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

<span class="kw">let </span><font color="Maroon" title="c1">X</font> be   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="pre_topc.html#V1" title="PRE_TOPC:attr.1">strict</a>   <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of  <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="pscomp_1.html#NM1" title="PSCOMP_1:NM.1">RealMap</a> of <font color="Maroon" title="c1">X</font><br/>  for <font color="Olive" title="b2">g</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> holds <br/>( <font color="Olive" title="b1">f</font> is  <a href="pscomp_1.html#V1" title="PSCOMP_1:attr.1">continuous</a>  iff <font color="Olive" title="b2">g</font> is  <a href="fcont_1.html#V1" title="FCONT_1:attr.1">continuous</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">f</font> be   <a href="pscomp_1.html#NM1" title="PSCOMP_1:NM.1">RealMap</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">g</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> holds <br/>( <font color="Maroon" title="c2">f</font> is  <a href="pscomp_1.html#V1" title="PSCOMP_1:attr.1">continuous</a>  iff <font color="Olive" title="b1">g</font> is  <a href="fcont_1.html#V1" title="FCONT_1:attr.1">continuous</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">g</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> implies ( <font color="Maroon" title="c2">f</font> is  <a href="pscomp_1.html#V1" title="PSCOMP_1:attr.1">continuous</a>  iff <font color="Maroon" title="c3">g</font> is  <a href="fcont_1.html#V1" title="FCONT_1:attr.1">continuous</a>  ) )</span><br/>





<span class="kw">assume </span><a NAME="E1:5"/><span class="lab"><font color="Green" title="E6">A1</font></span>: 
<font color="Maroon" title="c3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">f</font> is  <a href="pscomp_1.html#V1" title="PSCOMP_1:attr.1">continuous</a>  iff <font color="Maroon" title="c3">g</font> is  <a href="fcont_1.html#V1" title="FCONT_1:attr.1">continuous</a>  )</span><br/>

<a NAME="E2:5"/><span class="lab"><font color="Green" title="E7">A2</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">g</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" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a>, <a class="txt" href="#E1:5"><span class="lab"><font color="Green" title="E6">A1</font></span></a></span>;<br/>
<div><span class="kw">hereby </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">g</font> is  <a href="fcont_1.html#V1" title="FCONT_1:attr.1">continuous</a>  implies <font color="Maroon" title="c2">f</font> is  <a href="pscomp_1.html#V1" title="PSCOMP_1:attr.1">continuous</a>  )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:5_1"/><span class="lab"><font color="Green" title="E8">B3</font></span>: 
<font color="Maroon" title="c2">f</font> is  <a href="pscomp_1.html#V1" title="PSCOMP_1:attr.1">continuous</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">g</font> is  <a href="fcont_1.html#V1" title="FCONT_1:attr.1">continuous</a> </span><br/><a NAME="E2:5_1"/>
 for <font color="Olive" title="b1">x0</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st <font color="Olive" title="b1">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">g</font> holds <br/><font color="Maroon" title="c3">g</font> <a href="fcont_1.html#R1" title="FCONT_1:pred.1">is_continuous_in</a> <font color="Olive" title="b1">x0</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="5_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">x0</font> be   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">g</font> implies <font color="Maroon" title="c3">g</font> <a href="fcont_1.html#R1" title="FCONT_1:pred.1">is_continuous_in</a> <font color="Maroon" title="c4">x0</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:5_1_1"/>
<font color="Maroon" title="c4">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">g</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">g</font> <a href="fcont_1.html#R1" title="FCONT_1:pred.1">is_continuous_in</a> <font color="Maroon" title="c4">x0</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">x</font> = <font color="Maroon" title="c4">x0</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</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" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a>, <a class="txt" href="#E1:5"><span class="lab"><font color="Green" title="E6">A1</font></span></a></span>;<br/>
<a NAME="E3:5_1_1"/>
 for <font color="Olive" title="b1">V</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">V</font> &amp; <font color="Olive" title="b1">V</font> is  <a href="rcomp_1.html#V3" title="RCOMP_1: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="c5">x</font> <a href="tarski.html#R2" title="TARSKI: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="c2">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E1:5_1"><span class="lab"><font color="Green" title="E8">B3</font></span></a>, <a class="ref" href="c0sp2.html#T1" onmouseover="rs('c0sp2/T1')" onmouseout="rh()">C0SP2:1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:5_1_1"/>
<font color="Maroon" title="c3">g</font> <a href="fcont_1.html#R1" title="FCONT_1:pred.1">is_continuous_in</a> <font color="Maroon" title="c4">x0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:5"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="ref" href="dualsp05.html#T3" target="_self" onmouseover="rs('dualsp05/T3')" onmouseout="rh()">Th80a</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E3:5_1"/>
<font color="Maroon" title="c3">g</font> is  <a href="fcont_1.html#V1" title="FCONT_1:attr.1">continuous</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fcont_1.html#D2" onmouseover="rs('fcont_1/D2')" onmouseout="rh()">FCONT_1:def 2</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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">assume </span><a NAME="E4:5"/><span class="lab"><font color="Green" title="E8">B5</font></span>: 
<font color="Maroon" title="c3">g</font> is  <a href="fcont_1.html#V1" title="FCONT_1:attr.1">continuous</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">f</font> is  <a href="pscomp_1.html#V1" title="PSCOMP_1:attr.1">continuous</a> </span><br/>

<a NAME="E5:5"/>
 for <font color="Olive" title="b1">x</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="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b2">V</font> &amp; <font color="Olive" title="b2">V</font> is  <a href="rcomp_1.html#V3" title="RCOMP_1: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">x</font> <a href="tarski.html#R2" title="TARSKI: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="c2">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> )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="5_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">x</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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">V</font> &amp; <font color="Olive" title="b1">V</font> is  <a href="rcomp_1.html#V3" title="RCOMP_1: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">x</font> <a href="tarski.html#R2" title="TARSKI: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="c2">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="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">V</font> &amp; <font color="Maroon" title="c5">V</font> is  <a href="rcomp_1.html#V3" title="RCOMP_1: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">x</font> <a href="tarski.html#R2" title="TARSKI: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="c2">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="E1:5_2"/><span class="lab"><font color="Green" title="E9">A6</font></span>: 
( <font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">V</font> &amp; <font color="Maroon" title="c5">V</font> is  <a href="rcomp_1.html#V3" title="RCOMP_1:attr.3">open</a>  )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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">x</font> <a href="tarski.html#R2" title="TARSKI: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="c2">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">reconsider </span><font color="Maroon" title="c6">x0</font> = <font color="Maroon" title="c4">x</font> as   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> ;<br/>
<a NAME="E3:5_2"/>
<font color="Maroon" title="c3">g</font> <a href="fcont_1.html#R1" title="FCONT_1:pred.1">is_continuous_in</a> <font color="Maroon" title="c6">x0</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:5"><span class="lab"><font color="Green" title="E8">B5</font></span></a>, <a class="ref" href="fcont_1.html#D2" onmouseover="rs('fcont_1/D2')" onmouseout="rh()">FCONT_1:def 2</a>, <a class="txt" href="#E2:5"><span class="lab"><font color="Green" title="E7">A2</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:5_2"/>
 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">x</font> <a href="tarski.html#R2" title="TARSKI: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="c2">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="#E1:5"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="txt" href="#E1:5_2"><span class="lab"><font color="Green" title="E9">A6</font></span></a>, <a class="ref" href="dualsp05.html#T3" target="_self" onmouseover="rs('dualsp05/T3')" onmouseout="rh()">Th80a</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E6:5"/>
<font color="Maroon" title="c2">f</font> is  <a href="pscomp_1.html#V1" title="PSCOMP_1:attr.1">continuous</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="c0sp2.html#T1" onmouseover="rs('c0sp2/T1')" onmouseout="rh()">C0SP2:1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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