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<b>let </b><font color="Maroon" title="c2">S1</font>, <font color="Maroon" title="c3">S2</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a> <a href="yellow_0.html#V4" title="YELLOW_0:attr.4">full</a>  <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</a> of  <a href="yellow_1.html#K7" title="YELLOW_1:func.7">MonMaps</a> <font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c1">L</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:24_1_2"/><i><font color="Green" title="E10">A6</font></i>: 
for <font color="Olive" title="b1">f</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c1">L</font> holds <br/> ( <font color="Olive" title="b1">f</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">S1</font> iff <font color="Olive" title="b1">f</font> is <a href="waybel_1.html#V7" title="WAYBEL_1:attr.7">closure</a> )
 <b>and </b><br/><a NAME="E2:24_1_2"/><i><font color="Green" title="E11">A7</font></i>: 
for <font color="Olive" title="b1">f</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c1">L</font> holds <br/> ( <font color="Olive" title="b1">f</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">S2</font> iff <font color="Olive" title="b1">f</font> is <a href="waybel_1.html#V7" title="WAYBEL_1:attr.7">closure</a> )
 ;<br/>

<a NAME="E3:24_1_2"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">S2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_2_1"><b>proof </b></a><div class="add">

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a>,<a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a>
<div class="add"><b>let </b><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:24_1_2_1_1"/>
<font color="Maroon" title="c4">x</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 <font color="Maroon" title="c2">S1</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c5">y</font> = <font color="Maroon" title="c4">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">S1</font> ;<br/><a NAME="E3:24_1_2_1_1"/>
<font color="Maroon" title="c5">y</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K7" title="YELLOW_1:func.7">MonMaps</a> <font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c1">L</font></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_0.html#T59" title="YELLOW_0:th.59">YELLOW_0:59</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c6">y</font> = <font color="Maroon" title="c5">y</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c1">L</font> <b>by </b><i><a class="ref" href="waybel10.html#T10" target="_self" title="WAYBEL10:th.10">Th10</a></i>;<br/><a NAME="E5:24_1_2_1_1"/>
<font color="Maroon" title="c6">y</font> is <a href="waybel_1.html#V7" title="WAYBEL_1:attr.7">closure</a>
 <b>by </b><i><a class="txt" href="waybel10.html#E1:24_1_2"><i><font color="Green" title="E10">A6</font></i></a></i>;<br/><a NAME="E6:24_1_2_1_1"/><b>then </b>
<font color="Maroon" title="c6">y</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">S2</font>
 <b>by </b><i><a class="txt" href="waybel10.html#E2:24_1_2"><i><font color="Green" title="E11">A7</font></i></a></i>;<br/><b>hence </b><a NAME="E7:24_1_2_1_1"/>
<font color="Maroon" title="c4">x</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 <font color="Maroon" title="c3">S2</font>
 ;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon" title="c4">x</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="E2:24_1_2_1"/>
<font color="Maroon" title="c4">x</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 <font color="Maroon" title="c3">S2</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c5">y</font> = <font color="Maroon" title="c4">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">S2</font> ;<br/>
<a NAME="E4:24_1_2_1"/>
<font color="Maroon" title="c5">y</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K7" title="YELLOW_1:func.7">MonMaps</a> <font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c1">L</font></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T59" title="YELLOW_0:th.59">YELLOW_0:59</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c6">y</font> = <font color="Maroon" title="c5">y</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c1">L</font> <b>by </b><i><a class="ref" href="waybel10.html#T10" target="_self" title="WAYBEL10:th.10">Th10</a></i>;<br/>
<a NAME="E6:24_1_2_1"/>
<font color="Maroon" title="c6">y</font> is <a href="waybel_1.html#V7" title="WAYBEL_1:attr.7">closure</a>
 
<b>by </b><i><a class="txt" href="waybel10.html#E2:24_1_2"><i><font color="Green" title="E11">A7</font></i></a></i>;<br/>
<a NAME="E7:24_1_2_1"/><b>then </b>
<font color="Maroon" title="c6">y</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">S1</font>
 
<b>by </b><i><a class="txt" href="waybel10.html#E1:24_1_2"><i><font color="Green" title="E10">A6</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:24_1_2_1"/>
<font color="Maroon" title="c4">x</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 <font color="Maroon" title="c2">S1</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:24_1_2"/>
<font color="Maroon" title="c2">S1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">S2</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T58" title="YELLOW_0:th.58">YELLOW_0:58</a></i>;<br/>


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