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

<b>let </b><font color="Maroon" title="c1">S1</font>, <font color="Maroon" title="c2">S2</font> be    <a href="pre_topc.html#L1" title="PRE_TOPC:struct.1">TopStruct</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <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="c1">S1</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">S1</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="c2">S2</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">S2</font> #) implies for <font color="Olive" title="b1">T1</font>, <font color="Olive" title="b2">T2</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="waybel_9.html#L1" title="WAYBEL_9:struct.1">TopRelStr</a>   st  <a href="waybel_9.html#G1" title="WAYBEL_9:aggr.1">TopRelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">T1</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Olive" title="b1">T1</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Olive" title="b1">T1</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel_9.html#G1" title="WAYBEL_9:aggr.1">TopRelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b2">T2</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Olive" title="b2">T2</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Olive" title="b2">T2</font> #) holds <br/> <a href="waybel24.html#K3" title="WAYBEL24:func.3">ContMaps</a> <font color="Maroon" title="c1">S1</font>,<font color="Olive" title="b1">T1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel24.html#K3" title="WAYBEL24:func.3">ContMaps</a> <font color="Maroon" title="c2">S2</font>,<font color="Olive" title="b2">T2</font> )</span><br/>

<b>assume </b><a NAME="E1:10"/><i><font color="Green" title="E9">A1</font></i>: 
 <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="c1">S1</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">S1</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="c2">S2</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">S2</font> #)
 ; <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">T1</font>, <font color="Olive" title="b2">T2</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="waybel_9.html#L1" title="WAYBEL_9:struct.1">TopRelStr</a>   st  <a href="waybel_9.html#G1" title="WAYBEL_9:aggr.1">TopRelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">T1</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Olive" title="b1">T1</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Olive" title="b1">T1</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel_9.html#G1" title="WAYBEL_9:aggr.1">TopRelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b2">T2</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Olive" title="b2">T2</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Olive" title="b2">T2</font> #) holds <br/> <a href="waybel24.html#K3" title="WAYBEL24:func.3">ContMaps</a> <font color="Maroon" title="c1">S1</font>,<font color="Olive" title="b1">T1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel24.html#K3" title="WAYBEL24:func.3">ContMaps</a> <font color="Maroon" title="c2">S2</font>,<font color="Olive" title="b2">T2</font></span><br/>

<b>let </b><font color="Maroon" title="c3">T1</font>, <font color="Maroon" title="c4">T2</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="waybel_9.html#L1" title="WAYBEL_9:struct.1">TopRelStr</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="waybel_9.html#G1" title="WAYBEL_9:aggr.1">TopRelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">T1</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c3">T1</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c3">T1</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel_9.html#G1" title="WAYBEL_9:aggr.1">TopRelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">T2</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c4">T2</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c4">T2</font> #) implies  <a href="waybel24.html#K3" title="WAYBEL24:func.3">ContMaps</a> <font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c3">T1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel24.html#K3" title="WAYBEL24:func.3">ContMaps</a> <font color="Maroon" title="c2">S2</font>,<font color="Maroon" title="c4">T2</font> )</span><br/>

<b>assume </b><a NAME="E2:10"/><i><font color="Green" title="E10">A2</font></i>: 
 <a href="waybel_9.html#G1" title="WAYBEL_9:aggr.1">TopRelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">T1</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c3">T1</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c3">T1</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel_9.html#G1" title="WAYBEL_9:aggr.1">TopRelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">T2</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c4">T2</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c4">T2</font> #)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="waybel24.html#K3" title="WAYBEL24:func.3">ContMaps</a> <font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c3">T1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel24.html#K3" title="WAYBEL24:func.3">ContMaps</a> <font color="Maroon" title="c2">S2</font>,<font color="Maroon" title="c4">T2</font></span><br/>

<a NAME="E3:10"/><i><font color="Green" title="E11">A3</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="waybel24.html#K3" title="WAYBEL24:func.3">ContMaps</a> <font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c3">T1</font></span>)</span> <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="waybel24.html#K3" title="WAYBEL24:func.3">ContMaps</a> <font color="Maroon" title="c2">S2</font>,<font color="Maroon" title="c4">T2</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:10_1"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel24.html#K3" title="WAYBEL24:func.3">ContMaps</a> <font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c3">T1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel24.html#K3" title="WAYBEL24:func.3">ContMaps</a> <font color="Maroon" title="c2">S2</font>,<font color="Maroon" title="c4">T2</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> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel24.html#K3" title="WAYBEL24:func.3">ContMaps</a> <font color="Maroon" title="c2">S2</font>,<font color="Maroon" title="c4">T2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel24.html#K3" title="WAYBEL24:func.3">ContMaps</a> <font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c3">T1</font></span>)</span></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">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> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c5">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 <span class="p1">(<span class="default"><a href="waybel24.html#K3" title="WAYBEL24:func.3">ContMaps</a> <font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c3">T1</font></span>)</span> or <font color="Maroon" title="c5">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 <span class="p1">(<span class="default"><a href="waybel24.html#K3" title="WAYBEL24:func.3">ContMaps</a> <font color="Maroon" title="c2">S2</font>,<font color="Maroon" title="c4">T2</font></span>)</span> )</span><br/>

<b>assume </b><a NAME="E1:10_1_1"/>
<font color="Maroon" title="c5">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 <span class="p1">(<span class="default"><a href="waybel24.html#K3" title="WAYBEL24:func.3">ContMaps</a> <font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c3">T1</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">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 <span class="p1">(<span class="default"><a href="waybel24.html#K3" title="WAYBEL24:func.3">ContMaps</a> <font color="Maroon" title="c2">S2</font>,<font color="Maroon" title="c4">T2</font></span>)</span></span><br/>

<b>then consider </b><font color="Maroon" title="c6">f</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c3">T1</font><b> such that </b><br/><a NAME="E3:10_1_1"/><i><font color="Green" title="E12">A4</font></i>: 
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> &amp; <font color="Maroon" title="c6">f</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> )
 <b>by </b><i><a class="ref" href="waybel24.html#D3" title="WAYBEL24:def.3">WAYBEL24:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">f2</font> = <font color="Maroon" title="c6">f</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c2">S2</font>,<font color="Maroon" title="c4">T2</font> <b>by </b><i><a class="txt" href="#E1:10"><i><font color="Green" title="E9">A1</font></i></a>, <a class="txt" href="#E2:10"><i><font color="Green" title="E10">A2</font></i></a></i>;<br/>
<a NAME="E5:10_1_1"/>
<font color="Maroon" title="c7">f2</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">P2</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c4">T2</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="pre_topc.html#D12" title="PRE_TOPC:def.12">PRE_TOPC:def 12</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c8">P2</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> or <font color="Maroon" title="c7">f2</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c8">P2</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> )</span><br/>

<b>reconsider </b><font color="Maroon" title="c9">P1</font> = <font color="Maroon" title="c8">P2</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">T1</font> <b>by </b><i><a class="txt" href="#E2:10"><i><font color="Green" title="E10">A2</font></i></a></i>;<br/>
<b>assume </b><a NAME="E2:10_1_1_1"/>
<font color="Maroon" title="c8">P2</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">f2</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c8">P2</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a></span><br/>

<a NAME="E3:10_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c4">T2</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c8">P2</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D6" title="PRE_TOPC:def.6">PRE_TOPC:def 6</a></i>;<br/>
<a NAME="E4:10_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c4">T2</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c8">P2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c4">T2</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E5:10_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c3">T1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c9">P1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="txt" href="#E2:10"><i><font color="Green" title="E10">A2</font></i></a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E6:10_1_1_1"/><b>then </b>
<font color="Maroon" title="c9">P1</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D6" title="PRE_TOPC:def.6">PRE_TOPC:def 6</a></i>;<br/>
<a NAME="E7:10_1_1_1"/><b>then </b>
<font color="Maroon" title="c6">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c9">P1</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 
<b>by </b><i><a class="txt" href="#E3:10_1_1"><i><font color="Green" title="E12">A4</font></i></a>, <a class="ref" href="pre_topc.html#D12" title="PRE_TOPC:def.12">PRE_TOPC:def 12</a></i>;<br/>
<a NAME="E8:10_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">S1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c9">P1</font></span>)</span> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D6" title="PRE_TOPC:def.6">PRE_TOPC:def 6</a></i>;<br/>
<a NAME="E9:10_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">S1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f2</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c8">P2</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">S2</font>
 
<b>by </b><i><a class="txt" href="#E1:10"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E10:10_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">S2</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f2</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c8">P2</font></span>)</span> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="txt" href="#E1:10"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/>
<b>hence </b><a NAME="E11:10_1_1_1"/>
<font color="Maroon" title="c7">f2</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c8">P2</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 <b>by </b><i><a class="ref" href="pre_topc.html#D6" title="PRE_TOPC:def.6">PRE_TOPC:def 6</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>
<b>hence </b><a NAME="E6:10_1_1"/>
<font color="Maroon" title="c5">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 <span class="p1">(<span class="default"><a href="waybel24.html#K3" title="WAYBEL24:func.3">ContMaps</a> <font color="Maroon" title="c2">S2</font>,<font color="Maroon" title="c4">T2</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E3:10_1_1"><i><font color="Green" title="E12">A4</font></i></a>, <a class="ref" href="waybel24.html#D3" title="WAYBEL24:def.3">WAYBEL24:def 3</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>

<b>let </b><font color="Maroon" title="c5">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> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c5">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 <span class="p1">(<span class="default"><a href="waybel24.html#K3" title="WAYBEL24:func.3">ContMaps</a> <font color="Maroon" title="c2">S2</font>,<font color="Maroon" title="c4">T2</font></span>)</span> or <font color="Maroon" title="c5">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 <span class="p1">(<span class="default"><a href="waybel24.html#K3" title="WAYBEL24:func.3">ContMaps</a> <font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c3">T1</font></span>)</span> )</span><br/>

<b>assume </b><a NAME="E2:10_1"/>
<font color="Maroon" title="c5">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 <span class="p1">(<span class="default"><a href="waybel24.html#K3" title="WAYBEL24:func.3">ContMaps</a> <font color="Maroon" title="c2">S2</font>,<font color="Maroon" title="c4">T2</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">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 <span class="p1">(<span class="default"><a href="waybel24.html#K3" title="WAYBEL24:func.3">ContMaps</a> <font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c3">T1</font></span>)</span></span><br/>

<b>then consider </b><font color="Maroon" title="c6">f</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c2">S2</font>,<font color="Maroon" title="c4">T2</font><b> such that </b><br/><a NAME="E4:10_1"/><i><font color="Green" title="E12">A5</font></i>: 
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> &amp; <font color="Maroon" title="c6">f</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> )
 <b>by </b><i><a class="ref" href="waybel24.html#D3" title="WAYBEL24:def.3">WAYBEL24:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">f1</font> = <font color="Maroon" title="c6">f</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c3">T1</font> <b>by </b><i><a class="txt" href="#E1:10"><i><font color="Green" title="E9">A1</font></i></a>, <a class="txt" href="#E2:10"><i><font color="Green" title="E10">A2</font></i></a></i>;<br/>
<a NAME="E6:10_1"/>
<font color="Maroon" title="c7">f1</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">P1</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">T1</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="pre_topc.html#D12" title="PRE_TOPC:def.12">PRE_TOPC:def 12</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c8">P1</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> or <font color="Maroon" title="c7">f1</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c8">P1</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> )</span><br/>

<b>reconsider </b><font color="Maroon" title="c9">P2</font> = <font color="Maroon" title="c8">P1</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c4">T2</font> <b>by </b><i><a class="txt" href="#E2:10"><i><font color="Green" title="E10">A2</font></i></a></i>;<br/>
<b>assume </b><a NAME="E2:10_1_2"/>
<font color="Maroon" title="c8">P1</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">f1</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c8">P1</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a></span><br/>

<a NAME="E3:10_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c3">T1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c8">P1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D6" title="PRE_TOPC:def.6">PRE_TOPC:def 6</a></i>;<br/>
<a NAME="E4:10_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c3">T1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c9">P2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c4">T2</font>
 
<b>by </b><i><a class="txt" href="#E2:10"><i><font color="Green" title="E10">A2</font></i></a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E5:10_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c4">T2</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c9">P2</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="txt" href="#E2:10"><i><font color="Green" title="E10">A2</font></i></a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E6:10_1_2"/><b>then </b>
<font color="Maroon" title="c9">P2</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D6" title="PRE_TOPC:def.6">PRE_TOPC:def 6</a></i>;<br/>
<a NAME="E7:10_1_2"/><b>then </b>
<font color="Maroon" title="c6">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c9">P2</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 
<b>by </b><i><a class="txt" href="#E4:10_1"><i><font color="Green" title="E12">A5</font></i></a>, <a class="ref" href="pre_topc.html#D12" title="PRE_TOPC:def.12">PRE_TOPC:def 12</a></i>;<br/>
<a NAME="E8:10_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">S2</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c9">P2</font></span>)</span> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D6" title="PRE_TOPC:def.6">PRE_TOPC:def 6</a></i>;<br/>
<a NAME="E9:10_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">S2</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f1</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c8">P1</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">S1</font>
 
<b>by </b><i><a class="txt" href="#E1:10"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E10:10_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">S1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f1</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c8">P1</font></span>)</span> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="txt" href="#E1:10"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/>
<b>hence </b><a NAME="E11:10_1_2"/>
<font color="Maroon" title="c7">f1</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c8">P1</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 <b>by </b><i><a class="ref" href="pre_topc.html#D6" title="PRE_TOPC:def.6">PRE_TOPC:def 6</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>
<b>hence </b><a NAME="E7:10_1"/>
<font color="Maroon" title="c5">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 <span class="p1">(<span class="default"><a href="waybel24.html#K3" title="WAYBEL24:func.3">ContMaps</a> <font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c3">T1</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E4:10_1"><i><font color="Green" title="E12">A5</font></i></a>, <a class="ref" href="waybel24.html#D3" title="WAYBEL24:def.3">WAYBEL24:def 3</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>
<b>reconsider </b><font color="Maroon" title="c5">C1</font> =  <a href="waybel24.html#K3" title="WAYBEL24:func.3">ContMaps</a> <font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c3">T1</font> as  <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 <font color="Maroon" title="c3">T1</font> <a href="yellow_1.html#K6" title="YELLOW_1:func.6">|^</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S1</font> <b>by </b><i><a class="ref" href="waybel24.html#D3" title="WAYBEL24:def.3">WAYBEL24:def 3</a></i>;<br/>
<a NAME="E5:10"/>
(  <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="c3">T1</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c3">T1</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 <font color="Maroon" title="c4">T2</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c4">T2</font> #) &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">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="c2">S2</font> )
 
<b>by </b><i><a class="txt" href="#E1:10"><i><font color="Green" title="E9">A1</font></i></a>, <a class="txt" href="#E2:10"><i><font color="Green" title="E10">A2</font></i></a></i>;<br/>
<a NAME="E6:10"/><b>then </b>
<font color="Maroon" title="c3">T1</font> <a href="yellow_1.html#K6" title="YELLOW_1:func.6">|^</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">T2</font> <a href="yellow_1.html#K6" title="YELLOW_1:func.6">|^</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S2</font>
 
<b>by </b><i><a class="ref" href="waybel27.html#T15" title="WAYBEL27:th.15">WAYBEL27:15</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c6">C2</font> =  <a href="waybel24.html#K3" title="WAYBEL24:func.3">ContMaps</a> <font color="Maroon" title="c2">S2</font>,<font color="Maroon" title="c4">T2</font> as  <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 <font color="Maroon" title="c3">T1</font> <a href="yellow_1.html#K6" title="YELLOW_1:func.6">|^</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S1</font> <b>by </b><i><a class="ref" href="waybel24.html#D3" title="WAYBEL24:def.3">WAYBEL24:def 3</a></i>;<br/>
<a NAME="E8:10"/>
 <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="c5">C1</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c5">C1</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 <font color="Maroon" title="c6">C2</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c6">C2</font> #)
 
<b>by </b><i><a class="txt" href="#E3:10"><i><font color="Green" title="E11">A3</font></i></a>, <a class="ref" href="yellow_0.html#T58" title="YELLOW_0:th.58">YELLOW_0:58</a></i>;<br/>
<b>hence </b><a NAME="E9:10"/>
 <a href="waybel24.html#K3" title="WAYBEL24:func.3">ContMaps</a> <font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c3">T1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel24.html#K3" title="WAYBEL24:func.3">ContMaps</a> <font color="Maroon" title="c2">S2</font>,<font color="Maroon" title="c4">T2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
