font color="Maroon">p2</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E21:25"/>
 <a href="bhsp_5.html#K5" target="_self">setopfunc</a> <font color="Maroon">Z</font>,<font color="Maroon">DX</font>,<font color="Maroon">DK</font>,<font color="Maroon">F</font>,<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K5" target="_self">setopfunc</a> <font color="Maroon">Y1</font>,<font color="Maroon">DX</font>,<font color="Maroon">DK</font>,<font color="Maroon">F</font>,<font color="Maroon">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="bhsp_5.html#K5" target="_self">setopfunc</a> <font color="Maroon">Y2</font>,<font color="Maroon">DX</font>,<font color="Maroon">DK</font>,<font color="Maroon">F</font>,<font color="Maroon">f</font></span>)</span>
 <b>by </b><i>, , , , <a class="ref" href="finsop_1.html#T6">FINSOP_1:6</a></i>;<br/>


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