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<b>let </b><font color="Maroon" title="c1">X</font>, <font color="Maroon" title="c2">Y</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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">f</font> being <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="c2">Y</font>  st <font color="Olive" title="b1">f</font> is <a href="quantal1.html#V11" title="QUANTAL1:attr.11">idempotent</a> holds <br/> <a href="waybel26.html#K3" title="WAYBEL26:func.3">oContMaps</a> <font color="Maroon" title="c1">X</font>,<font color="Olive" title="b1">f</font> is <a href="quantal1.html#V11" title="QUANTAL1:attr.11">idempotent</a></span><br/><b>let </b><font color="Maroon" title="c3">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="c2">Y</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">f</font> is <a href="quantal1.html#V11" title="QUANTAL1:attr.11">idempotent</a> implies  <a href="waybel26.html#K3" title="WAYBEL26:func.3">oContMaps</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c3">f</font> is <a href="quantal1.html#V11" title="QUANTAL1:attr.11">idempotent</a> )</span><br/>



<b>assume </b><a NAME="E1:17"/><i><font color="Green" title="E11">A1</font></i>: 
<font color="Maroon" title="c3">f</font> is <a href="quantal1.html#V11" title="QUANTAL1:attr.11">idempotent</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="waybel26.html#K3" title="WAYBEL26:func.3">oContMaps</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c3">f</font> is <a href="quantal1.html#V11" title="QUANTAL1:attr.11">idempotent</a></span><br/>

<b>set </b><font color="Maroon" title="c4">Xf</font> =  <a href="waybel26.html#K3" title="WAYBEL26:func.3">oContMaps</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c3">f</font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">g</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel26.html#K3" title="WAYBEL26:func.3">oContMaps</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c3">f</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="waybel26.html#K3" title="WAYBEL26:func.3">oContMaps</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="waybel26.html#K3" title="WAYBEL26:func.3">oContMaps</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">g</font></span><br/><b>reconsider </b><font color="Maroon" title="c6">h</font> = <font color="Maroon" title="c5">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="c1">X</font>,<font color="Maroon" title="c2">Y</font> <b>by </b><i><a class="ref" href="waybel26.html#T2" target="_self" title="WAYBEL26:th.2">Th2</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel26.html#K3" title="WAYBEL26:func.3">oContMaps</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c3">f</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="waybel26.html#K3" title="WAYBEL26:func.3">oContMaps</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">g</font> = 
<span class="p1">(<span class="default"><a href="waybel26.html#K3" title="WAYBEL26:func.3">oContMaps</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel26.html#K3" title="WAYBEL26:func.3">oContMaps</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">g</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="waybel26.html#K3" title="WAYBEL26:func.3">oContMaps</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c6">h</font></span>)</span>
<b>by </b><i><a class="ref" href="waybel26.html#D2" target="_self" title="WAYBEL26:def.2">Def2</a></i>
<br/>.= 
<font color="Maroon" title="c3">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c6">h</font></span>)</span>
<b>by </b><i><a class="ref" href="waybel26.html#D2" target="_self" title="WAYBEL26:def.2">Def2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c6">h</font>
<b>by </b><i><a class="ref" href="relat_1.html#T55" title="RELAT_1:th.55">RELAT_1:55</a></i>
<br/>.= 
<font color="Maroon" title="c3">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c6">h</font>
<b>by </b><i><a class="txt" href="#E1:17"><i><font color="Green" title="E11">A1</font></i></a>, <a class="ref" href="quantal1.html#D9" title="QUANTAL1:def.9">QUANTAL1:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="waybel26.html#K3" title="WAYBEL26:func.3">oContMaps</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">g</font>
<b>by </b><i><a class="ref" href="waybel26.html#D2" target="_self" title="WAYBEL26:def.2">Def2</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E3:17"/><b>then </b>
<span class="p1">(<span class="default"><a href="waybel26.html#K3" title="WAYBEL26:func.3">oContMaps</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c3">f</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="waybel26.html#K3" title="WAYBEL26:func.3">oContMaps</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="waybel26.html#K3" title="WAYBEL26:func.3">oContMaps</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c3">f</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></i>;<br/>
<b>hence </b><a NAME="E4:17"/>
 <a href="waybel26.html#K3" title="WAYBEL26:func.3">oContMaps</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c3">f</font> is <a href="quantal1.html#V11" title="QUANTAL1:attr.11">idempotent</a>
 <b>by </b><i><a class="ref" href="quantal1.html#D9" title="QUANTAL1:def.9">QUANTAL1:def 9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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