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<span class="kw">let </span><font color="Maroon" title="c1">f</font> be   <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 <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</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">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="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> holds <br/><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="c2">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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">g</font> implies <font color="Maroon" title="c2">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="E1:12"/><span class="lab"><font color="Green" title="E10">A1</font></span>: 
<font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">g</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="c2">g</font> is  <a href="fcont_1.html#V1" title="FCONT_1:attr.1">continuous</a> </span><br/>

<a NAME="E2:12"/>
 for <font color="Olive" title="b1">x0</font> being   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</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="c2">g</font> holds <br/><font color="Maroon" title="c2">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" onclick="hs2(this)" href="javascript:()" title="12_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">x0</font> be   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</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">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="c2">g</font> implies <font color="Maroon" title="c2">g</font> <a href="fcont_1.html#R1" title="FCONT_1:pred.1">is_continuous_in</a> <font color="Maroon" title="c3">x0</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:12_1"/>
<font color="Maroon" title="c3">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="c2">g</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="c2">g</font> <a href="fcont_1.html#R1" title="FCONT_1:pred.1">is_continuous_in</a> <font color="Maroon" title="c3">x0</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c4">x0</font> = <font color="Maroon" title="c3">x0</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">x1</font> = <font color="Maroon" title="c4">x0</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="topmetr.html#T17" title="TOPMETR:th.17">TOPMETR:17</a></span>;<br/>
<a NAME="E4:12_1"/>
<font color="Maroon" title="c1">f</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c5">x1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tmap_1.html#T44" title="TMAP_1:th.44">TMAP_1:44</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:12_1"/>
<font color="Maroon" title="c2">g</font> <a href="fcont_1.html#R1" title="FCONT_1:pred.1">is_continuous_in</a> <font color="Maroon" title="c3">x0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:12"><span class="lab"><font color="Green" title="E10">A1</font></span></a>, <a class="ref" href="jordan5a.html#T6" target="_self" title="JORDAN5A:th.6">Th7</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="E3:12"/>
<font color="Maroon" title="c2">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" title="FCONT_1:def.2">FCONT_1:def 2</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/>


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