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<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <font color="Maroon">f</font> <a href="zf_fund1.html#K1">(#)</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<b>consider </b><font color="Maroon">F</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E2:15_2_2"/><i><font color="Green">E68</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel26.html#K1" target="_self">oContMaps</a> <font color="Maroon">Z</font>,<font color="Maroon">X</font></span>)</span>
 <b>and </b><br/><a NAME="E3:15_2_2"/><i><font color="Green">E70</font></i>: 
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> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel26.html#K1" target="_self">oContMaps</a> <font color="Maroon">Z</font>,<font color="Maroon">X</font></span>)</span> holds <br/><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">H<sub>1</sub></font>(<font color="Olive">x</font>)
 <b>from </b><i><a class="ref" href="funct_1.html#S3">FUNCT_1:sch 3</a>();<br/></i>
<a NAME="E4:15_2_2"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel26.html#K1" target="_self">oContMaps</a> <font color="Maroon">Y</font>,<font color="Maroon">X</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_2_2_1"><b>proof </b></a><div class="add">

<b>let </b><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="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:15_2_2_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:15_2_2_1"/><i><font color="Green">E73</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">g</font> = <font color="Maroon">x</font> as  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">Z</font>,<font color="Maroon">X</font> <b>by </b><i>, , </i>;<br/>
<a NAME="E5:15_2_2_1"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">g</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i><a class="ref" href="tops_2.html#T58">TOPS_2:58</a></i>;<br/>
<font color="Maroon">y</font> = 
<font color="Maroon">f</font> <a href="zf_fund1.html#K1">(#)</a> <font color="Maroon">g</font>
<b>by </b><i>, , </i>
<br/>.= 
<font color="Maroon">g</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font>
<b>by </b><i><a class="ref" href="yellow16.html#T1">YELLOW16:1</a></i>
;<br/>
<a NAME="E7:15_2_2_1"/><b>then </b>
<font color="Maroon">y</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel26.html#K1" target="_self">oContMaps</a> <font color="Maroon">Y</font>,<font color="Maroon">X</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E8:15_2_2_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel26.html#K1" target="_self">oContMaps</a> <font color="Maroon">Y</font>,<font color="Maroon">X</font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">F</font> = <font color="Maroon">F</font> as   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="waybel26.html#K1" target="_self">oContMaps</a> <font color="Maroon">Z</font>,<font color="Maroon">X</font></span>)</span>,<span class="p1">(<span class="default"><a href="waybel26.html#K1" target="_self">oContMaps</a> <font color="Maroon">Y</font>,<font color="Maroon">X</font></span>)</span> <b>by </b><i>, <a class="ref" href="funct_2.html#T4">FUNCT_2:4</a></i>;<br/>
<b>take </b>
<font color="Maroon">F</font>
;<br/>

<b>let </b><font color="Maroon">g</font> be  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">Z</font>,<font color="Maroon">X</font>;<br/>

<a NAME="E6:15_2_2"/>
<font color="Maroon">g</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel26.html#K1" target="_self">oContMaps</a> <font color="Maroon">Z</font>,<font color="Maroon">X</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">g</font> = 
<font color="Maroon">f</font> <a href="zf_fund1.html#K1">(#)</a> <font color="Maroon">g</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">g</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font>
<b>by </b><i><a class="ref" href="yellow16.html#T1">YELLOW16:1</a></i>
;<br/><br/>


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