<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/><span class="kw">let </span><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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>



<span class="kw">assume </span><a NAME="E1:17"/><span class="lab"><font color="Green" title="E11">A1</font></span>: 
<font color="Maroon" title="c3">f</font> is  <a href="quantal1.html#V11" title="QUANTAL1:attr.11">idempotent</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<span class="kw">set </span><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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="17_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">g</font> being   <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> holds  <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="Olive" title="b1">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="Olive" title="b1">g</font></span></a><div class="add"><span class="kw">let </span><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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/><span class="kw">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel26.html#T2" target="_self" onmouseover="rs('waybel26/T2')" onmouseout="rh()">Th2</a></span>;<br/><span class="kw">thus </span><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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T15" onmouseover="rs('funct_2/T15')" onmouseout="rh()">FUNCT_2:15</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel26.html#D2" target="_self" onmouseover="rs('waybel26/D2')" onmouseout="rh()">Def2</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel26.html#D2" target="_self" onmouseover="rs('waybel26/D2')" onmouseout="rh()">Def2</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T36" onmouseover="rs('relat_1/T36')" onmouseout="rh()">RELAT_1:36</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:17"><span class="lab"><font color="Green" title="E11">A1</font></span></a>, <a class="ref" href="quantal1.html#D9" onmouseover="rs('quantal1/D9')" onmouseout="rh()">QUANTAL1:def 9</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel26.html#D2" target="_self" onmouseover="rs('waybel26/D2')" onmouseout="rh()">Def2</a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E3:17"/><span class="kw">then </span>
<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>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T63" onmouseover="rs('funct_2/T63')" onmouseout="rh()">FUNCT_2:63</a></span>;<br/>
<span class="kw">hence </span><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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="quantal1.html#D9" onmouseover="rs('quantal1/D9')" onmouseout="rh()">QUANTAL1:def 9</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
