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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">p</font> be    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">D</font>, <font color="Maroon">q</font> be    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">D</font>;<br/><b>let </b><font color="Maroon">B</font> be   <a href="binop_1.html#NM2">BinOp</a> of <font color="Maroon">D</font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:23"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">p</font> is <a href="funct_1.html#V2">one-to-one</a> &amp; <font color="Maroon">q</font> is <a href="funct_1.html#V2">one-to-one</a> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">q</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">p</font> )
 <b>and </b><br/><a NAME="E2:23"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">B</font> is <a href="binop_1.html#V1">commutative</a> &amp; <font color="Maroon">B</font> is <a href="binop_1.html#V2">associative</a> &amp; ( <font color="Maroon">B</font> <a href="setwiseo.html#NR1" target="_self">has_a_unity</a>  or (  <a href="finseq_1.html#K3">len</a> <font color="Maroon">q</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">q</font> ) ) )
 ;<br/>

<b>consider </b><font color="Maroon">r</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E4:23"/><i><font color="Green">E61</font></i>: 
(  <a href="relat_1.html#K2">rng</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">p</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">q</font></span>)</span> &amp; <font color="Maroon">r</font> is <a href="funct_1.html#V2">one-to-one</a> )
 <b>by </b><i><a class="ref" href="finseq_4.html#T73">FINSEQ_4:73</a></i>;<br/>
<b>reconsider </b><font color="Maroon">r</font> = <font color="Maroon">r</font> as    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">D</font> <b>by </b><i>, <a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E6:23"/>
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">r</font> <a href="xboole_0.html#R1">misses</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">q</font> &amp;  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">r</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">q</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">p</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">q</font></span>)</span></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T44">FINSEQ_1:44</a>, <a class="ref" href="xboole_1.html#T79">XBOOLE_1:79</a></i>;<br/>
<a NAME="E7:23"/><b>then </b><i><font color="Green">E62</font></i>: 
( <font color="Maroon">q</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">r</font> is <a href="funct_1.html#V2">one-to-one</a> &amp;  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">r</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">p</font> )
 
<b>by </b><i>, , <a class="ref" href="finseq_3.html#T98">FINSEQ_3:98</a>, <a class="ref" href="xboole_1.html#T45">XBOOLE_1:45</a></i>;<br/>
<b>then consider </b><font color="Maroon">P</font> being   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">p</font><b> such that </b><br/><a NAME="E9:23"/><i><font color="Green">E63</font></i>: 
( <font color="Maroon">p</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">r</font> &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">P</font> <a href="hidden.html#R1">=</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">p</font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">P</font> <a href="hidden.html#R1">=</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">p</font> )
 <b>by </b><i>, <a class="ref" href="bhsp_5.html#T1">BHSP_5:1</a></i>;<br/>
<a NAME="E10:23"/>
(  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> &amp;  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">r</font></span>)</span> )
 
<b>by </b><i>, , <a class="ref" href="finseq_4.html#T77">FINSEQ_4:77</a></i>;<br/>
<a NAME="E11:23"/><b>then </b>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">q</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">q</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">r</font></span>)</span> or <font color="Maroon">B</font> <a href="setwiseo.html#NR1" target="_self">has_a_unity</a>  )
 
<b>by </b><i>, , <a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/>
<a NAME="E12:23"/><b>then </b>
( ( 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">r</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">q</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> ) or <font color="Maroon">B</font> <a href="setwiseo.html#NR1" target="_self">has_a_unity</a>  )
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a>, <a class="ref" href="nat_1.html#T59">NAT_1:59</a></i>;<br/>
<a NAME="E13:23"/><b>then </b>
( <font color="Maroon">B</font> <a href="finsop_1.html#K2">"**"</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">r</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="finsop_1.html#K2">"**"</a> <font color="Maroon">q</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="finsop_1.html#K2">"**"</a> <font color="Maroon">r</font></span>)</span> &amp; <font color="Maroon">B</font> <a href="finsop_1.html#K2">"**"</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> <a href="finsop_1.html#K2">"**"</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">r</font></span>)</span> )
 
<b>by </b><i>, , <a class="ref" href="finsop_1.html#T6">FINSOP_1:6</a>, <a class="ref" href="finsop_1.html#T8">FINSOP_1:8</a></i>;<br/>
<b>hence </b><a NAME="E14:23"/>
 ex <font color="Olive">r</font> being   <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">D</font> st <br/>( <font color="Olive">r</font> is <a href="funct_1.html#V2">one-to-one</a> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Olive">r</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">p</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">q</font></span>)</span> &amp; <font color="Maroon">B</font> <a href="finsop_1.html#K2">"**"</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="finsop_1.html#K2">"**"</a> <font color="Maroon">q</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="finsop_1.html#K2">"**"</a> <font color="Olive">r</font></span>)</span> )
 <b>by </b><i/>;<br/>


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