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<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">X</font>, <font color="Maroon" title="c2">Y</font> be   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">f</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font> holds <br/> ( <font color="Olive" title="b1">f</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  iff  for <font color="Olive" title="b2">p</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font><br/>  for <font color="Olive" title="b3">V</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">Y</font>  st <font color="Olive" title="b1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">V</font> &amp; <font color="Olive" title="b3">V</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  holds <br/> ex <font color="Olive" title="b4">W</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Olive" title="b2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b4">W</font> &amp; <font color="Olive" title="b4">W</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Olive" title="b1">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b4">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b3">V</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">f</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">f</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  iff  for <font color="Olive" title="b1">p</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font><br/>  for <font color="Olive" title="b2">V</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">Y</font>  st <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">V</font> &amp; <font color="Olive" title="b2">V</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  holds <br/> ex <font color="Olive" title="b3">W</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">W</font> &amp; <font color="Olive" title="b3">W</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b3">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">V</font> ) )</span><br/>



<a NAME="E1:8"/><span class="lab"><font color="Green" title="E11">A1</font></span>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">Y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
;<br/>
<a NAME="E2:8"/><span class="lab"><font color="Green" title="E12">A2</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">f</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="c1">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  for <font color="Olive" title="b1">p</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font><br/>  for <font color="Olive" title="b2">V</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">Y</font>  st <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">V</font> &amp; <font color="Olive" title="b2">V</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  holds <br/> ex <font color="Olive" title="b3">W</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">W</font> &amp; <font color="Olive" title="b3">W</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b3">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">V</font> ) ) implies <font color="Maroon" title="c3">f</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:8_1"/><span class="lab"><font color="Green" title="E13">A3</font></span>: 
<font color="Maroon" title="c3">f</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">p</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font><br/>  for <font color="Olive" title="b2">V</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">Y</font>  st <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">V</font> &amp; <font color="Olive" title="b2">V</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  holds <br/> ex <font color="Olive" title="b3">W</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">W</font> &amp; <font color="Olive" title="b3">W</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b3">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">V</font> )</span><br/><span class="kw">thus </span><a NAME="E2:8_1"/>
 for <font color="Olive" title="b1">p</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font><br/>  for <font color="Olive" title="b2">V</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">Y</font>  st <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">V</font> &amp; <font color="Olive" title="b2">V</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  holds <br/> ex <font color="Olive" title="b3">W</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">W</font> &amp; <font color="Olive" title="b3">W</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b3">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">V</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">p</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">V</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">Y</font>  st <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">V</font> &amp; <font color="Olive" title="b1">V</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  holds <br/> ex <font color="Olive" title="b2">W</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">W</font> &amp; <font color="Olive" title="b2">W</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b2">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">V</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c5">V</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">Y</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">V</font> &amp; <font color="Maroon" title="c5">V</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  implies  ex <font color="Olive" title="b1">W</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">W</font> &amp; <font color="Olive" title="b1">W</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">V</font> ) )</span><br/>



<span class="kw">assume </span><a NAME="E1:8_1_1"/>
( <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">V</font> &amp; <font color="Maroon" title="c5">V</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">W</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">W</font> &amp; <font color="Olive" title="b1">W</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">V</font> )</span><br/>

<a NAME="E2:8_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E14">A4</font></span>: 
( <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c5">V</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c5">V</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:8"><span class="lab"><font color="Green" title="E12">A2</font></span></a>, <a class="txt" href="#E1:8"><span class="lab"><font color="Green" title="E11">A1</font></span></a>, <a class="txt" href="#E1:8_1"><span class="lab"><font color="Green" title="E13">A3</font></span></a>, <a class="ref" href="funct_1.html#D7" title="FUNCT_1:def.7">FUNCT_1:def 7</a>, <a class="ref" href="tops_2.html#T43" title="TOPS_2:th.43">TOPS_2:43</a></span>;<br/>
<a NAME="E3:8_1_1"/>
<font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c5">V</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">V</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T75" title="FUNCT_1:th.75">FUNCT_1:75</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:8_1_1"/>
 ex <font color="Olive" title="b1">W</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">W</font> &amp; <font color="Olive" title="b1">W</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">V</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:8_1_1"><span class="lab"><font color="Green" title="E14">A4</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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

<span class="kw">assume </span><a NAME="E4:8"/><span class="lab"><font color="Green" title="E13">A5</font></span>: 
 for <font color="Olive" title="b1">p</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font><br/>  for <font color="Olive" title="b2">V</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">Y</font>  st <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">V</font> &amp; <font color="Olive" title="b2">V</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  holds <br/> ex <font color="Olive" title="b3">W</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">W</font> &amp; <font color="Olive" title="b3">W</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b3">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">V</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">f</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> </span><br/>

<a NAME="E5:8"/>
 for <font color="Olive" title="b1">G</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">Y</font>  st <font color="Olive" title="b1">G</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  holds <br/><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Olive" title="b1">G</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">G</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">Y</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">G</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  implies <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c4">G</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:8_2"/><span class="lab"><font color="Green" title="E14">A6</font></span>: 
<font color="Maroon" title="c4">G</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c4">G</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> </span><br/>

<a NAME="E2:8_2"/>
 for <font color="Olive" title="b1">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c4">G</font> iff  ex <font color="Olive" title="b2">Q</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Olive" title="b2">Q</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Olive" title="b2">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c4">G</font> &amp; <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">Q</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c4">G</font> iff  ex <font color="Olive" title="b1">Q</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Olive" title="b1">Q</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Olive" title="b1">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c4">G</font> &amp; <font color="Maroon" title="c5">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Q</font> ) )</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_2_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:8_2_1_1"/><span class="lab"><font color="Green" title="E15">A7</font></span>: 
<font color="Maroon" title="c5">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c4">G</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">Q</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Olive" title="b1">Q</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Olive" title="b1">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c4">G</font> &amp; <font color="Maroon" title="c5">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Q</font> )</span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">p</font> = <font color="Maroon" title="c5">z</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font> ;<br/><a NAME="E3:8_2_1_1"/>
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8_2_1_1"><span class="lab"><font color="Green" title="E15">A7</font></span></a>, <a class="ref" href="funct_1.html#D7" title="FUNCT_1:def.7">FUNCT_1:def 7</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">W</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><span class="kw"> such that </span><br/><a NAME="E5:8_2_1_1"/><span class="lab"><font color="Green" title="E16">A8</font></span>: 
( <font color="Maroon" title="c6">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">W</font> &amp; <font color="Maroon" title="c7">W</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  )
 <span class="kw">and </span><br/><a NAME="E6:8_2_1_1"/><span class="lab"><font color="Green" title="E17">A9</font></span>: 
<font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c7">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:8"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="txt" href="#E1:8_2"><span class="lab"><font color="Green" title="E14">A6</font></span></a></span>;<br/><a NAME="E7:8_2_1_1"/><span class="lab"><font color="Green" title="E18">A10</font></span>: 
<font color="Maroon" title="c7">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c7">W</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:8"><span class="lab"><font color="Green" title="E12">A2</font></span></a>, <a class="ref" href="funct_1.html#T76" title="FUNCT_1:th.76">FUNCT_1:76</a></span>;<br/><a NAME="E8:8_2_1_1"/>
<font color="Maroon" title="c3">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c7">W</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c4">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:8_2_1_1"><span class="lab"><font color="Green" title="E17">A9</font></span></a>, <a class="ref" href="relat_1.html#T143" title="RELAT_1:th.143">RELAT_1:143</a></span>;<br/><span class="kw">hence </span><a NAME="E9:8_2_1_1"/>
 ex <font color="Olive" title="b1">Q</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Olive" title="b1">Q</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Olive" title="b1">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c4">G</font> &amp; <font color="Maroon" title="c5">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Q</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:8_2_1_1"><span class="lab"><font color="Green" title="E16">A8</font></span></a>, <a class="txt" href="#E7:8_2_1_1"><span class="lab"><font color="Green" title="E18">A10</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:8_2_1"/>
( <font color="Maroon" title="c5">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c4">G</font> iff  ex <font color="Olive" title="b1">Q</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Olive" title="b1">Q</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Olive" title="b1">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c4">G</font> &amp; <font color="Maroon" title="c5">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Q</font> ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:8_2"/>
<font color="Maroon" title="c3">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c4">G</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tops_1.html#T25" title="TOPS_1:th.25">TOPS_1:25</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E6:8"/>
<font color="Maroon" title="c3">f</font> 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="#E1:8"><span class="lab"><font color="Green" title="E11">A1</font></span></a>, <a class="ref" href="tops_2.html#T43" title="TOPS_2:th.43">TOPS_2:43</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
