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<span class="kw">let </span><font color="Maroon" title="c1">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</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">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> <br/>  for <font color="Olive" title="b2">f</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>,<span class="p1">(<span class="default"><a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>  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="Olive" title="b2">f</font> &amp; <font color="Olive" title="b2">f</font> <a href="nfcont_4.html#R1" title="NFCONT_4:pred.1">is_continuous_in</a> <font color="Olive" title="b1">x0</font> holds <br/><span class="p1"><a href="nfcont_4.html#K1" title="NFCONT_4:func.1">|.</a><span class="default"><font color="Olive" title="b2">f</font></span><a href="nfcont_4.html#K1" title="NFCONT_4:func.1">.|</a></span> <a href="fcont_1.html#R1" title="FCONT_1:pred.1">is_continuous_in</a> <font color="Olive" title="b1">x0</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">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">  for <font color="Olive" title="b1">f</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>,<span class="p1">(<span class="default"><a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>  st <font color="Maroon" title="c2">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="Olive" title="b1">f</font> &amp; <font color="Olive" title="b1">f</font> <a href="nfcont_4.html#R1" title="NFCONT_4:pred.1">is_continuous_in</a> <font color="Maroon" title="c2">x0</font> holds <br/><span class="p1"><a href="nfcont_4.html#K1" title="NFCONT_4:func.1">|.</a><span class="default"><font color="Olive" title="b1">f</font></span><a href="nfcont_4.html#K1" title="NFCONT_4:func.1">.|</a></span> <a href="fcont_1.html#R1" title="FCONT_1:pred.1">is_continuous_in</a> <font color="Maroon" title="c2">x0</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">f</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>,<span class="p1">(<span class="default"><a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>; <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">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">f</font> &amp; <font color="Maroon" title="c3">f</font> <a href="nfcont_4.html#R1" title="NFCONT_4:pred.1">is_continuous_in</a> <font color="Maroon" title="c2">x0</font> implies <span class="p1"><a href="nfcont_4.html#K1" title="NFCONT_4:func.1">|.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="nfcont_4.html#K1" title="NFCONT_4:func.1">.|</a></span> <a href="fcont_1.html#R1" title="FCONT_1:pred.1">is_continuous_in</a> <font color="Maroon" title="c2">x0</font> )</span><br/>





<span class="kw">assume </span><a NAME="E1:20"/><span class="lab"><font color="Green" title="E13">AS</font></span>: 
( <font color="Maroon" title="c2">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">f</font> &amp; <font color="Maroon" title="c3">f</font> <a href="nfcont_4.html#R1" title="NFCONT_4:pred.1">is_continuous_in</a> <font color="Maroon" title="c2">x0</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="nfcont_4.html#K1" title="NFCONT_4:func.1">|.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="nfcont_4.html#K1" title="NFCONT_4:func.1">.|</a></span> <a href="fcont_1.html#R1" title="FCONT_1:pred.1">is_continuous_in</a> <font color="Maroon" title="c2">x0</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c4">g</font> = <font color="Maroon" title="c3">f</font> as   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">n</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="real_ns1.html#D4" title="REAL_NS1:def.4">REAL_NS1:def 4</a></span>;<br/>
<a NAME="E3:20"/>
<font color="Maroon" title="c4">g</font> <a href="nfcont_3.html#R1" title="NFCONT_3:pred.1">is_continuous_in</a> <font color="Maroon" title="c2">x0</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20"><span class="lab"><font color="Green" title="E13">AS</font></span></a>, <a class="ref" href="nfcont_4.html#T1" target="_self" title="NFCONT_4:th.1">Def1Th</a></span>;<br/>
<a NAME="E4:20"/><span class="kw">then </span>
<span class="p1"><a href="normsp_0.html#K3" title="NORMSP_0:func.3">||.</a><span class="default"><font color="Maroon" title="c4">g</font></span><a href="normsp_0.html#K3" title="NORMSP_0:func.3">.||</a></span> <a href="fcont_1.html#R1" title="FCONT_1:pred.1">is_continuous_in</a> <font color="Maroon" title="c2">x0</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20"><span class="lab"><font color="Green" title="E13">AS</font></span></a>, <a class="ref" href="nfcont_3.html#T14" title="NFCONT_3:th.14">NFCONT_3:14</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:20"/>
<span class="p1"><a href="nfcont_4.html#K1" title="NFCONT_4:func.1">|.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="nfcont_4.html#K1" title="NFCONT_4:func.1">.|</a></span> <a href="fcont_1.html#R1" title="FCONT_1:pred.1">is_continuous_in</a> <font color="Maroon" title="c2">x0</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nfcont_4.html#T9" target="_self" title="NFCONT_4:th.9">LM003F</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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