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<b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">P</font> be    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">X</font>, <font color="Maroon">Q</font> be    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">P</font>,<font color="Maroon">Q</font>;<br/>





<b>assume </b><a NAME="E1:15"/><i><font color="Green">E29</font></i>: 
for <font color="Olive">a</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> holds <br/><font color="Olive">a</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">a</font>
 ;<br/>

<b>let </b><font color="Maroon">p</font> be    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">P</font>;<br/><b>let </b><font color="Maroon">q</font> be    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">Q</font>;<br/>



<a NAME="E2:15"/><i><font color="Green">E36</font></i>: 
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">p</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">q</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Q</font> )
 
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:15_1"/>
<font color="Maroon">P</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><b>then reconsider </b><font color="Maroon">X</font> = <font color="Maroon">X</font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="ref" href="eqrel_1.html#D6">EQREL_1:def 6</a></i>;<br/><a NAME="E3:15_1"/>
<font color="Maroon">Q</font> is    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">X</font>
 ;<br/><b>hence </b><a NAME="E4:15_1"/>
<font color="Maroon">Q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/></div><b>end;</b></div>
<a NAME="E4:15"/><b>then </b>
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Q</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E5:15"/><b>then </b><i><font color="Green">E37</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font>
 
<b>by </b><i>, <a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:15_2"/>
 <a href="card_3.html#K4">product</a> <font color="Maroon">p</font> <a href="tarski.html#R1">c=</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">q</font>
 ;<br/><a NAME="E2:15_2"/><b>then </b><i><font color="Green">E45</font></i>: 
(  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">q</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">p</font> holds <br/><font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> ) )
 <b>by </b><i/>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_2_1"><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="E1:15_2_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font>
 ;<br/><a NAME="E2:15_2_1"/><b>then </b><i><font color="Green">E58</font></i>: 
( <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> &amp; <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">p</font> &amp; <font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">q</font> )
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>then reconsider </b><font color="Maroon">Y</font> = <font color="Maroon">X</font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="ref" href="eqrel_1.html#D6">EQREL_1:def 6</a></i>;<br/><b>reconsider </b><font color="Maroon">P'</font> = <font color="Maroon">P</font>, <font color="Maroon">Q'</font> = <font color="Maroon">Q</font> as    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">Y</font> ;<br/><b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">P'</font> <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">z</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">a</font>;<br/><a NAME="E7:15_2_1"/>
( <font color="Maroon">a</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> &amp; <font color="Maroon">f</font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">P'</font>,<font color="Maroon">Q'</font> )
 <b>by </b><i/>;<br/><a NAME="E8:15_2_1"/><b>then </b>
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q'</font> )
 <b>by </b><i>, <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/><a NAME="E9:15_2_1"/><b>then </b>
<font color="Maroon">q</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">a</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E10:15_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">p</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:15_2"/>
<font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E7:15"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_3"><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="E1:15_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font>
 ;<br/><a NAME="E2:15_3"/><b>then </b>
( <font color="Maroon">q</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> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span> &amp; <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">p</font> )
 <b>by </b><i>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E3:15_3"/>
<font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E9:15"/>
 <a href="card_3.html#K4">product</a> <font color="Maroon">p</font> <a href="tarski.html#R1">c=</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">q</font>
 <b>by </b><i>, , <a class="ref" href="card_3.html#T38">CARD_3:38</a></i>;<br/>


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