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<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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">X1</font>, <font color="Olive" title="b2">X2</font> being non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">X1</font>,<font color="Olive" title="b2">X2</font> <a href="tsep_1.html#R4" title="TSEP_1:pred.4">are_weakly_separated</a>  holds <br/>for <font color="Olive" title="b3">g</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><font color="Olive" title="b1">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Olive" title="b2">X2</font></span>)</span>,<font color="Maroon" title="c2">Y</font> holds <br/> ( <font color="Olive" title="b3">g</font> is   <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 <span class="p1">(<span class="default"><font color="Olive" title="b1">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Olive" title="b2">X2</font></span>)</span>,<font color="Maroon" title="c2">Y</font> iff ( <font color="Olive" title="b3">g</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <font color="Olive" title="b1">X1</font> is   <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="Olive" title="b1">X1</font>,<font color="Maroon" title="c2">Y</font> &amp; <font color="Olive" title="b3">g</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <font color="Olive" title="b2">X2</font> is   <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="Olive" title="b2">X2</font>,<font color="Maroon" title="c2">Y</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">X1</font>, <font color="Maroon" title="c4">X2</font> be  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">X1</font>,<font color="Maroon" title="c4">X2</font> <a href="tsep_1.html#R4" title="TSEP_1:pred.4">are_weakly_separated</a>  implies for <font color="Olive" title="b1">g</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font></span>)</span>,<font color="Maroon" title="c2">Y</font> holds <br/> ( <font color="Olive" title="b1">g</font> is   <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 <span class="p1">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font></span>)</span>,<font color="Maroon" title="c2">Y</font> iff ( <font color="Olive" title="b1">g</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <font color="Maroon" title="c3">X1</font> is   <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="c3">X1</font>,<font color="Maroon" title="c2">Y</font> &amp; <font color="Olive" title="b1">g</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <font color="Maroon" title="c4">X2</font> is   <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="c4">X2</font>,<font color="Maroon" title="c2">Y</font> ) ) )</span><br/>







<span class="kw">assume </span><a NAME="E1:136"/><span class="lab"><font color="Green" title="E82">A1</font></span>: 
<font color="Maroon" title="c3">X1</font>,<font color="Maroon" title="c4">X2</font> <a href="tsep_1.html#R4" title="TSEP_1:pred.4">are_weakly_separated</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">g</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font></span>)</span>,<font color="Maroon" title="c2">Y</font> holds <br/> ( <font color="Olive" title="b1">g</font> is   <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 <span class="p1">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font></span>)</span>,<font color="Maroon" title="c2">Y</font> iff ( <font color="Olive" title="b1">g</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <font color="Maroon" title="c3">X1</font> is   <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="c3">X1</font>,<font color="Maroon" title="c2">Y</font> &amp; <font color="Olive" title="b1">g</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <font color="Maroon" title="c4">X2</font> is   <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="c4">X2</font>,<font color="Maroon" title="c2">Y</font> ) )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c5">g</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font></span>)</span>,<font color="Maroon" title="c2">Y</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">g</font> is   <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 <span class="p1">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font></span>)</span>,<font color="Maroon" title="c2">Y</font> iff ( <font color="Maroon" title="c5">g</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <font color="Maroon" title="c3">X1</font> is   <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="c3">X1</font>,<font color="Maroon" title="c2">Y</font> &amp; <font color="Maroon" title="c5">g</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <font color="Maroon" title="c4">X2</font> is   <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="c4">X2</font>,<font color="Maroon" title="c2">Y</font> ) )</span><br/>

<a NAME="E2:136"/><span class="lab"><font color="Green" title="E83">A2</font></span>: 
<font color="Maroon" title="c4">X2</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tsep_1.html#T22" title="TSEP_1:th.22">TSEP_1:22</a></span>;<br/>
<a NAME="E3:136"/><span class="lab"><font color="Green" title="E84">A3</font></span>: 
<font color="Maroon" title="c3">X1</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tsep_1.html#T22" title="TSEP_1:th.22">TSEP_1:22</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:136"/>
( <font color="Maroon" title="c5">g</font> is   <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 <span class="p1">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font></span>)</span>,<font color="Maroon" title="c2">Y</font> implies ( <font color="Maroon" title="c5">g</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <font color="Maroon" title="c3">X1</font> is   <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="c3">X1</font>,<font color="Maroon" title="c2">Y</font> &amp; <font color="Maroon" title="c5">g</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <font color="Maroon" title="c4">X2</font> is   <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="c4">X2</font>,<font color="Maroon" title="c2">Y</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:136"><span class="lab"><font color="Green" title="E83">A2</font></span></a>, <a class="ref" href="tmap_1.html#T89" target="_self" title="TMAP_1:th.89">Th89</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">g</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <font color="Maroon" title="c3">X1</font> is   <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="c3">X1</font>,<font color="Maroon" title="c2">Y</font> &amp; <font color="Maroon" title="c5">g</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <font color="Maroon" title="c4">X2</font> is   <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="c4">X2</font>,<font color="Maroon" title="c2">Y</font> implies <font color="Maroon" title="c5">g</font> is   <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 <span class="p1">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font></span>)</span>,<font color="Maroon" title="c2">Y</font> )</span><br/>

<span class="kw">thus </span><a NAME="E5:136"/>
( <font color="Maroon" title="c5">g</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <font color="Maroon" title="c3">X1</font> is   <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="c3">X1</font>,<font color="Maroon" title="c2">Y</font> &amp; <font color="Maroon" title="c5">g</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <font color="Maroon" title="c4">X2</font> is   <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="c4">X2</font>,<font color="Maroon" title="c2">Y</font> implies <font color="Maroon" title="c5">g</font> is   <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 <span class="p1">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font></span>)</span>,<font color="Maroon" title="c2">Y</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="136_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:136_1"/><span class="lab"><font color="Green" title="E85">A4</font></span>: 
<font color="Maroon" title="c5">g</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <font color="Maroon" title="c3">X1</font> is   <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="c3">X1</font>,<font color="Maroon" title="c2">Y</font>
 <span class="kw">and </span><br/><a NAME="E2:136_1"/><span class="lab"><font color="Green" title="E86">A5</font></span>: 
<font color="Maroon" title="c5">g</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <font color="Maroon" title="c4">X2</font> is   <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="c4">X2</font>,<font color="Maroon" title="c2">Y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">g</font> is   <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 <span class="p1">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font></span>)</span>,<font color="Maroon" title="c2">Y</font></span><br/>

<a NAME="E3:136_1"/>
for <font color="Olive" title="b1">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font></span>)</span> holds  <font color="Maroon" title="c5">g</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="136_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c6">X0</font> = <font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font>;<br/>
<span class="kw">let </span><font color="Maroon" title="c7">x</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">g</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c7">x</font></span><br/>

<a NAME="E1:136_1_1"/><span class="lab"><font color="Green" title="E87">A6</font></span>: 
( <font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#NR2" title="TSEP_1:NR.2">meets</a> <font color="Maroon" title="c4">X2</font> implies <font color="Maroon" title="c5">g</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c7">x</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="136_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:136_1_1_1"/><span class="lab"><font color="Green" title="E88">A7</font></span>: 
<font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#NR2" title="TSEP_1:NR.2">meets</a> <font color="Maroon" title="c4">X2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">g</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c7">x</font></span><br/>

<div><span class="lab"><font color="Green" title="E89">A8</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="136_1_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:136_1_1_1_1"/><span class="lab"><font color="Green" title="E90">A9</font></span>: 
( <font color="Maroon" title="c3">X1</font> is  not    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c4">X2</font> &amp; <font color="Maroon" title="c4">X2</font> is  not    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c3">X1</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">g</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c7">x</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">Y1</font>, <font color="Maroon" title="c9">Y2</font> being  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="tsep_1.html#V1" title="TSEP_1:attr.1">open</a>   <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c1">X</font><span class="kw"> such that </span><br/><a NAME="E3:136_1_1_1_1"/><span class="lab"><font color="Green" title="E91">A10</font></span>: 
<font color="Maroon" title="c8">Y1</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font></span>)</span> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c3">X1</font>
 <span class="kw">and </span><br/><a NAME="E4:136_1_1_1_1"/><span class="lab"><font color="Green" title="E92">A11</font></span>: 
<font color="Maroon" title="c9">Y2</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font></span>)</span> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c4">X2</font>
 <span class="kw">and </span><br/><a NAME="E5:136_1_1_1_1"/><span class="lab"><font color="Green" title="E93">A12</font></span>: 
( <font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c8">Y1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c9">Y2</font> or  ex <font color="Olive" title="b1">Z</font> being non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="borsuk_1.html#V1" title="BORSUK_1:attr.1">closed</a>   <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c1">X</font> st <br/>(  <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">X</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">X</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">Y1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c9">Y2</font></span>)</span> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Olive" title="b1">Z</font> &amp; <font color="Olive" title="b1">Z</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font></span>)</span> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c4">X2</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:136"><span class="lab"><font color="Green" title="E82">A1</font></span></a>, <a class="txt" href="#E1:136_1_1_1"><span class="lab"><font color="Green" title="E88">A7</font></span></a>, <a class="ref" href="tsep_1.html#T96" title="TSEP_1:th.96">TSEP_1:96</a></span>;<br/><a NAME="E6:136_1_1_1_1"/><span class="lab"><font color="Green" title="E94">A13</font></span>: 
( <font color="Maroon" title="c9">Y2</font> <a href="tsep_1.html#NR2" title="TSEP_1:NR.2">meets</a> <font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font> implies <font color="Maroon" title="c9">Y2</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font></span>)</span> is   <a href="tsep_1.html#V1" title="TSEP_1:attr.1">open</a>   <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tmap_1.html#T44" target="_self" title="TMAP_1:th.44">Th44</a></span>;<br/><a NAME="E7:136_1_1_1_1"/><span class="lab"><font color="Green" title="E95">A14</font></span>: 
( <font color="Maroon" title="c8">Y1</font> <a href="tsep_1.html#NR2" title="TSEP_1:NR.2">meets</a> <font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font> implies <font color="Maroon" title="c8">Y1</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font></span>)</span> is   <a href="tsep_1.html#V1" title="TSEP_1:attr.1">open</a>   <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tmap_1.html#T44" target="_self" title="TMAP_1:th.44">Th44</a></span>;<br/><div><span class="lab"><font color="Green" title="E96">A15</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="136_1_1_1_1_1"><span class="kw">now </span></a><div class="add"><a NAME="E1:136_1_1_1_1_1"/>
<font color="Maroon" title="c1">X</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c1">X</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tsep_1.html#T2" title="TSEP_1:th.2">TSEP_1:2</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">X12</font> =  <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">X</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">X</font> #) as    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c1">X</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="tmap_1.html#T11" target="_self" title="TMAP_1:th.11">Th11</a></span>;<br/><span class="kw">assume </span><a NAME="E3:136_1_1_1_1_1"/><span class="lab"><font color="Green" title="E97">A16</font></span>: 
<font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font> is  not    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c8">Y1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c9">Y2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">g</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c7">x</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">Z</font> being  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="borsuk_1.html#V1" title="BORSUK_1:attr.1">closed</a>   <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c1">X</font><span class="kw"> such that </span><br/><a NAME="E5:136_1_1_1_1_1"/><span class="lab"><font color="Green" title="E98">A17</font></span>: 
 <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">X</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">X</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">Y1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c9">Y2</font></span>)</span> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c11">Z</font>
 <span class="kw">and </span><br/><a NAME="E6:136_1_1_1_1_1"/><span class="lab"><font color="Green" title="E99">A18</font></span>: 
<font color="Maroon" title="c11">Z</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font></span>)</span> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c4">X2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:136_1_1_1_1"><span class="lab"><font color="Green" title="E93">A12</font></span></a></span>;<br/><a NAME="E7:136_1_1_1_1_1"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</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 <font color="Maroon" title="c10">X12</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_1.html#T35" title="BORSUK_1:th.35">BORSUK_1:35</a></span>;<br/><a NAME="E8:136_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E100">A19</font></span>: 
<font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c10">X12</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tsep_1.html#T4" title="TSEP_1:th.4">TSEP_1:4</a></span>;<br/><a NAME="E9:136_1_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">X12</font> <a href="tsep_1.html#NR2" title="TSEP_1:NR.2">meets</a> <font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tmap_1.html#T22" target="_self" title="TMAP_1:th.22">Th22</a></span>;<br/><a NAME="E10:136_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E101">A20</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">Y1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c9">Y2</font></span>)</span> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c11">Z</font></span>)</span> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font></span>)</span> <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 <span class="p1">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font></span>)</span>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font></span>)</span> #)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:136_1_1_1_1_1"><span class="lab"><font color="Green" title="E98">A17</font></span></a>, <a class="txt" href="#E8:136_1_1_1_1_1"><span class="lab"><font color="Green" title="E100">A19</font></span></a>, <a class="ref" href="tsep_1.html#T31" title="TSEP_1:th.31">TSEP_1:31</a></span>;<br/><a NAME="E11:136_1_1_1_1_1"/><span class="lab"><font color="Green" title="E102">A21</font></span>: 
( <font color="Maroon" title="c8">Y1</font> <a href="tsep_1.html#NR2" title="TSEP_1:NR.2">meets</a> <font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font> &amp; <font color="Maroon" title="c9">Y2</font> <a href="tsep_1.html#NR2" title="TSEP_1:NR.2">meets</a> <font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:136_1_1_1"><span class="lab"><font color="Green" title="E88">A7</font></span></a>, <a class="txt" href="#E1:136_1_1_1_1"><span class="lab"><font color="Green" title="E90">A9</font></span></a>, <a class="txt" href="#E3:136_1_1_1_1"><span class="lab"><font color="Green" title="E91">A10</font></span></a>, <a class="txt" href="#E4:136_1_1_1_1"><span class="lab"><font color="Green" title="E92">A11</font></span></a>, <a class="txt" href="#E5:136_1_1_1_1_1"><span class="lab"><font color="Green" title="E98">A17</font></span></a>, <a class="txt" href="#E6:136_1_1_1_1_1"><span class="lab"><font color="Green" title="E99">A18</font></span></a>, <a class="ref" href="tmap_1.html#T37" target="_self" title="TMAP_1:th.37">Th37</a></span>;<br/><div><span class="lab"><font color="Green" title="E103">A22</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="136_1_1_1_1_1_1"><span class="kw">now </span></a><div class="add"><div><span class="lab"><font color="Green" title="E104">A23</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="136_1_1_1_1_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">given </span><font color="Maroon" title="c12">x2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c9">Y2</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font></span>)</span></span>)</span><span class="kw"> such that </span><a NAME="E2:136_1_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E105">A24</font></span>: 
<font color="Maroon" title="c12">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">g</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c7">x</font></span><br/><a NAME="E3:136_1_1_1_1_1_1_1"/>
<font color="Maroon" title="c5">g</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">Y2</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font></span>)</span></span>)</span> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:136"><span class="lab"><font color="Green" title="E83">A2</font></span></a>, <a class="txt" href="#E2:136_1"><span class="lab"><font color="Green" title="E86">A5</font></span></a>, <a class="txt" href="#E4:136_1_1_1_1"><span class="lab"><font color="Green" title="E92">A11</font></span></a>, <a class="ref" href="tmap_1.html#T90" target="_self" title="TMAP_1:th.90">Th90</a></span>;<br/><a NAME="E4:136_1_1_1_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">g</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">Y2</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font></span>)</span></span>)</span> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c12">x2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tmap_1.html#T49" target="_self" title="TMAP_1:th.49">Th49</a></span>;<br/><span class="kw">hence </span><a NAME="E5:136_1_1_1_1_1_1_1"/>
<font color="Maroon" title="c5">g</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c7">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:136_1_1_1"><span class="lab"><font color="Green" title="E88">A7</font></span></a>, <a class="txt" href="#E1:136_1_1_1_1"><span class="lab"><font color="Green" title="E90">A9</font></span></a>, <a class="txt" href="#E3:136_1_1_1_1"><span class="lab"><font color="Green" title="E91">A10</font></span></a>, <a class="txt" href="#E4:136_1_1_1_1"><span class="lab"><font color="Green" title="E92">A11</font></span></a>, <a class="txt" href="#E6:136_1_1_1_1"><span class="lab"><font color="Green" title="E94">A13</font></span></a>, <a class="txt" href="#E5:136_1_1_1_1_1"><span class="lab"><font color="Green" title="E98">A17</font></span></a>, <a class="txt" href="#E6:136_1_1_1_1_1"><span class="lab"><font color="Green" title="E99">A18</font></span></a>, <a class="txt" href="#E2:136_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E105">A24</font></span></a>, <a class="ref" href="tmap_1.html#T37" target="_self" title="TMAP_1:th.37">Th37</a>, <a class="ref" href="tmap_1.html#T86" target="_self" title="TMAP_1:th.86">Th86</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><span class="lab"><font color="Green" title="E105">A25</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="136_1_1_1_1_1_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">given </span><font color="Maroon" title="c12">x1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c8">Y1</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font></span>)</span></span>)</span><span class="kw"> such that </span><a NAME="E2:136_1_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E106">A26</font></span>: 
<font color="Maroon" title="c12">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">g</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c7">x</font></span><br/><a NAME="E3:136_1_1_1_1_1_1_2"/>
<font color="Maroon" title="c5">g</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">Y1</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font></span>)</span></span>)</span> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:136"><span class="lab"><font color="Green" title="E84">A3</font></span></a>, <a class="txt" href="#E1:136_1"><span class="lab"><font color="Green" title="E85">A4</font></span></a>, <a class="txt" href="#E3:136_1_1_1_1"><span class="lab"><font color="Green" title="E91">A10</font></span></a>, <a class="ref" href="tmap_1.html#T90" target="_self" title="TMAP_1:th.90">Th90</a></span>;<br/><a NAME="E4:136_1_1_1_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">g</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">Y1</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font></span>)</span></span>)</span> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c12">x1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tmap_1.html#T49" target="_self" title="TMAP_1:th.49">Th49</a></span>;<br/><span class="kw">hence </span><a NAME="E5:136_1_1_1_1_1_1_2"/>
<font color="Maroon" title="c5">g</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c7">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:136_1_1_1"><span class="lab"><font color="Green" title="E88">A7</font></span></a>, <a class="txt" href="#E1:136_1_1_1_1"><span class="lab"><font color="Green" title="E90">A9</font></span></a>, <a class="txt" href="#E3:136_1_1_1_1"><span class="lab"><font color="Green" title="E91">A10</font></span></a>, <a class="txt" href="#E4:136_1_1_1_1"><span class="lab"><font color="Green" title="E92">A11</font></span></a>, <a class="txt" href="#E7:136_1_1_1_1"><span class="lab"><font color="Green" title="E95">A14</font></span></a>, <a class="txt" href="#E5:136_1_1_1_1_1"><span class="lab"><font color="Green" title="E98">A17</font></span></a>, <a class="txt" href="#E6:136_1_1_1_1_1"><span class="lab"><font color="Green" title="E99">A18</font></span></a>, <a class="txt" href="#E2:136_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E106">A26</font></span></a>, <a class="ref" href="tmap_1.html#T37" target="_self" title="TMAP_1:th.37">Th37</a>, <a class="ref" href="tmap_1.html#T86" target="_self" title="TMAP_1:th.86">Th86</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">assume </span><a NAME="E3:136_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E106">A27</font></span>: 
 ex <font color="Olive" title="b1">x12</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">Y1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c9">Y2</font></span>)</span> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font></span>)</span></span>)</span> st <font color="Olive" title="b1">x12</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">g</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c7">x</font></span><br/><a NAME="E4:136_1_1_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c8">Y1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c9">Y2</font></span>)</span> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">Y1</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font></span>)</span></span>)</span> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">Y2</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:136_1_1_1_1_1"><span class="lab"><font color="Green" title="E102">A21</font></span></a>, <a class="ref" href="tsep_1.html#T35" title="TSEP_1:th.35">TSEP_1:35</a></span>;<br/><span class="kw">hence </span><a NAME="E5:136_1_1_1_1_1_1"/>
<font color="Maroon" title="c5">g</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c7">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:136_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E106">A27</font></span></a>, <a class="txt" href="#E2:136_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E105">A25</font></span></a>, <a class="txt" href="#E1:136_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E104">A23</font></span></a>, <a class="ref" href="tmap_1.html#T16" target="_self" title="TMAP_1:th.16">Th16</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><span class="lab"><font color="Green" title="E104">A28</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="136_1_1_1_1_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">given </span><font color="Maroon" title="c12">x0</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c11">Z</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font></span>)</span></span>)</span><span class="kw"> such that </span><a NAME="E2:136_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E105">A29</font></span>: 
<font color="Maroon" title="c12">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">g</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c7">x</font></span><br/><span class="kw">consider </span><font color="Maroon" title="c13">x00</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c4">X2</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E4:136_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E106">A30</font></span>: 
<font color="Maroon" title="c13">x00</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">x0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:136_1_1_1_1_1"><span class="lab"><font color="Green" title="E99">A18</font></span></a>, <a class="ref" href="tmap_1.html#T15" target="_self" title="TMAP_1:th.15">Th15</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c14">x1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c3">X1</font><span class="kw"> such that </span><br/><a NAME="E6:136_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E107">A31</font></span>: 
<font color="Maroon" title="c14">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">x00</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:136_1_1_1"><span class="lab"><font color="Green" title="E88">A7</font></span></a>, <a class="ref" href="tmap_1.html#T17" target="_self" title="TMAP_1:th.17">Th17</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c15">x2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c4">X2</font><span class="kw"> such that </span><br/><a NAME="E8:136_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E108">A32</font></span>: 
<font color="Maroon" title="c15">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">x00</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:136_1_1_1"><span class="lab"><font color="Green" title="E88">A7</font></span></a>, <a class="ref" href="tmap_1.html#T17" target="_self" title="TMAP_1:th.17">Th17</a></span>;<br/><a NAME="E9:136_1_1_1_1_1_2"/>
( <font color="Maroon" title="c5">g</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <font color="Maroon" title="c3">X1</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c14">x1</font> &amp; <font color="Maroon" title="c5">g</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <font color="Maroon" title="c4">X2</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c15">x2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:136_1"><span class="lab"><font color="Green" title="E85">A4</font></span></a>, <a class="txt" href="#E2:136_1"><span class="lab"><font color="Green" title="E86">A5</font></span></a>, <a class="ref" href="tmap_1.html#T49" target="_self" title="TMAP_1:th.49">Th49</a></span>;<br/><span class="kw">hence </span><a NAME="E10:136_1_1_1_1_1_2"/>
<font color="Maroon" title="c5">g</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c7">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:136_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E105">A29</font></span></a>, <a class="txt" href="#E4:136_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E106">A30</font></span></a>, <a class="txt" href="#E6:136_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E107">A31</font></span></a>, <a class="txt" href="#E8:136_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E108">A32</font></span></a>, <a class="ref" href="tmap_1.html#T123" target="_self" title="TMAP_1:th.123">Th123</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E14:136_1_1_1_1_1"/>
( <font color="Maroon" title="c8">Y1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c9">Y2</font> <a href="tsep_1.html#NR2" title="TSEP_1:NR.2">meets</a> <font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font> &amp; <font color="Maroon" title="c11">Z</font> <a href="tsep_1.html#NR2" title="TSEP_1:NR.2">meets</a> <font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:136_1_1_1"><span class="lab"><font color="Green" title="E88">A7</font></span></a>, <a class="txt" href="#E1:136_1_1_1_1"><span class="lab"><font color="Green" title="E90">A9</font></span></a>, <a class="txt" href="#E3:136_1_1_1_1"><span class="lab"><font color="Green" title="E91">A10</font></span></a>, <a class="txt" href="#E4:136_1_1_1_1"><span class="lab"><font color="Green" title="E92">A11</font></span></a>, <a class="txt" href="#E3:136_1_1_1_1_1"><span class="lab"><font color="Green" title="E97">A16</font></span></a>, <a class="txt" href="#E5:136_1_1_1_1_1"><span class="lab"><font color="Green" title="E98">A17</font></span></a>, <a class="txt" href="#E6:136_1_1_1_1_1"><span class="lab"><font color="Green" title="E99">A18</font></span></a>, <a class="ref" href="tmap_1.html#T38" target="_self" title="TMAP_1:th.38">Th38</a></span>;<br/><a NAME="E15:136_1_1_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">Y1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c9">Y2</font></span>)</span> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font></span>)</span></span>)</span> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">Z</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font></span>)</span></span>)</span> <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 <span class="p1">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font></span>)</span>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font></span>)</span> #)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:136_1_1_1_1_1"><span class="lab"><font color="Green" title="E101">A20</font></span></a>, <a class="ref" href="tsep_1.html#T35" title="TSEP_1:th.35">TSEP_1:35</a></span>;<br/><span class="kw">hence </span><a NAME="E16:136_1_1_1_1_1"/>
<font color="Maroon" title="c5">g</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c7">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:136_1_1_1_1_1"><span class="lab"><font color="Green" title="E103">A22</font></span></a>, <a class="txt" href="#E13:136_1_1_1_1_1"><span class="lab"><font color="Green" title="E104">A28</font></span></a>, <a class="ref" href="tmap_1.html#T16" target="_self" title="TMAP_1:th.16">Th16</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="136_1_1_1_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:136_1_1_1_1_2"/><span class="lab"><font color="Green" title="E97">A33</font></span>: 
<font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c8">Y1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c9">Y2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">g</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c7">x</font></span><br/><a NAME="E2:136_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E98">A34</font></span>: 
<font color="Maroon" title="c8">Y1</font> <a href="tsep_1.html#NR2" title="TSEP_1:NR.2">meets</a> <font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:136_1_1_1_1"><span class="lab"><font color="Green" title="E90">A9</font></span></a>, <a class="txt" href="#E3:136_1_1_1_1"><span class="lab"><font color="Green" title="E91">A10</font></span></a>, <a class="txt" href="#E4:136_1_1_1_1"><span class="lab"><font color="Green" title="E92">A11</font></span></a>, <a class="ref" href="tmap_1.html#T36" target="_self" title="TMAP_1:th.36">Th36</a></span>;<br/><div><span class="lab"><font color="Green" title="E99">A35</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="136_1_1_1_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">given </span><font color="Maroon" title="c10">x2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c9">Y2</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font></span>)</span></span>)</span><span class="kw"> such that </span><a NAME="E2:136_1_1_1_1_2_1"/><span class="lab"><font color="Green" title="E100">A36</font></span>: 
<font color="Maroon" title="c10">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">g</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c7">x</font></span><br/><a NAME="E3:136_1_1_1_1_2_1"/>
<font color="Maroon" title="c5">g</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">Y2</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font></span>)</span></span>)</span> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:136"><span class="lab"><font color="Green" title="E83">A2</font></span></a>, <a class="txt" href="#E2:136_1"><span class="lab"><font color="Green" title="E86">A5</font></span></a>, <a class="txt" href="#E4:136_1_1_1_1"><span class="lab"><font color="Green" title="E92">A11</font></span></a>, <a class="ref" href="tmap_1.html#T90" target="_self" title="TMAP_1:th.90">Th90</a></span>;<br/><a NAME="E4:136_1_1_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">g</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">Y2</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font></span>)</span></span>)</span> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c10">x2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tmap_1.html#T49" target="_self" title="TMAP_1:th.49">Th49</a></span>;<br/><span class="kw">hence </span><a NAME="E5:136_1_1_1_1_2_1"/>
<font color="Maroon" title="c5">g</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c7">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:136_1_1_1_1"><span class="lab"><font color="Green" title="E90">A9</font></span></a>, <a class="txt" href="#E3:136_1_1_1_1"><span class="lab"><font color="Green" title="E91">A10</font></span></a>, <a class="txt" href="#E4:136_1_1_1_1"><span class="lab"><font color="Green" title="E92">A11</font></span></a>, <a class="txt" href="#E6:136_1_1_1_1"><span class="lab"><font color="Green" title="E94">A13</font></span></a>, <a class="txt" href="#E1:136_1_1_1_1_2"><span class="lab"><font color="Green" title="E97">A33</font></span></a>, <a class="txt" href="#E2:136_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E100">A36</font></span></a>, <a class="ref" href="tmap_1.html#T36" target="_self" title="TMAP_1:th.36">Th36</a>, <a class="ref" href="tmap_1.html#T86" target="_self" title="TMAP_1:th.86">Th86</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><span class="lab"><font color="Green" title="E100">A37</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="136_1_1_1_1_2_2"><span class="kw">now </span></a><div class="add"><span class="kw">given </span><font color="Maroon" title="c10">x1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c8">Y1</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font></span>)</span></span>)</span><span class="kw"> such that </span><a NAME="E2:136_1_1_1_1_2_2"/><span class="lab"><font color="Green" title="E101">A38</font></span>: 
<font color="Maroon" title="c10">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">g</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c7">x</font></span><br/><a NAME="E3:136_1_1_1_1_2_2"/>
<font color="Maroon" title="c5">g</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">Y1</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font></span>)</span></span>)</span> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:136"><span class="lab"><font color="Green" title="E84">A3</font></span></a>, <a class="txt" href="#E1:136_1"><span class="lab"><font color="Green" title="E85">A4</font></span></a>, <a class="txt" href="#E3:136_1_1_1_1"><span class="lab"><font color="Green" title="E91">A10</font></span></a>, <a class="ref" href="tmap_1.html#T90" target="_self" title="TMAP_1:th.90">Th90</a></span>;<br/><a NAME="E4:136_1_1_1_1_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">g</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">Y1</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font></span>)</span></span>)</span> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c10">x1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tmap_1.html#T49" target="_self" title="TMAP_1:th.49">Th49</a></span>;<br/><span class="kw">hence </span><a NAME="E5:136_1_1_1_1_2_2"/>
<font color="Maroon" title="c5">g</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c7">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:136_1_1_1_1"><span class="lab"><font color="Green" title="E90">A9</font></span></a>, <a class="txt" href="#E3:136_1_1_1_1"><span class="lab"><font color="Green" title="E91">A10</font></span></a>, <a class="txt" href="#E4:136_1_1_1_1"><span class="lab"><font color="Green" title="E92">A11</font></span></a>, <a class="txt" href="#E7:136_1_1_1_1"><span class="lab"><font color="Green" title="E95">A14</font></span></a>, <a class="txt" href="#E1:136_1_1_1_1_2"><span class="lab"><font color="Green" title="E97">A33</font></span></a>, <a class="txt" href="#E2:136_1_1_1_1_2_2"><span class="lab"><font color="Green" title="E101">A38</font></span></a>, <a class="ref" href="tmap_1.html#T36" target="_self" title="TMAP_1:th.36">Th36</a>, <a class="ref" href="tmap_1.html#T86" target="_self" title="TMAP_1:th.86">Th86</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E5:136_1_1_1_1_2"/>
<font color="Maroon" title="c8">Y1</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c8">Y1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c9">Y2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tsep_1.html#T22" title="TSEP_1:th.22">TSEP_1:22</a></span>;<br/><a NAME="E6:136_1_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c8">Y1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c9">Y2</font> <a href="tsep_1.html#NR2" title="TSEP_1:NR.2">meets</a> <font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:136_1_1_1_1_2"><span class="lab"><font color="Green" title="E98">A34</font></span></a>, <a class="ref" href="tmap_1.html#T23" target="_self" title="TMAP_1:th.23">Th23</a></span>;<br/><a NAME="E7:136_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E101">A39</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">Y1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c9">Y2</font></span>)</span> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:136_1_1_1_1_2"><span class="lab"><font color="Green" title="E97">A33</font></span></a>, <a class="ref" href="tsep_1.html#T31" title="TSEP_1:th.31">TSEP_1:31</a></span>;<br/><a NAME="E8:136_1_1_1_1_2"/>
<font color="Maroon" title="c9">Y2</font> <a href="tsep_1.html#NR2" title="TSEP_1:NR.2">meets</a> <font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:136_1_1_1_1"><span class="lab"><font color="Green" title="E90">A9</font></span></a>, <a class="txt" href="#E3:136_1_1_1_1"><span class="lab"><font color="Green" title="E91">A10</font></span></a>, <a class="txt" href="#E4:136_1_1_1_1"><span class="lab"><font color="Green" title="E92">A11</font></span></a>, <a class="txt" href="#E1:136_1_1_1_1_2"><span class="lab"><font color="Green" title="E97">A33</font></span></a>, <a class="ref" href="tmap_1.html#T36" target="_self" title="TMAP_1:th.36">Th36</a></span>;<br/><a NAME="E9:136_1_1_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c8">Y1</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font></span>)</span></span>)</span> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">Y2</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:136_1_1_1_1_2"><span class="lab"><font color="Green" title="E98">A34</font></span></a>, <a class="txt" href="#E7:136_1_1_1_1_2"><span class="lab"><font color="Green" title="E101">A39</font></span></a>, <a class="ref" href="tsep_1.html#T35" title="TSEP_1:th.35">TSEP_1:35</a></span>;<br/><span class="kw">hence </span><a NAME="E10:136_1_1_1_1_2"/>
<font color="Maroon" title="c5">g</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c7">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:136_1_1_1_1_2"><span class="lab"><font color="Green" title="E100">A37</font></span></a>, <a class="txt" href="#E3:136_1_1_1_1_2"><span class="lab"><font color="Green" title="E99">A35</font></span></a>, <a class="ref" href="tmap_1.html#T16" target="_self" title="TMAP_1:th.16">Th16</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E10:136_1_1_1_1"/>
<font color="Maroon" title="c5">g</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c7">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:136_1_1_1_1"><span class="lab"><font color="Green" title="E96">A15</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="136_1_1_1_2"><span class="kw">now </span></a><div class="add"><div><span class="lab"><font color="Green" title="E90">A40</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="136_1_1_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:136_1_1_1_2_1"/>
<font color="Maroon" title="c4">X2</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c3">X1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">g</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c7">x</font></span><br/><a NAME="E2:136_1_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E91">A41</font></span>: 
 <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="c3">X1</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c3">X1</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tsep_1.html#T23" title="TSEP_1:th.23">TSEP_1:23</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">x1</font> = <font color="Maroon" title="c7">x</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c3">X1</font> ;<br/><a NAME="E4:136_1_1_1_2_1"/>
<font color="Maroon" title="c5">g</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <font color="Maroon" title="c3">X1</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c8">x1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:136_1"><span class="lab"><font color="Green" title="E85">A4</font></span></a>, <a class="ref" href="tmap_1.html#T49" target="_self" title="TMAP_1:th.49">Th49</a></span>;<br/><span class="kw">hence </span><a NAME="E5:136_1_1_1_2_1"/>
<font color="Maroon" title="c5">g</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c7">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:136_1_1_1_2_1"><span class="lab"><font color="Green" title="E91">A41</font></span></a>, <a class="ref" href="tmap_1.html#T88" target="_self" title="TMAP_1:th.88">Th88</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><span class="lab"><font color="Green" title="E91">A42</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="136_1_1_1_2_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:136_1_1_1_2_2"/>
<font color="Maroon" title="c3">X1</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c4">X2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">g</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c7">x</font></span><br/><a NAME="E2:136_1_1_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E92">A43</font></span>: 
 <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="c4">X2</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c4">X2</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tsep_1.html#T23" title="TSEP_1:th.23">TSEP_1:23</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">x2</font> = <font color="Maroon" title="c7">x</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c4">X2</font> ;<br/><a NAME="E4:136_1_1_1_2_2"/>
<font color="Maroon" title="c5">g</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <font color="Maroon" title="c4">X2</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c8">x2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:136_1"><span class="lab"><font color="Green" title="E86">A5</font></span></a>, <a class="ref" href="tmap_1.html#T49" target="_self" title="TMAP_1:th.49">Th49</a></span>;<br/><span class="kw">hence </span><a NAME="E5:136_1_1_1_2_2"/>
<font color="Maroon" title="c5">g</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c7">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:136_1_1_1_2_2"><span class="lab"><font color="Green" title="E92">A43</font></span></a>, <a class="ref" href="tmap_1.html#T88" target="_self" title="TMAP_1:th.88">Th88</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">assume </span><a NAME="E3:136_1_1_1_2"/>
( <font color="Maroon" title="c3">X1</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c4">X2</font> or <font color="Maroon" title="c4">X2</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c3">X1</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">g</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c7">x</font></span><br/><span class="kw">hence </span><a NAME="E4:136_1_1_1_2"/>
<font color="Maroon" title="c5">g</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c7">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:136_1_1_1_2"><span class="lab"><font color="Green" title="E91">A42</font></span></a>, <a class="txt" href="#E1:136_1_1_1_2"><span class="lab"><font color="Green" title="E90">A40</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:136_1_1_1"/>
<font color="Maroon" title="c5">g</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c7">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:136_1_1_1"><span class="lab"><font color="Green" title="E89">A8</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:136_1_1"/>
( <font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#R1" title="TSEP_1:pred.1">misses</a> <font color="Maroon" title="c4">X2</font> implies <font color="Maroon" title="c5">g</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c7">x</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="136_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:136_1_1_2"/>
<font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#R1" title="TSEP_1:pred.1">misses</a> <font color="Maroon" title="c4">X2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">g</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c7">x</font></span><br/>

<a NAME="E2:136_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">X1</font>,<font color="Maroon" title="c4">X2</font> <a href="tsep_1.html#R3" title="TSEP_1:pred.3">are_separated</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:136"><span class="lab"><font color="Green" title="E82">A1</font></span></a>, <a class="ref" href="tsep_1.html#T85" title="TSEP_1:th.85">TSEP_1:85</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">Y1</font>, <font color="Maroon" title="c9">Y2</font> being  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="tsep_1.html#V1" title="TSEP_1:attr.1">open</a>   <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c1">X</font><span class="kw"> such that </span><br/><a NAME="E4:136_1_1_2"/><span class="lab"><font color="Green" title="E88">A44</font></span>: 
<font color="Maroon" title="c3">X1</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c8">Y1</font>
 <span class="kw">and </span><br/><a NAME="E5:136_1_1_2"/><span class="lab"><font color="Green" title="E89">A45</font></span>: 
<font color="Maroon" title="c4">X2</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c9">Y2</font>
 <span class="kw">and </span><br/><a NAME="E6:136_1_1_2"/><span class="lab"><font color="Green" title="E90">A46</font></span>: 
( <font color="Maroon" title="c8">Y1</font> <a href="tsep_1.html#R1" title="TSEP_1:pred.1">misses</a> <font color="Maroon" title="c9">Y2</font> or <font color="Maroon" title="c8">Y1</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c9">Y2</font> <a href="tsep_1.html#R1" title="TSEP_1:pred.1">misses</a> <font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tsep_1.html#T83" title="TSEP_1:th.83">TSEP_1:83</a></span>;<br/>
<a NAME="E7:136_1_1_2"/>
<font color="Maroon" title="c9">Y2</font> <a href="tsep_1.html#R1" title="TSEP_1:pred.1">misses</a> <font color="Maroon" title="c3">X1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:136_1_1_2"><span class="lab"><font color="Green" title="E88">A44</font></span></a>, <a class="txt" href="#E5:136_1_1_2"><span class="lab"><font color="Green" title="E89">A45</font></span></a>, <a class="txt" href="#E6:136_1_1_2"><span class="lab"><font color="Green" title="E90">A46</font></span></a>, <a class="ref" href="tmap_1.html#T35" target="_self" title="TMAP_1:th.35">Th35</a></span>;<br/>
<a NAME="E8:136_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E91">A47</font></span>: 
<font color="Maroon" title="c4">X2</font> is   <a href="tsep_1.html#V1" title="TSEP_1:attr.1">open</a>   <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:136_1_1_2"><span class="lab"><font color="Green" title="E89">A45</font></span></a>, <a class="ref" href="tmap_1.html#T46" target="_self" title="TMAP_1:th.46">Th46</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E92">A48</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="136_1_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">given </span><font color="Maroon" title="c10">x2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c4">X2</font><span class="kw"> such that </span><a NAME="E2:136_1_1_2_1"/><span class="lab"><font color="Green" title="E93">A49</font></span>: 
<font color="Maroon" title="c10">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">g</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c7">x</font></span><br/><a NAME="E3:136_1_1_2_1"/>
<font color="Maroon" title="c5">g</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <font color="Maroon" title="c4">X2</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c10">x2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:136_1"><span class="lab"><font color="Green" title="E86">A5</font></span></a>, <a class="ref" href="tmap_1.html#T49" target="_self" title="TMAP_1:th.49">Th49</a></span>;<br/><span class="kw">hence </span><a NAME="E4:136_1_1_2_1"/>
<font color="Maroon" title="c5">g</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c7">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:136_1_1_2"><span class="lab"><font color="Green" title="E91">A47</font></span></a>, <a class="txt" href="#E2:136_1_1_2_1"><span class="lab"><font color="Green" title="E93">A49</font></span></a>, <a class="ref" href="tmap_1.html#T86" target="_self" title="TMAP_1:th.86">Th86</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E10:136_1_1_2"/>
<font color="Maroon" title="c8">Y1</font> <a href="tsep_1.html#R1" title="TSEP_1:pred.1">misses</a> <font color="Maroon" title="c4">X2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:136_1_1_2"><span class="lab"><font color="Green" title="E88">A44</font></span></a>, <a class="txt" href="#E5:136_1_1_2"><span class="lab"><font color="Green" title="E89">A45</font></span></a>, <a class="txt" href="#E6:136_1_1_2"><span class="lab"><font color="Green" title="E90">A46</font></span></a>, <a class="ref" href="tmap_1.html#T35" target="_self" title="TMAP_1:th.35">Th35</a></span>;<br/>
<a NAME="E11:136_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E93">A50</font></span>: 
<font color="Maroon" title="c3">X1</font> is   <a href="tsep_1.html#V1" title="TSEP_1:attr.1">open</a>   <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:136_1_1_2"><span class="lab"><font color="Green" title="E88">A44</font></span></a>, <a class="ref" href="tmap_1.html#T46" target="_self" title="TMAP_1:th.46">Th46</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="136_1_1_2_2"><span class="kw">now </span></a><div class="add"><span class="kw">given </span><font color="Maroon" title="c10">x1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c3">X1</font><span class="kw"> such that </span><a NAME="E2:136_1_1_2_2"/><span class="lab"><font color="Green" title="E94">A51</font></span>: 
<font color="Maroon" title="c10">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">g</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c7">x</font></span><br/><a NAME="E3:136_1_1_2_2"/>
<font color="Maroon" title="c5">g</font> <a href="tmap_1.html#K3" title="TMAP_1:func.3">|</a> <font color="Maroon" title="c3">X1</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c10">x1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:136_1"><span class="lab"><font color="Green" title="E85">A4</font></span></a>, <a class="ref" href="tmap_1.html#T49" target="_self" title="TMAP_1:th.49">Th49</a></span>;<br/><span class="kw">hence </span><a NAME="E4:136_1_1_2_2"/>
<font color="Maroon" title="c5">g</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c7">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:136_1_1_2"><span class="lab"><font color="Green" title="E93">A50</font></span></a>, <a class="txt" href="#E2:136_1_1_2_2"><span class="lab"><font color="Green" title="E94">A51</font></span></a>, <a class="ref" href="tmap_1.html#T86" target="_self" title="TMAP_1:th.86">Th86</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E13:136_1_1_2"/>
<font color="Maroon" title="c5">g</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c7">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:136_1_1_2"><span class="lab"><font color="Green" title="E92">A48</font></span></a>, <a class="ref" href="tmap_1.html#T16" target="_self" title="TMAP_1:th.16">Th16</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:136_1_1"/>
<font color="Maroon" title="c5">g</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c7">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:136_1_1"><span class="lab"><font color="Green" title="E87">A6</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:136_1"/>
<font color="Maroon" title="c5">g</font> is   <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 <span class="p1">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="tsep_1.html#K1" title="TSEP_1:func.1">union</a> <font color="Maroon" title="c4">X2</font></span>)</span>,<font color="Maroon" title="c2">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tmap_1.html#T49" target="_self" title="TMAP_1:th.49">Th49</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>


</div>
