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<b>let </b><font color="Maroon" title="c1">T</font>, <font color="Maroon" title="c2">S</font> be   <a href="normsp_1.html#NM1" title="NORMSP_1:NM.1">RealNormSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">f1</font>, <font color="Olive" title="b2">f2</font> being  <a href="struct_0.html#NM10" title="STRUCT_0:NM.10">PartFunc</a> of <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">T</font><br/> for <font color="Olive" title="b3">x0</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">S</font>  st <font color="Olive" title="b1">f1</font> <a href="nfcont_1.html#R1" title="NFCONT_1:pred.1">is_continuous_in</a> <font color="Olive" title="b3">x0</font> &amp; <font color="Olive" title="b2">f2</font> <a href="nfcont_1.html#R1" title="NFCONT_1:pred.1">is_continuous_in</a> <font color="Olive" title="b3">x0</font> holds <br/>( <font color="Olive" title="b1">f1</font> <a href="vfunct_1.html#K1" title="VFUNCT_1:func.1">+</a> <font color="Olive" title="b2">f2</font> <a href="nfcont_1.html#R1" title="NFCONT_1:pred.1">is_continuous_in</a> <font color="Olive" title="b3">x0</font> &amp; <font color="Olive" title="b1">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Olive" title="b2">f2</font> <a href="nfcont_1.html#R1" title="NFCONT_1:pred.1">is_continuous_in</a> <font color="Olive" title="b3">x0</font> )</span><br/><b>let </b><font color="Maroon" title="c3">f1</font>, <font color="Maroon" title="c4">f2</font> be   <a href="struct_0.html#NM10" title="STRUCT_0:NM.10">PartFunc</a> of <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">T</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">x0</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">S</font>  st <font color="Maroon" title="c3">f1</font> <a href="nfcont_1.html#R1" title="NFCONT_1:pred.1">is_continuous_in</a> <font color="Olive" title="b1">x0</font> &amp; <font color="Maroon" title="c4">f2</font> <a href="nfcont_1.html#R1" title="NFCONT_1:pred.1">is_continuous_in</a> <font color="Olive" title="b1">x0</font> holds <br/>( <font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K1" title="VFUNCT_1:func.1">+</a> <font color="Maroon" title="c4">f2</font> <a href="nfcont_1.html#R1" title="NFCONT_1:pred.1">is_continuous_in</a> <font color="Olive" title="b1">x0</font> &amp; <font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c4">f2</font> <a href="nfcont_1.html#R1" title="NFCONT_1:pred.1">is_continuous_in</a> <font color="Olive" title="b1">x0</font> )</span><br/><b>let </b><font color="Maroon" title="c5">x0</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">f1</font> <a href="nfcont_1.html#R1" title="NFCONT_1:pred.1">is_continuous_in</a> <font color="Maroon" title="c5">x0</font> &amp; <font color="Maroon" title="c4">f2</font> <a href="nfcont_1.html#R1" title="NFCONT_1:pred.1">is_continuous_in</a> <font color="Maroon" title="c5">x0</font> implies ( <font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K1" title="VFUNCT_1:func.1">+</a> <font color="Maroon" title="c4">f2</font> <a href="nfcont_1.html#R1" title="NFCONT_1:pred.1">is_continuous_in</a> <font color="Maroon" title="c5">x0</font> &amp; <font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c4">f2</font> <a href="nfcont_1.html#R1" title="NFCONT_1:pred.1">is_continuous_in</a> <font color="Maroon" title="c5">x0</font> ) )</span><br/>









<b>assume </b><a NAME="E1:31"/><i><font color="Green" title="E27">A1</font></i>: 
( <font color="Maroon" title="c3">f1</font> <a href="nfcont_1.html#R1" title="NFCONT_1:pred.1">is_continuous_in</a> <font color="Maroon" title="c5">x0</font> &amp; <font color="Maroon" title="c4">f2</font> <a href="nfcont_1.html#R1" title="NFCONT_1:pred.1">is_continuous_in</a> <font color="Maroon" title="c5">x0</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K1" title="VFUNCT_1:func.1">+</a> <font color="Maroon" title="c4">f2</font> <a href="nfcont_1.html#R1" title="NFCONT_1:pred.1">is_continuous_in</a> <font color="Maroon" title="c5">x0</font> &amp; <font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c4">f2</font> <a href="nfcont_1.html#R1" title="NFCONT_1:pred.1">is_continuous_in</a> <font color="Maroon" title="c5">x0</font> )</span><br/>

<a NAME="E2:31"/><b>then </b><i><font color="Green" title="E28">A2</font></i>: 
( <font color="Maroon" title="c5">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f1</font> &amp; ( for <font color="Olive" title="b1">s1</font> being  <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c2">S</font>  st  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b1">s1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f1</font> &amp; <font color="Olive" title="b1">s1</font> is <a href="normsp_1.html#V4" title="NORMSP_1:attr.4">convergent</a> &amp;  <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <font color="Olive" title="b1">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x0</font> holds <br/>( <font color="Maroon" title="c3">f1</font> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <font color="Olive" title="b1">s1</font> is <a href="normsp_1.html#V4" title="NORMSP_1:attr.4">convergent</a> &amp; <font color="Maroon" title="c3">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <font color="Olive" title="b1">s1</font></span>)</span> ) ) )
 
<b>by </b><i><a class="ref" href="nfcont_1.html#D9" target="_self" title="NFCONT_1:def.9">Def9</a></i>;<br/>
<a NAME="E3:31"/><i><font color="Green" title="E29">A3</font></i>: 
( <font color="Maroon" title="c5">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f2</font> &amp; ( for <font color="Olive" title="b1">s1</font> being  <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c2">S</font>  st  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b1">s1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f2</font> &amp; <font color="Olive" title="b1">s1</font> is <a href="normsp_1.html#V4" title="NORMSP_1:attr.4">convergent</a> &amp;  <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <font color="Olive" title="b1">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x0</font> holds <br/>( <font color="Maroon" title="c4">f2</font> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <font color="Olive" title="b1">s1</font> is <a href="normsp_1.html#V4" title="NORMSP_1:attr.4">convergent</a> &amp; <font color="Maroon" title="c4">f2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f2</font> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <font color="Olive" title="b1">s1</font></span>)</span> ) ) )
 
<b>by </b><i><a class="txt" href="nfcont_1.html#E1:31"><i><font color="Green" title="E27">A1</font></i></a>, <a class="ref" href="nfcont_1.html#D9" target="_self" title="NFCONT_1:def.9">Def9</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1"><b>now </b></a><div class="add"><a NAME="E1:31_1"/>
<font color="Maroon" title="c5">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">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#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f2</font></span>)</span>
 <b>by </b><i><a class="txt" href="nfcont_1.html#E2:31"><i><font color="Green" title="E28">A2</font></i></a>, <a class="txt" href="nfcont_1.html#E3:31"><i><font color="Green" title="E29">A3</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E2:31_1"/><i><font color="Green" title="E30">A4</font></i>: 
<font color="Maroon" title="c5">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K1" title="VFUNCT_1:func.1">+</a> <font color="Maroon" title="c4">f2</font></span>)</span>
 <b>by </b><i><a class="ref" href="vfunct_1.html#D1" title="VFUNCT_1:def.1">VFUNCT_1:def 1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">s1</font> being  <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c2">S</font>  st  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b1">s1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K1" title="VFUNCT_1:func.1">+</a> <font color="Maroon" title="c4">f2</font></span>)</span> &amp; <font color="Olive" title="b1">s1</font> is <a href="normsp_1.html#V4" title="NORMSP_1:attr.4">convergent</a> &amp;  <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <font color="Olive" title="b1">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x0</font> holds <br/>( <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K1" title="VFUNCT_1:func.1">+</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <font color="Olive" title="b1">s1</font> is <a href="normsp_1.html#V4" title="NORMSP_1:attr.4">convergent</a> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K1" title="VFUNCT_1:func.1">+</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K1" title="VFUNCT_1:func.1">+</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <font color="Olive" title="b1">s1</font></span>)</span> )</span><br/><b>let </b><font color="Maroon" title="c6">s1</font> be   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c2">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">s1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K1" title="VFUNCT_1:func.1">+</a> <font color="Maroon" title="c4">f2</font></span>)</span> &amp; <font color="Maroon" title="c6">s1</font> is <a href="normsp_1.html#V4" title="NORMSP_1:attr.4">convergent</a> &amp;  <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <font color="Maroon" title="c6">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x0</font> implies ( <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K1" title="VFUNCT_1:func.1">+</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <font color="Maroon" title="c6">s1</font> is <a href="normsp_1.html#V4" title="NORMSP_1:attr.4">convergent</a> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K1" title="VFUNCT_1:func.1">+</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K1" title="VFUNCT_1:func.1">+</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <font color="Maroon" title="c6">s1</font></span>)</span> ) )</span><br/><b>assume </b><a NAME="E3:31_1"/><i><font color="Green" title="E31">A5</font></i>: 
(  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">s1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K1" title="VFUNCT_1:func.1">+</a> <font color="Maroon" title="c4">f2</font></span>)</span> &amp; <font color="Maroon" title="c6">s1</font> is <a href="normsp_1.html#V4" title="NORMSP_1:attr.4">convergent</a> &amp;  <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <font color="Maroon" title="c6">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x0</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K1" title="VFUNCT_1:func.1">+</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <font color="Maroon" title="c6">s1</font> is <a href="normsp_1.html#V4" title="NORMSP_1:attr.4">convergent</a> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K1" title="VFUNCT_1:func.1">+</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K1" title="VFUNCT_1:func.1">+</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <font color="Maroon" title="c6">s1</font></span>)</span> )</span><br/><a NAME="E4:31_1"/><b>then </b><i><font color="Green" title="E32">A6</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">s1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">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#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f2</font></span>)</span>
 <b>by </b><i><a class="ref" href="vfunct_1.html#D1" title="VFUNCT_1:def.1">VFUNCT_1:def 1</a></i>;<br/><a NAME="E5:31_1"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K1" title="VFUNCT_1:func.1">+</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">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#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f2</font></span>)</span>
 <b>by </b><i><a class="ref" href="vfunct_1.html#D1" title="VFUNCT_1:def.1">VFUNCT_1:def 1</a></i>;<br/><a NAME="E6:31_1"/><b>then </b>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K1" title="VFUNCT_1:func.1">+</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f1</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/><a NAME="E7:31_1"/><b>then </b>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">s1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f1</font>
 <b>by </b><i><a class="txt" href="nfcont_1.html#E3:31_1"><i><font color="Green" title="E31">A5</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><a NAME="E8:31_1"/><b>then </b><i><font color="Green" title="E33">A7</font></i>: 
( <font color="Maroon" title="c3">f1</font> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <font color="Maroon" title="c6">s1</font> is <a href="normsp_1.html#V4" title="NORMSP_1:attr.4">convergent</a> &amp; <font color="Maroon" title="c3">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <font color="Maroon" title="c6">s1</font></span>)</span> )
 <b>by </b><i><a class="txt" href="nfcont_1.html#E1:31"><i><font color="Green" title="E27">A1</font></i></a>, <a class="txt" href="nfcont_1.html#E3:31_1"><i><font color="Green" title="E31">A5</font></i></a>, <a class="ref" href="nfcont_1.html#D9" target="_self" title="NFCONT_1:def.9">Def9</a></i>;<br/><a NAME="E9:31_1"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K1" title="VFUNCT_1:func.1">+</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">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#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f2</font></span>)</span>
 <b>by </b><i><a class="ref" href="vfunct_1.html#D1" title="VFUNCT_1:def.1">VFUNCT_1:def 1</a></i>;<br/><a NAME="E10:31_1"/><b>then </b>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K1" title="VFUNCT_1:func.1">+</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f2</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/><a NAME="E11:31_1"/><b>then </b>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">s1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f2</font>
 <b>by </b><i><a class="txt" href="nfcont_1.html#E3:31_1"><i><font color="Green" title="E31">A5</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><a NAME="E12:31_1"/><b>then </b><i><font color="Green" title="E34">A8</font></i>: 
( <font color="Maroon" title="c4">f2</font> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <font color="Maroon" title="c6">s1</font> is <a href="normsp_1.html#V4" title="NORMSP_1:attr.4">convergent</a> &amp; <font color="Maroon" title="c4">f2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f2</font> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <font color="Maroon" title="c6">s1</font></span>)</span> )
 <b>by </b><i><a class="txt" href="nfcont_1.html#E1:31"><i><font color="Green" title="E27">A1</font></i></a>, <a class="txt" href="nfcont_1.html#E3:31_1"><i><font color="Green" title="E31">A5</font></i></a>, <a class="ref" href="nfcont_1.html#D9" target="_self" title="NFCONT_1:def.9">Def9</a></i>;<br/><a NAME="E13:31_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <font color="Maroon" title="c6">s1</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f2</font> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <font color="Maroon" title="c6">s1</font></span>)</span> is <a href="normsp_1.html#V4" title="NORMSP_1:attr.4">convergent</a>
 <b>by </b><i><a class="txt" href="nfcont_1.html#E8:31_1"><i><font color="Green" title="E33">A7</font></i></a>, <a class="ref" href="normsp_1.html#T34" title="NORMSP_1:th.34">NORMSP_1:34</a></i>;<br/><b>hence </b><a NAME="E14:31_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K1" title="VFUNCT_1:func.1">+</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <font color="Maroon" title="c6">s1</font> is <a href="normsp_1.html#V4" title="NORMSP_1:attr.4">convergent</a>
 <b>by </b><i><a class="txt" href="nfcont_1.html#E4:31_1"><i><font color="Green" title="E32">A6</font></i></a>, <a class="ref" href="nfcont_1.html#T19" target="_self" title="NFCONT_1:th.19">Th19</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K1" title="VFUNCT_1:func.1">+</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K1" title="VFUNCT_1:func.1">+</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <font color="Maroon" title="c6">s1</font></span>)</span></span><br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K1" title="VFUNCT_1:func.1">+</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">x0</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">x0</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">x0</font></span>)</span>
<b>by </b><i><a class="txt" href="nfcont_1.html#E2:31_1"><i><font color="Green" title="E30">A4</font></i></a>, <a class="ref" href="vfunct_1.html#D1" title="VFUNCT_1:def.1">VFUNCT_1:def 1</a></i>
<br/>.= 
 <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <font color="Maroon" title="c6">s1</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f2</font> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <font color="Maroon" title="c6">s1</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="nfcont_1.html#E8:31_1"><i><font color="Green" title="E33">A7</font></i></a>, <a class="txt" href="nfcont_1.html#E12:31_1"><i><font color="Green" title="E34">A8</font></i></a>, <a class="ref" href="normsp_1.html#T42" title="NORMSP_1:th.42">NORMSP_1:42</a></i>
<br/>.= 
 <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K1" title="VFUNCT_1:func.1">+</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <font color="Maroon" title="c6">s1</font></span>)</span>
<b>by </b><i><a class="txt" href="nfcont_1.html#E4:31_1"><i><font color="Green" title="E32">A6</font></i></a>, <a class="ref" href="nfcont_1.html#T19" target="_self" title="NFCONT_1:th.19">Th19</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:31"/>
<font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K1" title="VFUNCT_1:func.1">+</a> <font color="Maroon" title="c4">f2</font> <a href="nfcont_1.html#R1" title="NFCONT_1:pred.1">is_continuous_in</a> <font color="Maroon" title="c5">x0</font>
 <b>by </b><i><a class="ref" href="nfcont_1.html#D9" target="_self" title="NFCONT_1:def.9">Def9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c4">f2</font> <a href="nfcont_1.html#R1" title="NFCONT_1:pred.1">is_continuous_in</a> <font color="Maroon" title="c5">x0</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2"><b>now </b></a><div class="add"><a NAME="E1:31_2"/>
<font color="Maroon" title="c5">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">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#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f2</font></span>)</span>
 <b>by </b><i><a class="txt" href="nfcont_1.html#E2:31"><i><font color="Green" title="E28">A2</font></i></a>, <a class="txt" href="nfcont_1.html#E3:31"><i><font color="Green" title="E29">A3</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E2:31_2"/><i><font color="Green" title="E30">A9</font></i>: 
<font color="Maroon" title="c5">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c4">f2</font></span>)</span>
 <b>by </b><i><a class="ref" href="vfunct_1.html#D2" title="VFUNCT_1:def.2">VFUNCT_1:def 2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">s1</font> being  <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c2">S</font>  st  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b1">s1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c4">f2</font></span>)</span> &amp; <font color="Olive" title="b1">s1</font> is <a href="normsp_1.html#V4" title="NORMSP_1:attr.4">convergent</a> &amp;  <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <font color="Olive" title="b1">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x0</font> holds <br/>( <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <font color="Olive" title="b1">s1</font> is <a href="normsp_1.html#V4" title="NORMSP_1:attr.4">convergent</a> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <font color="Olive" title="b1">s1</font></span>)</span> )</span><br/><b>let </b><font color="Maroon" title="c6">s1</font> be   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c2">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">s1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c4">f2</font></span>)</span> &amp; <font color="Maroon" title="c6">s1</font> is <a href="normsp_1.html#V4" title="NORMSP_1:attr.4">convergent</a> &amp;  <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <font color="Maroon" title="c6">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x0</font> implies ( <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <font color="Maroon" title="c6">s1</font> is <a href="normsp_1.html#V4" title="NORMSP_1:attr.4">convergent</a> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <font color="Maroon" title="c6">s1</font></span>)</span> ) )</span><br/><b>assume </b><a NAME="E3:31_2"/><i><font color="Green" title="E31">A10</font></i>: 
(  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">s1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c4">f2</font></span>)</span> &amp; <font color="Maroon" title="c6">s1</font> is <a href="normsp_1.html#V4" title="NORMSP_1:attr.4">convergent</a> &amp;  <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <font color="Maroon" title="c6">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x0</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <font color="Maroon" title="c6">s1</font> is <a href="normsp_1.html#V4" title="NORMSP_1:attr.4">convergent</a> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <font color="Maroon" title="c6">s1</font></span>)</span> )</span><br/><a NAME="E4:31_2"/><b>then </b><i><font color="Green" title="E32">A11</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">s1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">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#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f2</font></span>)</span>
 <b>by </b><i><a class="ref" href="vfunct_1.html#D2" title="VFUNCT_1:def.2">VFUNCT_1:def 2</a></i>;<br/><a NAME="E5:31_2"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">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#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f2</font></span>)</span>
 <b>by </b><i><a class="ref" href="vfunct_1.html#D2" title="VFUNCT_1:def.2">VFUNCT_1:def 2</a></i>;<br/><a NAME="E6:31_2"/><b>then </b>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f1</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/><a NAME="E7:31_2"/><b>then </b>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">s1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f1</font>
 <b>by </b><i><a class="txt" href="nfcont_1.html#E3:31_2"><i><font color="Green" title="E31">A10</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><a NAME="E8:31_2"/><b>then </b><i><font color="Green" title="E33">A12</font></i>: 
( <font color="Maroon" title="c3">f1</font> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <font color="Maroon" title="c6">s1</font> is <a href="normsp_1.html#V4" title="NORMSP_1:attr.4">convergent</a> &amp; <font color="Maroon" title="c3">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <font color="Maroon" title="c6">s1</font></span>)</span> )
 <b>by </b><i><a class="txt" href="nfcont_1.html#E1:31"><i><font color="Green" title="E27">A1</font></i></a>, <a class="txt" href="nfcont_1.html#E3:31_2"><i><font color="Green" title="E31">A10</font></i></a>, <a class="ref" href="nfcont_1.html#D9" target="_self" title="NFCONT_1:def.9">Def9</a></i>;<br/><a NAME="E9:31_2"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">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#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f2</font></span>)</span>
 <b>by </b><i><a class="ref" href="vfunct_1.html#D2" title="VFUNCT_1:def.2">VFUNCT_1:def 2</a></i>;<br/><a NAME="E10:31_2"/><b>then </b>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f2</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/><a NAME="E11:31_2"/><b>then </b>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">s1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f2</font>
 <b>by </b><i><a class="txt" href="nfcont_1.html#E3:31_2"><i><font color="Green" title="E31">A10</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><a NAME="E12:31_2"/><b>then </b><i><font color="Green" title="E34">A13</font></i>: 
( <font color="Maroon" title="c4">f2</font> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <font color="Maroon" title="c6">s1</font> is <a href="normsp_1.html#V4" title="NORMSP_1:attr.4">convergent</a> &amp; <font color="Maroon" title="c4">f2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f2</font> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <font color="Maroon" title="c6">s1</font></span>)</span> )
 <b>by </b><i><a class="txt" href="nfcont_1.html#E1:31"><i><font color="Green" title="E27">A1</font></i></a>, <a class="txt" href="nfcont_1.html#E3:31_2"><i><font color="Green" title="E31">A10</font></i></a>, <a class="ref" href="nfcont_1.html#D9" target="_self" title="NFCONT_1:def.9">Def9</a></i>;<br/><a NAME="E13:31_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <font color="Maroon" title="c6">s1</font></span>)</span> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f2</font> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <font color="Maroon" title="c6">s1</font></span>)</span> is <a href="normsp_1.html#V4" title="NORMSP_1:attr.4">convergent</a>
 <b>by </b><i><a class="txt" href="nfcont_1.html#E8:31_2"><i><font color="Green" title="E33">A12</font></i></a>, <a class="ref" href="normsp_1.html#T35" title="NORMSP_1:th.35">NORMSP_1:35</a></i>;<br/><b>hence </b><a NAME="E14:31_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <font color="Maroon" title="c6">s1</font> is <a href="normsp_1.html#V4" title="NORMSP_1:attr.4">convergent</a>
 <b>by </b><i><a class="txt" href="nfcont_1.html#E4:31_2"><i><font color="Green" title="E32">A11</font></i></a>, <a class="ref" href="nfcont_1.html#T19" target="_self" title="NFCONT_1:th.19">Th19</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <font color="Maroon" title="c6">s1</font></span>)</span></span><br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">x0</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">x0</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">x0</font></span>)</span>
<b>by </b><i><a class="txt" href="nfcont_1.html#E2:31_2"><i><font color="Green" title="E30">A9</font></i></a>, <a class="ref" href="vfunct_1.html#D2" title="VFUNCT_1:def.2">VFUNCT_1:def 2</a></i>
<br/>.= 
 <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <font color="Maroon" title="c6">s1</font></span>)</span> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f2</font> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <font color="Maroon" title="c6">s1</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="nfcont_1.html#E8:31_2"><i><font color="Green" title="E33">A12</font></i></a>, <a class="txt" href="nfcont_1.html#E12:31_2"><i><font color="Green" title="E34">A13</font></i></a>, <a class="ref" href="normsp_1.html#T43" title="NORMSP_1:th.43">NORMSP_1:43</a></i>
<br/>.= 
 <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <font color="Maroon" title="c6">s1</font></span>)</span>
<b>by </b><i><a class="txt" href="nfcont_1.html#E4:31_2"><i><font color="Green" title="E32">A11</font></i></a>, <a class="ref" href="nfcont_1.html#T19" target="_self" title="NFCONT_1:th.19">Th19</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:31"/>
<font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c4">f2</font> <a href="nfcont_1.html#R1" title="NFCONT_1:pred.1">is_continuous_in</a> <font color="Maroon" title="c5">x0</font>
 <b>by </b><i><a class="ref" href="nfcont_1.html#D9" target="_self" title="NFCONT_1:def.9">Def9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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