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<span class="kw">let </span><font color="Maroon" title="c1">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</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">x0</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><br/>  for <font color="Olive" title="b2">f1</font>, <font color="Olive" title="b3">f2</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> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b2">f1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b3">f2</font></span>)</span> &amp; <font color="Olive" title="b2">f1</font> <a href="nfcont_4.html#R1" title="NFCONT_4:pred.1">is_continuous_in</a> <font color="Olive" title="b1">x0</font> &amp; <font color="Olive" title="b3">f2</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/><font color="Olive" title="b2">f1</font> <a href="integr15.html#K7" title="INTEGR15:func.7">+</a> <font color="Olive" title="b3">f2</font> <a href="nfcont_4.html#R1" title="NFCONT_4: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#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">  for <font color="Olive" title="b1">f1</font>, <font color="Olive" title="b2">f2</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> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b1">f1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b2">f2</font></span>)</span> &amp; <font color="Olive" title="b1">f1</font> <a href="nfcont_4.html#R1" title="NFCONT_4:pred.1">is_continuous_in</a> <font color="Maroon" title="c2">x0</font> &amp; <font color="Olive" title="b2">f2</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/><font color="Olive" title="b1">f1</font> <a href="integr15.html#K7" title="INTEGR15:func.7">+</a> <font color="Olive" title="b2">f2</font> <a href="nfcont_4.html#R1" title="NFCONT_4: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">f1</font>, <font color="Maroon" title="c4">f2</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" 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">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">f1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">f2</font></span>)</span> &amp; <font color="Maroon" title="c3">f1</font> <a href="nfcont_4.html#R1" title="NFCONT_4:pred.1">is_continuous_in</a> <font color="Maroon" title="c2">x0</font> &amp; <font color="Maroon" title="c4">f2</font> <a href="nfcont_4.html#R1" title="NFCONT_4:pred.1">is_continuous_in</a> <font color="Maroon" title="c2">x0</font> implies <font color="Maroon" title="c3">f1</font> <a href="integr15.html#K7" title="INTEGR15:func.7">+</a> <font color="Maroon" title="c4">f2</font> <a href="nfcont_4.html#R1" title="NFCONT_4:pred.1">is_continuous_in</a> <font color="Maroon" title="c2">x0</font> )</span><br/>







<span class="kw">assume </span><a NAME="E1:16"/><span class="lab"><font color="Green" title="E11">A1</font></span>: 
( <font color="Maroon" title="c2">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">f1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">f2</font></span>)</span> &amp; <font color="Maroon" title="c3">f1</font> <a href="nfcont_4.html#R1" title="NFCONT_4:pred.1">is_continuous_in</a> <font color="Maroon" title="c2">x0</font> &amp; <font color="Maroon" title="c4">f2</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" 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">f1</font> <a href="integr15.html#K7" title="INTEGR15:func.7">+</a> <font color="Maroon" title="c4">f2</font> <a href="nfcont_4.html#R1" title="NFCONT_4:pred.1">is_continuous_in</a> <font color="Maroon" title="c2">x0</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c5">g1</font> = <font color="Maroon" title="c3">f1</font>, <font color="Maroon" title="c6">g2</font> = <font color="Maroon" title="c4">f2</font> as   <a href="struct_0.html#NM11" title="STRUCT_0:NM.11">PartFunc</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>,<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" onmouseover="rs('real_ns1/D4')" onmouseout="rh()">REAL_NS1:def 4</a></span>;<br/>
<a NAME="E3:16"/><span class="lab"><font color="Green" title="E12">A2</font></span>: 
( <font color="Maroon" title="c5">g1</font> <a href="vfunct_1.html#K6" title="VFUNCT_1:func.6">+</a> <font color="Maroon" title="c6">g2</font> <a href="nfcont_3.html#R1" title="NFCONT_3:pred.1">is_continuous_in</a> <font color="Maroon" title="c2">x0</font> &amp; <font color="Maroon" title="c5">g1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c6">g2</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="#E1:16"><span class="lab"><font color="Green" title="E11">A1</font></span></a>, <a class="ref" href="nfcont_3.html#T12" onmouseover="rs('nfcont_3/T12')" onmouseout="rh()">NFCONT_3:12</a></span>;<br/>
<a NAME="E4:16"/>
<font color="Maroon" title="c5">g1</font> <a href="vfunct_1.html#K6" title="VFUNCT_1:func.6">+</a> <font color="Maroon" title="c6">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f1</font> <a href="integr15.html#K7" title="INTEGR15:func.7">+</a> <font color="Maroon" title="c4">f2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nfcont_4.html#T5" target="_self" onmouseover="rs('nfcont_4/T5')" onmouseout="rh()">Th5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:16"/>
<font color="Maroon" title="c3">f1</font> <a href="integr15.html#K7" title="INTEGR15:func.7">+</a> <font color="Maroon" title="c4">f2</font> <a href="nfcont_4.html#R1" title="NFCONT_4: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="#E3:16"><span class="lab"><font color="Green" title="E12">A2</font></span></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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