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<b>let </b><font color="Maroon" title="c1">X</font>, <font color="Maroon" title="c2">Y</font> be   <a href="normsp_1.html#NM1" title="NORMSP_1:NM.1">RealNormSpace</a>;<br/><b>let </b><font color="Maroon" title="c3">f</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K3" title="NORMSP_2:func.3">TopSpaceNorm</a> <font color="Maroon" title="c1">X</font></span>)</span>,<span class="p1">(<span class="default"><a href="normsp_2.html#K3" title="NORMSP_2:func.3">TopSpaceNorm</a> <font color="Maroon" title="c2">Y</font></span>)</span>;<br/><b>let </b><font color="Maroon" title="c4">ft</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K4" title="NORMSP_2:func.4">LinearTopSpaceNorm</a> <font color="Maroon" title="c1">X</font></span>)</span>,<span class="p1">(<span class="default"><a href="normsp_2.html#K4" title="NORMSP_2:func.4">LinearTopSpaceNorm</a> <font color="Maroon" title="c2">Y</font></span>)</span>;<br/>





<b>assume </b><a NAME="E1:49"/><i><font color="Green" title="E31">A1</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c4">ft</font>
 ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:49_1"/><i><font color="Green" title="E32">A2</font></i>: 
<font color="Maroon" title="c3">f</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">xt</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K4" title="NORMSP_2:func.4">LinearTopSpaceNorm</a> <font color="Maroon" title="c1">X</font></span>)</span>;<br/><b>reconsider </b><font color="Maroon" title="c6">x</font> = <font color="Maroon" title="c5">xt</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K3" title="NORMSP_2:func.3">TopSpaceNorm</a> <font color="Maroon" title="c1">X</font></span>)</span> <b>by </b><i><a class="ref" href="normsp_2.html#D4" target="_self" title="NORMSP_2:def.4">Def4</a></i>;<br/><a NAME="E2:49_1_1"/>
<font color="Maroon" title="c3">f</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c6">x</font>
 <b>by </b><i><a class="txt" href="normsp_2.html#E1:49_1"><i><font color="Green" title="E32">A2</font></i></a>, <a class="ref" href="tmap_1.html#T49" title="TMAP_1:th.49">TMAP_1:49</a></i>;<br/><b>hence </b><a NAME="E3:49_1_1"/>
<font color="Maroon" title="c4">ft</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c5">xt</font>
 <b>by </b><i><a class="ref" href="normsp_2.html#T35" target="_self" title="NORMSP_2:th.35">Th35</a>, <a class="txt" href="normsp_2.html#E1:49"><i><font color="Green" title="E31">A1</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:49_1"/>
<font color="Maroon" title="c4">ft</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>by </b><i><a class="ref" href="tmap_1.html#T49" title="TMAP_1:th.49">TMAP_1:49</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E3:49"/><i><font color="Green" title="E32">A3</font></i>: 
<font color="Maroon" title="c4">ft</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">x</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K3" title="NORMSP_2:func.3">TopSpaceNorm</a> <font color="Maroon" title="c1">X</font></span>)</span>;<br/><b>reconsider </b><font color="Maroon" title="c6">xt</font> = <font color="Maroon" title="c5">x</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K4" title="NORMSP_2:func.4">LinearTopSpaceNorm</a> <font color="Maroon" title="c1">X</font></span>)</span> <b>by </b><i><a class="ref" href="normsp_2.html#D4" target="_self" title="NORMSP_2:def.4">Def4</a></i>;<br/><a NAME="E2:49_2"/>
<font color="Maroon" title="c4">ft</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c6">xt</font>
 <b>by </b><i><a class="txt" href="normsp_2.html#E3:49"><i><font color="Green" title="E32">A3</font></i></a>, <a class="ref" href="tmap_1.html#T49" title="TMAP_1:th.49">TMAP_1:49</a></i>;<br/><b>hence </b><a NAME="E3:49_2"/>
<font color="Maroon" title="c3">f</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c5">x</font>
 <b>by </b><i><a class="ref" href="normsp_2.html#T35" target="_self" title="NORMSP_2:th.35">Th35</a>, <a class="txt" href="normsp_2.html#E1:49"><i><font color="Green" title="E31">A1</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:49"/>
<font color="Maroon" title="c3">f</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>by </b><i><a class="ref" href="tmap_1.html#T49" title="TMAP_1:th.49">TMAP_1:49</a></i>;<br/>


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