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<b>let </b><font color="Maroon" title="c1">X</font>, <font color="Maroon" title="c2">Y</font>, <font color="Maroon" title="c3">Z</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>;<br/><b>let </b><font color="Maroon" title="c4">f</font> be  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c2">Y</font>,<font color="Maroon" title="c3">Z</font>;<br/>



<b>set </b><font color="Maroon" title="c5">fX</font> =  <a href="waybel26.html#K4" title="WAYBEL26:func.4">oContMaps</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c1">X</font>;<br/>
<b>let </b><font color="Maroon" title="c6">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">Y</font>;<br/><b>let </b><font color="Maroon" title="c7">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> <font color="Maroon" title="c3">Z</font>,<font color="Maroon" title="c1">X</font></span>)</span>;<br/>



<b>thus </b><a NAME="E1:23"/>
 <a href="waybel26.html#K2" title="WAYBEL26:func.2">pi</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel26.html#K4" title="WAYBEL26:func.4">oContMaps</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c7">A</font></span>)</span>,<font color="Maroon" title="c6">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="waybel26.html#K2" title="WAYBEL26:func.2">pi</a> <font color="Maroon" title="c7">A</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x</font></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:23_1"/>
<font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel26.html#K2" title="WAYBEL26:func.2">pi</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel26.html#K4" title="WAYBEL26:func.4">oContMaps</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c7">A</font></span>)</span>,<font color="Maroon" title="c6">x</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c9">h</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E3:23_1"/><i><font color="Green" title="E16">A1</font></i>: 
( <font color="Maroon" title="c9">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="waybel26.html#K4" title="WAYBEL26:func.4">oContMaps</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c7">A</font> &amp; <font color="Maroon" title="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> )
 <b>by </b><i><a class="ref" href="card_3.html#D6" title="CARD_3:def.6">CARD_3:def 6</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c10">g</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:23_1"/><i><font color="Green" title="E17">A2</font></i>: 
( <font color="Maroon" title="c10">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> <font color="Maroon" title="c3">Z</font>,<font color="Maroon" title="c1">X</font></span>)</span> &amp; <font color="Maroon" title="c10">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">A</font> &amp; <font color="Maroon" title="c9">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="waybel26.html#K4" title="WAYBEL26:func.4">oContMaps</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">g</font> )
 <b>by </b><i><a class="txt" href="waybel26.html#E3:23_1"><i><font color="Green" title="E16">A1</font></i></a>, <a class="ref" href="funct_2.html#T115" title="FUNCT_2:th.115">FUNCT_2:115</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">g</font> = <font color="Maroon" title="c10">g</font> as  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c3">Z</font>,<font color="Maroon" title="c1">X</font> <b>by </b><i><a class="txt" href="waybel26.html#E5:23_1"><i><font color="Green" title="E17">A2</font></i></a>, <a class="ref" href="waybel26.html#T2" target="_self" title="WAYBEL26:th.2">Th2</a></i>;<br/>
<a NAME="E7:23_1"/>
<font color="Maroon" title="c9">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</font>
 
<b>by </b><i><a class="txt" href="waybel26.html#E5:23_1"><i><font color="Green" title="E17">A2</font></i></a>, <a class="ref" href="waybel26.html#D3" target="_self" title="WAYBEL26:def.3">Def3</a></i>;<br/>
<a NAME="E8:23_1"/><b>then </b>
<font color="Maroon" title="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x</font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel26.html#E3:23_1"><i><font color="Green" title="E16">A1</font></i></a>, <a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>;<br/>
<b>hence </b><a NAME="E9:23_1"/>
<font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel26.html#K2" title="WAYBEL26:func.2">pi</a> <font color="Maroon" title="c7">A</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x</font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel26.html#E5:23_1"><i><font color="Green" title="E17">A2</font></i></a>, <a class="ref" href="card_3.html#D6" title="CARD_3:def.6">CARD_3:def 6</a></i>;<br/>


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

<b>let </b><font color="Maroon" title="c8">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:23"/>
<font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel26.html#K2" title="WAYBEL26:func.2">pi</a> <font color="Maroon" title="c7">A</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c9">g</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E4:23"/><i><font color="Green" title="E16">A3</font></i>: 
( <font color="Maroon" title="c9">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">A</font> &amp; <font color="Maroon" title="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x</font></span>)</span> )
 <b>by </b><i><a class="ref" href="card_3.html#D6" title="CARD_3:def.6">CARD_3:def 6</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">g</font> = <font color="Maroon" title="c9">g</font> as  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c3">Z</font>,<font color="Maroon" title="c1">X</font> <b>by </b><i><a class="txt" href="waybel26.html#E4:23"><i><font color="Green" title="E16">A3</font></i></a>, <a class="ref" href="waybel26.html#T2" target="_self" title="WAYBEL26:th.2">Th2</a></i>;<br/>
<a NAME="E6:23"/><i><font color="Green" title="E17">A4</font></i>: 
<font color="Maroon" title="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x</font>
 
<b>by </b><i><a class="txt" href="waybel26.html#E4:23"><i><font color="Green" title="E16">A3</font></i></a>, <a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>;<br/>
<a NAME="E7:23"/>
( <font color="Maroon" title="c10">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="waybel26.html#K4" title="WAYBEL26:func.4">oContMaps</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">g</font> &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> <font color="Maroon" title="c2">Y</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
<b>by </b><i><a class="ref" href="waybel26.html#D3" target="_self" title="WAYBEL26:def.3">Def3</a></i>;<br/>
<a NAME="E8:23"/><b>then </b>
<font color="Maroon" title="c10">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="waybel26.html#K4" title="WAYBEL26:func.4">oContMaps</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c7">A</font>
 
<b>by </b><i><a class="txt" href="waybel26.html#E4:23"><i><font color="Green" title="E16">A3</font></i></a>, <a class="ref" href="funct_2.html#T43" title="FUNCT_2:th.43">FUNCT_2:43</a></i>;<br/>
<b>hence </b><a NAME="E9:23"/>
<font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel26.html#K2" title="WAYBEL26:func.2">pi</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel26.html#K4" title="WAYBEL26:func.4">oContMaps</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c7">A</font></span>)</span>,<font color="Maroon" title="c6">x</font>
 <b>by </b><i><a class="txt" href="waybel26.html#E6:23"><i><font color="Green" title="E17">A4</font></i></a>, <a class="ref" href="card_3.html#D6" title="CARD_3:def.6">CARD_3:def 6</a></i>;<br/>


</div>
