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<b>let </b><font color="Maroon">i</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">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">f</font> be    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">D</font>;<br/>





<b>assume </b><a NAME="E1:42"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</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">m</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E1:42_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">i</font></span>)</span>
 <b>and </b><br/><a NAME="E2:42_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">i</font></span>)</span>
 <b>and </b><br/><a NAME="E3:42_1"/><i><font color="Green">E32</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">i</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">i</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">m</font>
 ;<br/><a NAME="E4:42_1"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font> )
 <b>by </b><i><a class="ref" href="finseq_5.html#T3" target="_self">Th3</a>, , </i>;<br/><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">i</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font>
<b>by </b><i><a class="ref" href="finseq_5.html#T3" target="_self">Th3</a>, </i>
<br/>.= 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font></span>)</span>
<b>by </b><i>, , </i>
;<br/><a NAME="E6:42_1"/><b>then </b>
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="txt" href="finseq_5.html#E2:42_1"><i><font color="Green">E25</font></i></a>, , <a class="ref" href="partfun2.html#T17">PARTFUN2:17</a></i>;<br/><b>hence </b><a NAME="E7:42_1"/>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:42"/>
<font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">i</font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i><a class="ref" href="partfun2.html#T16">PARTFUN2:16</a></i>;<br/>


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