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<b>let </b><font color="Maroon">S</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">D1</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> , <font color="Maroon">D2</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">f1</font> be   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">S</font>,<font color="Maroon">D1</font>;<br/><b>let </b><font color="Maroon">f2</font> be   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">D1</font>,<font color="Maroon">D2</font>;<br/>







<b>assume </b><a NAME="E1:2"/>
( <font color="Maroon">f1</font> is <a href="funct_2.html#V3">bijective</a> &amp; <font color="Maroon">f2</font> is <a href="funct_2.html#V3">bijective</a> )
 ;<br/>

<a NAME="E2:2"/><b>then </b><i><font color="Green">E22</font></i>: 
( <font color="Maroon">f1</font> is <a href="funct_1.html#V2">one-to-one</a> &amp; <font color="Maroon">f1</font> is <a href="funct_2.html#V2">onto</a> &amp; <font color="Maroon">f2</font> is <a href="funct_1.html#V2">one-to-one</a> &amp; <font color="Maroon">f2</font> is <a href="funct_2.html#V2">onto</a> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D4">FUNCT_2:def 4</a></i>;<br/>
<b>set </b><font color="Maroon">f3</font> = <font color="Maroon">f2</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f1</font>;<br/>
<a NAME="E3:2"/>
(  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D2</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D1</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D1</font> )
 
<b>by </b><i>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="funct_2.html#D3">FUNCT_2:def 3</a></i>;<br/>
<a NAME="E4:2"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D2</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T47">RELAT_1:47</a></i>;<br/>
<a NAME="E5:2"/><b>then </b>
( <font color="Maroon">f2</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f1</font> is <a href="funct_1.html#V2">one-to-one</a> &amp; <font color="Maroon">f2</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f1</font> is <a href="funct_2.html#V2">onto</a> )
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T46">FUNCT_1:46</a>, <a class="ref" href="funct_2.html#D3">FUNCT_2:def 3</a></i>;<br/>
<b>hence </b><a NAME="E6:2"/>
<font color="Maroon">f2</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f1</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></i>;<br/>


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