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<b>let </b><font color="Maroon">n</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="finseq_1.html#NM1">FinSequence</a>, <font color="Maroon">g</font> be   <a href="finseq_1.html#NM1">FinSequence</a>;<br/>



<b>assume </b><a NAME="E1:27"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">f</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">g</font> <a href="hidden.html#R2">in</a>  <a href="matrix_2.html#K12">Permutations</a> <font color="Maroon">n</font>
 ;<br/>

<a NAME="E2:27"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">f</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">g</font> is   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="matrix_2.html#D11">MATRIX_2:def 11</a>, </i>;<br/>
<a NAME="E3:27"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">f</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">g</font> is   <a href="funct_2.html#NM1">Function</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>, <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> &amp;  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D3">FUNCT_2:def 3</a>, </i>;<br/>
<a NAME="E4:27"/><b>then </b><i><font color="Green">E38</font></i>: 
 <a href="setwop_2.html#K3">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T67">FUNCT_2:67</a></i>;<br/>
<i><font color="Green">E39</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">g</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</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">g</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">f</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
;<br/>
<a NAME="E6:27"/>
 <a href="setwop_2.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="setwop_2.html#K3">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">g</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E7:27"/><b>then </b><i><font color="Green">E40</font></i>: 
 <a href="setwop_2.html#K3">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="setwop_2.html#K3">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<i><font color="Green">E41</font></i>:  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">g</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T44">FINSEQ_1:44</a></i>
<br/>.= 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">f</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T44">FINSEQ_1:44</a></i>
;<br/>
<a NAME="E9:27"/><i><font color="Green">E42</font></i>: 
(  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="xboole_0.html#R1">misses</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font> &amp; <font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a> &amp; <font color="Maroon">g</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/>
<b>reconsider </b><font color="Maroon">h</font> = <font color="Maroon">g</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">f</font> as  <a href="finseq_1.html#V1">FinSequence-like</a> <a href="funct_2.html#NM1">Function</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>, <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> <b>by </b><i><a class="ref" href="funct_2.html#T4">FUNCT_2:4</a>, , , , </i>;<br/>
<a NAME="E11:27"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">h</font> is <a href="funct_2.html#V2">onto</a>
 
<b>by </b><i><a class="ref" href="funct_2.html#D3">FUNCT_2:def 3</a>, , </i>;<br/>
<a NAME="E12:27"/>
<font color="Maroon">h</font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="matrix_9.html#E1"><i><font color="Green">Lemma27</font></i></a>, <a class="ref" href="finseq_3.html#T98">FINSEQ_3:98</a></i>;<br/>
<a NAME="E13:27"/><b>then </b>
<font color="Maroon">h</font> is <a href="funct_2.html#V3">bijective</a>
 
<b>by </b><i><a class="ref" href="funct_2.html#D4">FUNCT_2:def 4</a>, <a class="ref" href="matrix_9.html#T1" target="_self">Th1</a></i>;<br/>
<b>hence </b><a NAME="E14:27"/>
<font color="Maroon">g</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <a href="matrix_2.html#K12">Permutations</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="matrix_2.html#D11">MATRIX_2:def 11</a></i>;<br/>


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