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

<b>set </b><font color="Maroon" title="c1">S</font> =  <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> ;<br/>
<b>set </b><font color="Maroon" title="c2">B</font> =  <a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1;<br/>
<b>let </b><font color="Maroon" title="c3">T</font> be  <a href="waybel18.html#V2" title="WAYBEL18:attr.2">injective</a> <a href="t_0topsp.html#NM1" title="T_0TOPSP:NM.1">T_0-TopSpace</a>;<br/>

<b>consider </b><font color="Maroon" title="c4">M</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E2:37"/><i><font color="Green" title="E20">A1</font></i>: 
<font color="Maroon" title="c3">T</font> <a href="waybel18.html#R1" title="WAYBEL18:pred.1">is_Retract_of</a>  <a href="waybel18.html#K3" title="WAYBEL18:func.3">product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span>
 <b>by </b><i><a class="ref" href="waybel18.html#T20" title="WAYBEL18:th.20">WAYBEL18:20</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c5">f</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="waybel18.html#K3" title="WAYBEL18:func.3">product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="waybel18.html#K3" title="WAYBEL18:func.3">product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span><b> such that </b><br/><a NAME="E4:37"/><i><font color="Green" title="E21">A2</font></i>: 
<font color="Maroon" title="c5">f</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>and </b><br/><a NAME="E5:37"/><i><font color="Green" title="E22">A3</font></i>: 
<font color="Maroon" title="c5">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">f</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c5">f</font>
 <b>and </b><br/><a NAME="E6:37"/><i><font color="Green" title="E23">A4</font></i>: 
 <a href="waybel18.html#K7" title="WAYBEL18:func.7">Image</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c3">T</font> <a href="borsuk_3.html#R2" title="BORSUK_3:pred.2">are_homeomorphic</a> 
 <b>by </b><i><a class="txt" href="waybel25.html#E2:37"><i><font color="Green" title="E20">A1</font></i></a>, <a class="ref" href="waybel18.html#D8" title="WAYBEL18:def.8">WAYBEL18:def 8</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">TL</font> being  <a href="waybel11.html#V4" title="WAYBEL11:attr.4">Scott</a>  <a href="yellow_9.html#M1" title="YELLOW_9:mode.1">TopAugmentation</a> of  <a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1</span>)</span></span>)</span>;<br/>
<a NAME="E8:37"/><i><font color="Green" title="E24">A5</font></i>: 
the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c6">TL</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="waybel18.html#K3" title="WAYBEL18:func.3">product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="waybel18.html#T16" title="WAYBEL18:th.16">WAYBEL18:16</a></i>;<br/>
<a NAME="E9:37"/><i><font color="Green" title="E25">A6</font></i>: 
 <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">TL</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c6">TL</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1</span>)</span></span>)</span></span>)</span>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1</span>)</span></span>)</span></span>)</span> #)
 
<b>by </b><i><a class="ref" href="yellow_9.html#D4" title="YELLOW_9:def.4">YELLOW_9:def 4</a></i>;<br/>
<a NAME="E10:37"/><i><font color="Green" title="E26">A7</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">TL</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 <span class="p1">(<span class="default"><a href="waybel18.html#K3" title="WAYBEL18:func.3">product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="waybel25.html#T3" target="_self" title="WAYBEL25:th.3">Th3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c7">ff</font> = <font color="Maroon" title="c5">f</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c6">TL</font>,<font color="Maroon" title="c6">TL</font> ;<br/>
<a NAME="E12:37"/>
(  <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">TL</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c6">TL</font> #),the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">TL</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c6">TL</font> #) #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">TL</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c6">TL</font> #) implies  <a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">TL</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c6">TL</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c6">TL</font> )
 
<b>by </b><i><a class="ref" href="waybel25.html#T13" target="_self" title="WAYBEL25:th.13">Th13</a></i>;<br/>
<a NAME="E13:37"/><b>then </b><i><font color="Green" title="E27">A8</font></i>: 
 <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel18.html#K3" title="WAYBEL18:func.3">product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="waybel18.html#K3" title="WAYBEL18:func.3">product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> #)</span>)</span>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel18.html#K3" title="WAYBEL18:func.3">product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="waybel18.html#K3" title="WAYBEL18:func.3">product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> #)</span>)</span> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1</span>)</span></span>)</span></span>)</span>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1</span>)</span></span>)</span></span>)</span> #)
 
<b>by </b><i><a class="txt" href="waybel25.html#E8:37"><i><font color="Green" title="E24">A5</font></i></a>, <a class="txt" href="waybel25.html#E10:37"><i><font color="Green" title="E26">A7</font></i></a>, <a class="ref" href="waybel25.html#T16" target="_self" title="WAYBEL25:th.16">Th16</a></i>;<br/>
<a NAME="E14:37"/>
<font color="Maroon" title="c7">ff</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 
<b>by </b><i><a class="txt" href="waybel25.html#E4:37"><i><font color="Green" title="E21">A2</font></i></a>, <a class="txt" href="waybel25.html#E8:37"><i><font color="Green" title="E24">A5</font></i></a>, <a class="txt" href="waybel25.html#E10:37"><i><font color="Green" title="E26">A7</font></i></a>, <a class="ref" href="yellow12.html#T36" title="YELLOW12:th.36">YELLOW12:36</a></i>;<br/>
<a NAME="E15:37"/><b>then </b><i><font color="Green" title="E28">A9</font></i>: 
<font color="Maroon" title="c7">ff</font> is <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a>
 
<b>by </b><i><a class="ref" href="waybel17.html#T22" title="WAYBEL17:th.22">WAYBEL17:22</a></i>;<br/>
<a NAME="E16:37"/>
<font color="Maroon" title="c7">ff</font> is <a href="waybel_1.html#V6" title="WAYBEL_1:attr.6">projection</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:37_1"/>
( <font color="Maroon" title="c7">ff</font> is <a href="quantal1.html#V11" title="QUANTAL1:attr.11">idempotent</a> &amp; <font color="Maroon" title="c7">ff</font> is <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a> )
 <b>by </b><i><a class="txt" href="waybel25.html#E5:37"><i><font color="Green" title="E22">A3</font></i></a>, <a class="txt" href="waybel25.html#E15:37"><i><font color="Green" title="E28">A9</font></i></a>, <a class="ref" href="quantal1.html#D9" title="QUANTAL1:def.9">QUANTAL1:def 9</a>, <a class="ref" href="waybel17.html#T3" title="WAYBEL17:th.3">WAYBEL17:3</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_1.html#D13" title="WAYBEL_1:def.13">WAYBEL_1:def 13</a><br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c8">g</font> = <font color="Maroon" title="c7">ff</font> as  <a href="waybel_1.html#V6" title="WAYBEL_1:attr.6">projection</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1</span>)</span></span>)</span></span>)</span> <b>by </b><i><a class="txt" href="waybel25.html#E9:37"><i><font color="Green" title="E25">A6</font></i></a>, <a class="txt" href="waybel25.html#E24"><i><font color="Green" title="E18">Lm3</font></i></a></i>;<br/>
<a NAME="E18:37"/>
<font color="Maroon" title="c8">g</font> is <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a>
 
<b>by </b><i><a class="txt" href="waybel25.html#E9:37"><i><font color="Green" title="E25">A6</font></i></a>, <a class="txt" href="waybel25.html#E15:37"><i><font color="Green" title="E28">A9</font></i></a>, <a class="ref" href="waybel21.html#T6" title="WAYBEL21:th.6">WAYBEL21:6</a></i>;<br/>
<a NAME="E19:37"/><b>then </b><i><font color="Green" title="E29">A10</font></i>: 
 <a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c8">g</font> is  <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a> <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a> <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a>
 
<b>by </b><i><a class="ref" href="waybel15.html#T17" title="WAYBEL15:th.17">WAYBEL15:17</a>, <a class="ref" href="yellow_2.html#T37" title="YELLOW_2:th.37">YELLOW_2:37</a></i>;<br/>
<a NAME="E20:37"/><i><font color="Green" title="E30">A11</font></i>: 
 <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <span class="p2">(<span class="default"><a href="waybel18.html#K7" title="WAYBEL18:func.7">Image</a> <font color="Maroon" title="c5">f</font></span>)</span></span>)</span>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <span class="p2">(<span class="default"><a href="waybel18.html#K7" title="WAYBEL18:func.7">Image</a> <font color="Maroon" title="c5">f</font></span>)</span></span>)</span> #), <a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <span class="p1">(<span class="default"><a href="waybel18.html#K7" title="WAYBEL18:func.7">Image</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="waybel_1.html#R5" title="WAYBEL_1:pred.5">are_isomorphic</a> 
 
<b>by </b><i><a class="ref" href="waybel13.html#T26" title="WAYBEL13:th.26">WAYBEL13:26</a></i>;<br/>
<a NAME="E21:37"/>
 <a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <span class="p1">(<span class="default"><a href="waybel18.html#K7" title="WAYBEL18:func.7">Image</a> <font color="Maroon" title="c5">f</font></span>)</span>, <a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c3">T</font> <a href="waybel_1.html#R5" title="WAYBEL_1:pred.5">are_isomorphic</a> 
 
<b>by </b><i><a class="txt" href="waybel25.html#E6:37"><i><font color="Green" title="E23">A4</font></i></a>, <a class="ref" href="waybel25.html#T19" target="_self" title="WAYBEL25:th.19">Th19</a></i>;<br/>
<a NAME="E22:37"/><b>then </b><i><font color="Green" title="E31">A12</font></i>: 
 <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <span class="p2">(<span class="default"><a href="waybel18.html#K7" title="WAYBEL18:func.7">Image</a> <font color="Maroon" title="c5">f</font></span>)</span></span>)</span>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <span class="p2">(<span class="default"><a href="waybel18.html#K7" title="WAYBEL18:func.7">Image</a> <font color="Maroon" title="c5">f</font></span>)</span></span>)</span> #), <a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c3">T</font> <a href="waybel_1.html#R5" title="WAYBEL_1:pred.5">are_isomorphic</a> 
 
<b>by </b><i><a class="txt" href="waybel25.html#E20:37"><i><font color="Green" title="E30">A11</font></i></a>, <a class="ref" href="waybel_1.html#T8" title="WAYBEL_1:th.8">WAYBEL_1:8</a></i>;<br/>
<i><font color="Green" title="E32">A13</font></i>: the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c8">g</font></span>)</span> = 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c8">g</font>
<b>by </b><i><a class="ref" href="yellow_0.html#D15" title="YELLOW_0:def.15">YELLOW_0:def 15</a></i>
<br/>.= 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel18.html#K7" title="WAYBEL18:func.7">Image</a> <font color="Maroon" title="c5">f</font></span>)</span>
<b>by </b><i><a class="ref" href="waybel18.html#T10" title="WAYBEL18:th.10">WAYBEL18:10</a></i>
<br/>.= 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <span class="p2">(<span class="default"><a href="waybel18.html#K7" title="WAYBEL18:func.7">Image</a> <font color="Maroon" title="c5">f</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="waybel25.html#E15"><i><font color="Green" title="E10">Lm1</font></i></a></i>
;<br/>
<a NAME="E24:37"/><i><font color="Green" title="E33">A14</font></i>: 
the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c8">g</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1</span>)</span></span>)</span></span>)</span> <a href="toler_1.html#K1" title="TOLER_1:func.1">|_2</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c8">g</font></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_0.html#D14" title="YELLOW_0:def.14">YELLOW_0:def 14</a></i>;<br/>
<a NAME="E25:37"/>
 <a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <span class="p1">(<span class="default"><a href="waybel18.html#K7" title="WAYBEL18:func.7">Image</a> <font color="Maroon" title="c5">f</font></span>)</span> is  <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="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <span class="p1">(<span class="default"><a href="waybel18.html#K3" title="WAYBEL18:func.3">product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="waybel25.html#T17" target="_self" title="WAYBEL25:th.17">Th17</a></i>;<br/>
<a NAME="E26:37"/><b>then </b>
the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <span class="p2">(<span class="default"><a href="waybel18.html#K7" title="WAYBEL18:func.7">Image</a> <font color="Maroon" title="c5">f</font></span>)</span></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <span class="p2">(<span class="default"><a href="waybel18.html#K3" title="WAYBEL18:func.3">product</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span></span>)</span> <a href="toler_1.html#K1" title="TOLER_1:func.1">|_2</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <span class="p2">(<span class="default"><a href="waybel18.html#K7" title="WAYBEL18:func.7">Image</a> <font color="Maroon" title="c5">f</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_0.html#D14" title="YELLOW_0:def.14">YELLOW_0:def 14</a></i>;<br/>
<b>hence </b><a NAME="E27:37"/>
 <a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c3">T</font> is  <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a> <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a> <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a>
 <b>by </b><i><a class="txt" href="waybel25.html#E13:37"><i><font color="Green" title="E27">A8</font></i></a>, <a class="txt" href="waybel25.html#E19:37"><i><font color="Green" title="E29">A10</font></i></a>, <a class="txt" href="waybel25.html#E22:37"><i><font color="Green" title="E31">A12</font></i></a>, <a class="txt" href="waybel25.html#E23:37"><i><font color="Green" title="E32">A13</font></i></a>, <a class="txt" href="waybel25.html#E24:37"><i><font color="Green" title="E33">A14</font></i></a>, <a class="ref" href="waybel15.html#T11" title="WAYBEL15:th.11">WAYBEL15:11</a>, <a class="ref" href="waybel20.html#T18" title="WAYBEL20:th.18">WAYBEL20:18</a>, <a class="ref" href="yellow14.html#T12" title="YELLOW14:th.12">YELLOW14:12</a>, <a class="ref" href="yellow14.html#T13" title="YELLOW14:th.13">YELLOW14:13</a></i>;<br/>


</div>
