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<a NAME="E1:13_1_1"/><i><font color="Green">E35</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>thus </b><a NAME="E2:13_1_1"/>
( <font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a> implies for <font color="Olive">a1</font>, <font color="Olive">a2</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font><br/> for <font color="Olive">b1</font>, <font color="Olive">b2</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">B</font>  st <font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Olive">a1</font>,<font color="Olive">b1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Olive">a2</font>,<font color="Olive">b2</font> holds <br/>( <font color="Olive">a1</font> <a href="hidden.html#R1">=</a> <font color="Olive">a2</font> &amp; <font color="Olive">b1</font> <a href="hidden.html#R1">=</a> <font color="Olive">b2</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:13_1_1_1"/><i><font color="Green">E36</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font> holds <br/><font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">y</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a><br/>

<b>let </b><font color="Maroon">a1</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font>, <font color="Maroon">a2</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font>;<br/><b>let </b><font color="Maroon">b1</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">B</font>, <font color="Maroon">b2</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">B</font>;<br/>



<a NAME="E2:13_1_1_1"/>
( <font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a1</font>,<font color="Maroon">b1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a2</font>,<font color="Maroon">b2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="tarski.html#K4">]</a></span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="zfmisc_1.html#K2">:]</a></span> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<a NAME="E3:13_1_1_1"/><b>then </b>
( <font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a1</font>,<font color="Maroon">b1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a2</font>,<font color="Maroon">b2</font> implies <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="tarski.html#K4">]</a></span> )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E4:13_1_1_1"/>
( <font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a1</font>,<font color="Maroon">b1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a2</font>,<font color="Maroon">b2</font> implies ( <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a2</font> &amp; <font color="Maroon">b1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b2</font> ) )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E3:13_1_1"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">a1</font>, <font color="Olive">a2</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font><br/> for <font color="Olive">b1</font>, <font color="Olive">b2</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">B</font>  st <font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Olive">a1</font>,<font color="Olive">b1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Olive">a2</font>,<font color="Olive">b2</font> holds <br/>( <font color="Olive">a1</font> <a href="hidden.html#R1">=</a> <font color="Olive">a2</font> &amp; <font color="Olive">b1</font> <a href="hidden.html#R1">=</a> <font color="Olive">b2</font> )
 ;<br/>

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a><br/>

<b>assume </b><a NAME="E4:13_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 ;<br/>

<b>then consider </b><font color="Maroon">a1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">b1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:13_1_1"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">a1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">b1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<b>assume </b><a NAME="E7:13_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 ;<br/>

<b>then consider </b><font color="Maroon">a2</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">b2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E9:13_1_1"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">a2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">b2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">a1</font> = <font color="Maroon">a1</font>, <font color="Maroon">a2</font> = <font color="Maroon">a2</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> <b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">b1</font> = <font color="Maroon">b1</font>, <font color="Maroon">b2</font> = <font color="Maroon">b2</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">B</font> <b>by </b><i>, </i>;<br/>
<b>assume </b><a NAME="E12:13_1_1"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
 ;<br/>

<a NAME="E13:13_1_1"/><b>then </b>
<font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a1</font>,<font color="Maroon">b1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a2</font>,<font color="Maroon">b2</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E14:13_1_1"/><b>then </b>
( <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a2</font> &amp; <font color="Maroon">b1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b2</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E15:13_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 <b>by </b><i>, </i>;<br/>


</div>
