<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="prvect_1.html#NM1">Domain-Sequence</a>;<br/><b>let </b><font color="Maroon">b</font> be    <a href="prvect_1.html#M1" target="_self">BinOps</a> of <font color="Maroon">a</font>;<br/><b>let </b><font color="Maroon">u</font> be    <a href="prvect_1.html#M2" target="_self">UnOps</a> of <font color="Maroon">a</font>;<br/>





<b>assume </b><a NAME="E1:50"/><i><font color="Green">E22</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">a</font> holds <br/> ( <font color="Maroon">u</font> <a href="prvect_1.html#K11" target="_self">.</a> <font color="Olive">i</font> <a href="finseqop.html#R1">is_an_inverseOp_wrt</a> <font color="Maroon">b</font> <a href="prvect_1.html#K10" target="_self">.</a> <font color="Olive">i</font> &amp; <font color="Maroon">b</font> <a href="prvect_1.html#K10" target="_self">.</a> <font color="Olive">i</font> <a href="setwiseo.html#NR1" target="_self">has_a_unity</a>  )
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">a</font> st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">i</font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="binop_1.html#K3">the_unity_wrt</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="prvect_1.html#K10" target="_self">.</a> <font color="Olive">i</font></span>)</span> );<br/>
<a NAME="E2:50"/><i><font color="Green">E25</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">a</font> holds <br/> ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:50_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">a</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">i</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">a</font> ;<br/>
<b>take </b>
 <a href="binop_1.html#K3">the_unity_wrt</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="prvect_1.html#K10" target="_self">.</a> <font color="Maroon">i</font></span>)</span>
;<br/>

<b>thus </b><a NAME="E3:50_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>, <a href="binop_1.html#K3">the_unity_wrt</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="prvect_1.html#K10" target="_self">.</a> <font color="Maroon">i</font></span>)</span>]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:50"/><i><font color="Green">E26</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y1</font>, <font color="Olive">y2</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">a</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y2</font>] holds <br/><font color="Olive">y1</font> <a href="hidden.html#R1">=</a> <font color="Olive">y2</font>
 
;<br/>
<b>consider </b><font color="Maroon">f</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E5:50"/><i><font color="Green">E27</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">a</font> &amp; ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">a</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>] ) )
 <b>from </b><i><a class="ref" href="funct_1.html#S2">FUNCT_1:sch 2</a>(, <a class="txt" href="prvect_1.html#E8"><i><font color="Green">Lemma32</font></i></a>);<br/></i>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:50_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">a</font>
 ;<br/><a NAME="E2:50_2"/><b>then </b>
 ex <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">a</font> st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">i</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="binop_1.html#K3">the_unity_wrt</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="prvect_1.html#K10" target="_self">.</a> <font color="Olive">i</font></span>)</span> )
 <b>by </b><i><a class="txt" href="prvect_1.html#E9"><i><font color="Green">Lemma34</font></i></a></i>;<br/><b>hence </b><a NAME="E3:50_2"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 ;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="card_3.html#K4">product</a> <font color="Maroon">a</font> <b>by </b><i><a class="txt" href="prvect_1.html#E9"><i><font color="Green">Lemma34</font></i></a>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">a</font>;<br/><a NAME="E1:50_3"/>
 ex <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">a</font> st <br/>( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Olive">j</font> &amp; <font color="Maroon">f</font> <a href="prvect_1.html#K9" target="_self">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="binop_1.html#K3">the_unity_wrt</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="prvect_1.html#K10" target="_self">.</a> <font color="Olive">j</font></span>)</span> )
 <b>by </b><i><a class="txt" href="prvect_1.html#E9"><i><font color="Green">Lemma34</font></i></a></i>;<br/><a NAME="E2:50_3"/><b>then </b>
( <font color="Maroon">f</font> <a href="prvect_1.html#K9" target="_self">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="binop_1.html#K3">the_unity_wrt</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="prvect_1.html#K10" target="_self">.</a> <font color="Maroon">i</font></span>)</span> &amp; <font color="Maroon">b</font> <a href="prvect_1.html#K10" target="_self">.</a> <font color="Maroon">i</font> <a href="setwiseo.html#NR1" target="_self">has_a_unity</a>  )
 <b>by </b><i><a class="txt" href="prvect_1.html#E7"><i><font color="Green">Lemma31</font></i></a></i>;<br/><b>hence </b><a NAME="E3:50_3"/>
<font color="Maroon">f</font> <a href="prvect_1.html#K9" target="_self">.</a> <font color="Maroon">i</font> <a href="binop_1.html#R3">is_a_unity_wrt</a> <font color="Maroon">b</font> <a href="prvect_1.html#K10" target="_self">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="setwiseo.html#T22">SETWISEO:22</a></i>;<br/></div><b>end;</b></div>
<a NAME="E9:50"/><b>then </b>
<font color="Maroon">f</font> <a href="binop_1.html#R3">is_a_unity_wrt</a> <span class="p1"><a href="prvect_1.html#K14" target="_self">[:</a><span class="default"><font color="Maroon">b</font></span><a href="prvect_1.html#K14" target="_self">:]</a></span>
 
<b>by </b><i/>;<br/>
<a NAME="E10:50"/><b>then </b><i><font color="Green">E55</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="binop_1.html#K3">the_unity_wrt</a> <span class="p1"><a href="prvect_1.html#K14" target="_self">[:</a><span class="default"><font color="Maroon">b</font></span><a href="prvect_1.html#K14" target="_self">:]</a></span>
 
<b>by </b><i><a class="ref" href="binop_1.html#D8">BINOP_1:def 8</a></i>;<br/>
<b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="card_3.html#K4">product</a> <font color="Maroon">a</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="finseqop.html#D1">FINSEQOP:def 1</a><br/>

<a NAME="E11:50"/><i><font color="Green">E60</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="prvect_1.html#K14" target="_self">[:</a><span class="default"><font color="Maroon">b</font></span><a href="prvect_1.html#K14" target="_self">:]</a></span> <a href="prvect_1.html#K13" target="_self">.</a> <font color="Maroon">x</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="prvect_1.html#K12" target="_self">Frege</a> <font color="Maroon">u</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">a</font> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="prvect_1.html#K14" target="_self">[:</a><span class="default"><font color="Maroon">b</font></span><a href="prvect_1.html#K14" target="_self">:]</a></span> <a href="prvect_1.html#K13" target="_self">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="prvect_1.html#K12" target="_self">Frege</a> <font color="Maroon">u</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>,<font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">a</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">a</font> )
 
<b>by </b><i><a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:50_4"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">a</font>
 ;<br/><b>then reconsider </b><font color="Maroon">i</font> = <font color="Maroon">y</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">a</font> ;<br/><a NAME="E3:50_4"/>
 ex <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">a</font> st <br/>( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Olive">j</font> &amp; <font color="Maroon">f</font> <a href="prvect_1.html#K9" target="_self">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="binop_1.html#K3">the_unity_wrt</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="prvect_1.html#K10" target="_self">.</a> <font color="Olive">j</font></span>)</span> )
 <b>by </b><i><a class="txt" href="prvect_1.html#E9"><i><font color="Green">Lemma34</font></i></a></i>;<br/><a NAME="E4:50_4"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2"><a href="prvect_1.html#K14" target="_self">[:</a><span class="default"><font color="Maroon">b</font></span><a href="prvect_1.html#K14" target="_self">:]</a></span> <a href="prvect_1.html#K13" target="_self">.</a> <font color="Maroon">x</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="prvect_1.html#K12" target="_self">Frege</a> <font color="Maroon">u</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="prvect_1.html#K9" target="_self">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="prvect_1.html#K10" target="_self">.</a> <font color="Maroon">i</font></span>)</span> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="prvect_1.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="prvect_1.html#K12" target="_self">Frege</a> <font color="Maroon">u</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="prvect_1.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="prvect_1.html#K12" target="_self">Frege</a> <font color="Maroon">u</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="prvect_1.html#K9" target="_self">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="prvect_1.html#K11" target="_self">.</a> <font color="Maroon">i</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="prvect_1.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span> &amp; <font color="Maroon">f</font> <a href="prvect_1.html#K9" target="_self">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="binop_1.html#K3">the_unity_wrt</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="prvect_1.html#K10" target="_self">.</a> <font color="Maroon">i</font></span>)</span> &amp; <font color="Maroon">u</font> <a href="prvect_1.html#K11" target="_self">.</a> <font color="Maroon">i</font> <a href="finseqop.html#R1">is_an_inverseOp_wrt</a> <font color="Maroon">b</font> <a href="prvect_1.html#K10" target="_self">.</a> <font color="Maroon">i</font> )
 <b>by </b><i><a class="txt" href="prvect_1.html#E7"><i><font color="Green">Lemma31</font></i></a>, <a class="txt" href="prvect_1.html#E3:50"><i><font color="Green">E26</font></i></a>, </i>;<br/><b>hence </b><a NAME="E5:50_4"/>
<span class="p1">(<span class="default"><span class="p2"><a href="prvect_1.html#K14" target="_self">[:</a><span class="default"><font color="Maroon">b</font></span><a href="prvect_1.html#K14" target="_self">:]</a></span> <a href="prvect_1.html#K13" target="_self">.</a> <font color="Maroon">x</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="prvect_1.html#K12" target="_self">Frege</a> <font color="Maroon">u</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="finseqop.html#D1">FINSEQOP:def 1</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E13:50"/>
<span class="p1"><a href="prvect_1.html#K14" target="_self">[:</a><span class="default"><font color="Maroon">b</font></span><a href="prvect_1.html#K14" target="_self">:]</a></span> <a href="prvect_1.html#K13" target="_self">.</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="prvect_1.html#K12" target="_self">Frege</a> <font color="Maroon">u</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="binop_1.html#K3">the_unity_wrt</a> <span class="p1"><a href="prvect_1.html#K14" target="_self">[:</a><span class="default"><font color="Maroon">b</font></span><a href="prvect_1.html#K14" target="_self">:]</a></span>
 <b>by </b><i><a class="ref" href="prvect_1.html#T11" target="_self">Th11</a>, <a class="txt" href="prvect_1.html#E16"><i><font color="Green">Lemma36</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:50_5"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">a</font>
 ;<br/><b>then reconsider </b><font color="Maroon">i</font> = <font color="Maroon">y</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">a</font> ;<br/><a NAME="E3:50_5"/>
 ex <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">a</font> st <br/>( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Olive">j</font> &amp; <font color="Maroon">f</font> <a href="prvect_1.html#K9" target="_self">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="binop_1.html#K3">the_unity_wrt</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="prvect_1.html#K10" target="_self">.</a> <font color="Olive">j</font></span>)</span> )
 <b>by </b><i><a class="txt" href="prvect_1.html#E9"><i><font color="Green">Lemma34</font></i></a></i>;<br/><a NAME="E4:50_5"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2"><a href="prvect_1.html#K14" target="_self">[:</a><span class="default"><font color="Maroon">b</font></span><a href="prvect_1.html#K14" target="_self">:]</a></span> <a href="prvect_1.html#K13" target="_self">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="prvect_1.html#K12" target="_self">Frege</a> <font color="Maroon">u</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>,<font color="Maroon">x</font></span>)</span> <a href="prvect_1.html#K9" target="_self">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="prvect_1.html#K10" target="_self">.</a> <font color="Maroon">i</font></span>)</span> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="prvect_1.html#K12" target="_self">Frege</a> <font color="Maroon">u</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="prvect_1.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="prvect_1.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="prvect_1.html#K12" target="_self">Frege</a> <font color="Maroon">u</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="prvect_1.html#K9" target="_self">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="prvect_1.html#K11" target="_self">.</a> <font color="Maroon">i</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="prvect_1.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span> &amp; <font color="Maroon">f</font> <a href="prvect_1.html#K9" target="_self">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="binop_1.html#K3">the_unity_wrt</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="prvect_1.html#K10" target="_self">.</a> <font color="Maroon">i</font></span>)</span> &amp; <font color="Maroon">u</font> <a href="prvect_1.html#K11" target="_self">.</a> <font color="Maroon">i</font> <a href="finseqop.html#R1">is_an_inverseOp_wrt</a> <font color="Maroon">b</font> <a href="prvect_1.html#K10" target="_self">.</a> <font color="Maroon">i</font> )
 <b>by </b><i><a class="txt" href="prvect_1.html#E7"><i><font color="Green">Lemma31</font></i></a>, <a class="txt" href="prvect_1.html#E3:50"><i><font color="Green">E26</font></i></a>, </i>;<br/><b>hence </b><a NAME="E5:50_5"/>
<span class="p1">(<span class="default"><span class="p2"><a href="prvect_1.html#K14" target="_self">[:</a><span class="default"><font color="Maroon">b</font></span><a href="prvect_1.html#K14" target="_self">:]</a></span> <a href="prvect_1.html#K13" target="_self">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="prvect_1.html#K12" target="_self">Frege</a> <font color="Maroon">u</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>,<font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="finseqop.html#D1">FINSEQOP:def 1</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E15:50"/>
<span class="p1"><a href="prvect_1.html#K14" target="_self">[:</a><span class="default"><font color="Maroon">b</font></span><a href="prvect_1.html#K14" target="_self">:]</a></span> <a href="prvect_1.html#K13" target="_self">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="prvect_1.html#K12" target="_self">Frege</a> <font color="Maroon">u</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>,<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="binop_1.html#K3">the_unity_wrt</a> <span class="p1"><a href="prvect_1.html#K14" target="_self">[:</a><span class="default"><font color="Maroon">b</font></span><a href="prvect_1.html#K14" target="_self">:]</a></span>
 <b>by </b><i><a class="ref" href="prvect_1.html#T11" target="_self">Th11</a>, <a class="txt" href="prvect_1.html#E16"><i><font color="Green">Lemma36</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


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