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<b>let </b><font color="Maroon" title="c1">X</font> be   <a href="normsp_1.html#NM1" title="NORMSP_1:NM.1">RealNormSpace</a>;<br/><b>let </b><font color="Maroon" title="c2">S</font> be   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</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>let </b><font color="Maroon" title="c3">St</font> be   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</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>let </b><font color="Maroon" title="c4">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>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>assume </b><a NAME="E1:36"/><i><font color="Green" title="E25">A1</font></i>: 
( <font color="Maroon" title="c2">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">St</font> &amp; <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">xt</font> )
 ;<br/>

<a NAME="E2:36"/><b>then </b><i><font color="Green" title="E26">A2</font></i>: 
 <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c2">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c3">St</font>
 
<b>by </b><i><a class="ref" href="normsp_2.html#D4" target="_self" title="NORMSP_2:def.4">Def4</a></i>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:36_1"/><i><font color="Green" title="E27">A3</font></i>: 
<font color="Maroon" title="c3">St</font> is    <a href="yellow_8.html#M1" title="YELLOW_8:mode.1">Basis</a> of <font color="Maroon" title="c5">xt</font>
 ;<br/><a NAME="E2:36_1"/><b>then </b>
<font color="Maroon" title="c3">St</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</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="yellow_8.html#D2" title="YELLOW_8:def.2">YELLOW_8:def 2</a></i>;<br/><a NAME="E3:36_1"/><b>then </b><i><font color="Green" title="E28">A4</font></i>: 
<font color="Maroon" title="c2">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</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="txt" href="normsp_2.html#E1:36"><i><font color="Green" title="E25">A1</font></i></a>, <a class="ref" href="normsp_2.html#D4" target="_self" title="NORMSP_2:def.4">Def4</a></i>;<br/><a NAME="E4:36_1"/><i><font color="Green" title="E29">A5</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c2">S</font>
 <b>by </b><i><a class="txt" href="normsp_2.html#E1:36"><i><font color="Green" title="E25">A1</font></i></a>, <a class="txt" href="normsp_2.html#E2:36"><i><font color="Green" title="E26">A2</font></i></a>, <a class="txt" href="normsp_2.html#E1:36_1"><i><font color="Green" title="E27">A3</font></i></a>, <a class="ref" href="yellow_8.html#D2" title="YELLOW_8:def.2">YELLOW_8:def 2</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">U</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</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>assume </b><a NAME="E1:36_1_1"/><i><font color="Green" title="E30">A6</font></i>: 
( <font color="Maroon" title="c6">U</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">U</font> )
 ;<br/><b>reconsider </b><font color="Maroon" title="c7">Ut</font> = <font color="Maroon" title="c6">U</font> as  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</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="txt" href="normsp_2.html#E1:36_1_1"><i><font color="Green" title="E30">A6</font></i></a>, <a class="ref" href="normsp_2.html#D4" target="_self" title="NORMSP_2:def.4">Def4</a>, <a class="ref" href="normsp_2.html#T20" target="_self" title="NORMSP_2:th.20">Th20</a></i>;<br/><b>consider </b><font color="Maroon" title="c8">Vt</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</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> such that </b><br/><a NAME="E4:36_1_1"/><i><font color="Green" title="E31">A7</font></i>: 
( <font color="Maroon" title="c8">Vt</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">St</font> &amp; <font color="Maroon" title="c8">Vt</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">Ut</font> )
 <b>by </b><i><a class="txt" href="normsp_2.html#E1:36"><i><font color="Green" title="E25">A1</font></i></a>, <a class="txt" href="normsp_2.html#E1:36_1"><i><font color="Green" title="E27">A3</font></i></a>, <a class="txt" href="normsp_2.html#E1:36_1_1"><i><font color="Green" title="E30">A6</font></i></a>, <a class="ref" href="yellow_8.html#D2" title="YELLOW_8:def.2">YELLOW_8:def 2</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c9">V</font> = <font color="Maroon" title="c8">Vt</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</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/><b>take </b><font color="Maroon" title="c10">V</font> = <font color="Maroon" title="c9">V</font>;<br/><b>thus </b><a NAME="E6:36_1_1"/>
( <font color="Maroon" title="c10">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> &amp; <font color="Maroon" title="c10">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">U</font> )
 <b>by </b><i><a class="txt" href="normsp_2.html#E1:36"><i><font color="Green" title="E25">A1</font></i></a>, <a class="txt" href="normsp_2.html#E4:36_1_1"><i><font color="Green" title="E31">A7</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E6:36_1"/>
<font color="Maroon" title="c2">S</font> is    <a href="yellow_8.html#M1" title="YELLOW_8:mode.1">Basis</a> of <font color="Maroon" title="c4">x</font>
 <b>by </b><i><a class="txt" href="normsp_2.html#E3:36_1"><i><font color="Green" title="E28">A4</font></i></a>, <a class="txt" href="normsp_2.html#E4:36_1"><i><font color="Green" title="E29">A5</font></i></a>, <a class="ref" href="yellow_8.html#D2" title="YELLOW_8:def.2">YELLOW_8:def 2</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E4:36"/><i><font color="Green" title="E27">A8</font></i>: 
<font color="Maroon" title="c2">S</font> is    <a href="yellow_8.html#M1" title="YELLOW_8:mode.1">Basis</a> of <font color="Maroon" title="c4">x</font>
 ;<br/>

<a NAME="E5:36"/><b>then </b>
<font color="Maroon" title="c2">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</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="yellow_8.html#D2" title="YELLOW_8:def.2">YELLOW_8:def 2</a></i>;<br/>
<a NAME="E6:36"/><b>then </b><i><font color="Green" title="E28">A9</font></i>: 
<font color="Maroon" title="c3">St</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</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="txt" href="normsp_2.html#E1:36"><i><font color="Green" title="E25">A1</font></i></a>, <a class="ref" href="normsp_2.html#D4" target="_self" title="NORMSP_2:def.4">Def4</a></i>;<br/>
<a NAME="E7:36"/><i><font color="Green" title="E29">A10</font></i>: 
<font color="Maroon" title="c5">xt</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c3">St</font>
 
<b>by </b><i><a class="txt" href="normsp_2.html#E1:36"><i><font color="Green" title="E25">A1</font></i></a>, <a class="txt" href="normsp_2.html#E2:36"><i><font color="Green" title="E26">A2</font></i></a>, <a class="txt" href="normsp_2.html#E4:36"><i><font color="Green" title="E27">A8</font></i></a>, <a class="ref" href="yellow_8.html#D2" title="YELLOW_8:def.2">YELLOW_8:def 2</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">Ut</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</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>assume </b><a NAME="E1:36_2"/><i><font color="Green" title="E30">A11</font></i>: 
( <font color="Maroon" title="c6">Ut</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c5">xt</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Ut</font> )
 ;<br/><b>reconsider </b><font color="Maroon" title="c7">U</font> = <font color="Maroon" title="c6">Ut</font> as  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</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="txt" href="normsp_2.html#E1:36_2"><i><font color="Green" title="E30">A11</font></i></a>, <a class="ref" href="normsp_2.html#D4" target="_self" title="NORMSP_2:def.4">Def4</a>, <a class="ref" href="normsp_2.html#T20" target="_self" title="NORMSP_2:th.20">Th20</a></i>;<br/><b>consider </b><font color="Maroon" title="c8">V</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</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> such that </b><br/><a NAME="E4:36_2"/><i><font color="Green" title="E31">A12</font></i>: 
( <font color="Maroon" title="c8">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> &amp; <font color="Maroon" title="c8">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">U</font> )
 <b>by </b><i><a class="txt" href="normsp_2.html#E1:36"><i><font color="Green" title="E25">A1</font></i></a>, <a class="txt" href="normsp_2.html#E4:36"><i><font color="Green" title="E27">A8</font></i></a>, <a class="txt" href="normsp_2.html#E1:36_2"><i><font color="Green" title="E30">A11</font></i></a>, <a class="ref" href="yellow_8.html#D2" title="YELLOW_8:def.2">YELLOW_8:def 2</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c9">Vt</font> = <font color="Maroon" title="c8">V</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</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/><b>take </b><font color="Maroon" title="c10">Vt</font> = <font color="Maroon" title="c9">Vt</font>;<br/><b>thus </b><a NAME="E6:36_2"/>
( <font color="Maroon" title="c10">Vt</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">St</font> &amp; <font color="Maroon" title="c10">Vt</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">Ut</font> )
 <b>by </b><i><a class="txt" href="normsp_2.html#E1:36"><i><font color="Green" title="E25">A1</font></i></a>, <a class="txt" href="normsp_2.html#E4:36_2"><i><font color="Green" title="E31">A12</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E9:36"/>
<font color="Maroon" title="c3">St</font> is    <a href="yellow_8.html#M1" title="YELLOW_8:mode.1">Basis</a> of <font color="Maroon" title="c5">xt</font>
 <b>by </b><i><a class="txt" href="normsp_2.html#E6:36"><i><font color="Green" title="E28">A9</font></i></a>, <a class="txt" href="normsp_2.html#E7:36"><i><font color="Green" title="E29">A10</font></i></a>, <a class="ref" href="yellow_8.html#D2" title="YELLOW_8:def.2">YELLOW_8:def 2</a></i>;<br/>


</div>
