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<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">X</font> be   <a href="normsp_1.html#NM1" title="NORMSP_1:NM.1">RealNormSpace</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">S</font> being   <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/>  for <font color="Olive" title="b2">St</font> being   <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/>  for <font color="Olive" title="b3">x</font> being   <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/>  for <font color="Olive" title="b4">xt</font> being   <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>  st <font color="Olive" title="b1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">St</font> &amp; <font color="Olive" title="b3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">xt</font> holds <br/>( <font color="Olive" title="b2">St</font> is   <a href="yellow_8.html#NM1" title="YELLOW_8:NM.1">Basis</a> of  iff <font color="Olive" title="b1">S</font> is   <a href="yellow_8.html#NM1" title="YELLOW_8:NM.1">Basis</a> of  )</span><br/><span class="kw">let </span><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>; <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">St</font> being   <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/>  for <font color="Olive" title="b2">x</font> being   <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/>  for <font color="Olive" title="b3">xt</font> being   <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>  st <font color="Maroon" title="c2">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">St</font> &amp; <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">xt</font> holds <br/>( <font color="Olive" title="b1">St</font> is   <a href="yellow_8.html#NM1" title="YELLOW_8:NM.1">Basis</a> of  iff <font color="Maroon" title="c2">S</font> is   <a href="yellow_8.html#NM1" title="YELLOW_8:NM.1">Basis</a> of  )</span><br/><span class="kw">let </span><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>; <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">x</font> being   <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/>  for <font color="Olive" title="b2">xt</font> being   <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>  st <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="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">xt</font> holds <br/>( <font color="Maroon" title="c3">St</font> is   <a href="yellow_8.html#NM1" title="YELLOW_8:NM.1">Basis</a> of  iff <font color="Maroon" title="c2">S</font> is   <a href="yellow_8.html#NM1" title="YELLOW_8:NM.1">Basis</a> of  )</span><br/><span class="kw">let </span><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>; <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">xt</font> being   <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>  st <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="Olive" title="b1">xt</font> holds <br/>( <font color="Maroon" title="c3">St</font> is   <a href="yellow_8.html#NM1" title="YELLOW_8:NM.1">Basis</a> of  iff <font color="Maroon" title="c2">S</font> is   <a href="yellow_8.html#NM1" title="YELLOW_8:NM.1">Basis</a> of  )</span><br/><span class="kw">let </span><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>; <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">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> implies ( <font color="Maroon" title="c3">St</font> is   <a href="yellow_8.html#NM1" title="YELLOW_8:NM.1">Basis</a> of  iff <font color="Maroon" title="c2">S</font> is   <a href="yellow_8.html#NM1" title="YELLOW_8:NM.1">Basis</a> of  ) )</span><br/>









<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:35"/><span class="lab"><font color="Green" title="E25">A1</font></span>: 
<font color="Maroon" title="c2">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">St</font>
 <span class="kw">and </span><br/><a NAME="E2:35"/><span class="lab"><font color="Green" title="E26">A2</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">xt</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">St</font> is   <a href="yellow_8.html#NM1" title="YELLOW_8:NM.1">Basis</a> of  iff <font color="Maroon" title="c2">S</font> is   <a href="yellow_8.html#NM1" title="YELLOW_8:NM.1">Basis</a> of  )</span><br/>

<a NAME="E3:35"/><span class="lab"><font color="Green" title="E27">A3</font></span>: 
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35"><span class="lab"><font color="Green" title="E25">A1</font></span></a>, <a class="ref" href="normsp_2.html#D4" target="_self" onmouseover="rs('normsp_2/D4')" onmouseout="rh()">Def4</a></span>;<br/>
<div><span class="kw">hereby </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">S</font> is   <a href="yellow_8.html#NM1" title="YELLOW_8:NM.1">Basis</a> of  implies <font color="Maroon" title="c3">St</font> is   <a href="yellow_8.html#NM1" title="YELLOW_8:NM.1">Basis</a> of  )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:35_1"/><span class="lab"><font color="Green" title="E28">A4</font></span>: 
<font color="Maroon" title="c3">St</font> is   <a href="yellow_8.html#NM1" title="YELLOW_8:NM.1">Basis</a> of 
 ; <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">S</font> is   <a href="yellow_8.html#NM1" title="YELLOW_8:NM.1">Basis</a> of </span><br/><a NAME="E2:35_1"/><span class="kw">then </span>
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tops_2.html#T64" onmouseover="rs('tops_2/T64')" onmouseout="rh()">TOPS_2:64</a></span>;<br/><a NAME="E3:35_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A5</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35"><span class="lab"><font color="Green" title="E25">A1</font></span></a>, <a class="ref" href="normsp_2.html#D4" target="_self" onmouseover="rs('normsp_2/D4')" onmouseout="rh()">Def4</a></span>;<br/><div><span class="lab"><font color="Green" title="E30">A6</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="35_1_1"><span class="kw">now </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">  for <font color="Olive" title="b1">U</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>  st <font color="Olive" title="b1">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="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">U</font> holds <br/> ex <font color="Olive" title="b2">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> st <br/>( <font color="Olive" title="b2">V</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">S</font> &amp; <font color="Olive" title="b2">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">U</font> )</span></a><div class="add"><span class="kw">let </span><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>; <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="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="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">U</font> implies  ex <font color="Olive" title="b1">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> st <br/>( <font color="Olive" title="b1">V</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">S</font> &amp; <font color="Olive" title="b1">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">U</font> ) )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:35_1_1"/><span class="lab"><font color="Green" title="E31">A7</font></span>: 
<font color="Maroon" title="c6">U</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 <span class="kw">and </span><br/><a NAME="E2:35_1_1"/><span class="lab"><font color="Green" title="E32">A8</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">U</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">  ex <font color="Olive" title="b1">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> st <br/>( <font color="Olive" title="b1">V</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">S</font> &amp; <font color="Olive" title="b1">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">U</font> )</span><br/><span class="kw">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1"><span class="lab"><font color="Green" title="E31">A7</font></span></a>, <a class="ref" href="normsp_2.html#D4" target="_self" onmouseover="rs('normsp_2/D4')" onmouseout="rh()">Def4</a>, <a class="ref" href="normsp_2.html#T20" target="_self" onmouseover="rs('normsp_2/T20')" onmouseout="rh()">Th20</a></span>;<br/><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E5:35_1_1"/><span class="lab"><font color="Green" title="E33">A9</font></span>: 
( <font color="Maroon" title="c8">Vt</font> <a href="tarski.html#R2" title="TARSKI: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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35"><span class="lab"><font color="Green" title="E26">A2</font></span></a>, <a class="txt" href="#E1:35_1"><span class="lab"><font color="Green" title="E28">A4</font></span></a>, <a class="txt" href="#E2:35_1_1"><span class="lab"><font color="Green" title="E32">A8</font></span></a>, <a class="ref" href="yellow_8.html#D1" onmouseover="rs('yellow_8/D1')" onmouseout="rh()">YELLOW_8:def 1</a></span>;<br/><span class="kw">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="ref" href="normsp_2.html#D4" target="_self" onmouseover="rs('normsp_2/D4')" onmouseout="rh()">Def4</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c10">V</font> = <font color="Maroon" title="c9">V</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="c10">V</font> <a href="tarski.html#R2" title="TARSKI: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> )</span><br/><span class="kw">thus </span><a NAME="E7:35_1_1"/>
( <font color="Maroon" title="c10">V</font> <a href="tarski.html#R2" title="TARSKI: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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35"><span class="lab"><font color="Green" title="E25">A1</font></span></a>, <a class="txt" href="#E5:35_1_1"><span class="lab"><font color="Green" title="E33">A9</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/></div><span class="kw">end;</span></div><a NAME="E5:35_1"/>
<font color="Maroon" title="c4">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c2">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35"><span class="lab"><font color="Green" title="E26">A2</font></span></a>, <a class="txt" href="#E3:35"><span class="lab"><font color="Green" title="E27">A3</font></span></a>, <a class="txt" href="#E1:35_1"><span class="lab"><font color="Green" title="E28">A4</font></span></a>, <a class="ref" href="yellow_8.html#D1" onmouseover="rs('yellow_8/D1')" onmouseout="rh()">YELLOW_8:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E6:35_1"/>
<font color="Maroon" title="c2">S</font> is   <a href="yellow_8.html#NM1" title="YELLOW_8:NM.1">Basis</a> of 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1"><span class="lab"><font color="Green" title="E29">A5</font></span></a>, <a class="txt" href="#E4:35_1"><span class="lab"><font color="Green" title="E30">A6</font></span></a>, <a class="ref" href="tops_2.html#T64" onmouseover="rs('tops_2/T64')" onmouseout="rh()">TOPS_2:64</a>, <a class="ref" href="yellow_8.html#D1" onmouseover="rs('yellow_8/D1')" onmouseout="rh()">YELLOW_8:def 1</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/></div>
<span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E5:35"/><span class="lab"><font color="Green" title="E28">A10</font></span>: 
<font color="Maroon" title="c2">S</font> is   <a href="yellow_8.html#NM1" title="YELLOW_8:NM.1">Basis</a> of 
 ; <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">St</font> is   <a href="yellow_8.html#NM1" title="YELLOW_8:NM.1">Basis</a> of </span><br/>

<a NAME="E6:35"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tops_2.html#T64" onmouseover="rs('tops_2/T64')" onmouseout="rh()">TOPS_2:64</a></span>;<br/>
<a NAME="E7:35"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A11</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35"><span class="lab"><font color="Green" title="E25">A1</font></span></a>, <a class="ref" href="normsp_2.html#D4" target="_self" onmouseover="rs('normsp_2/D4')" onmouseout="rh()">Def4</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E30">A12</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="35_2"><span class="kw">now </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">  for <font color="Olive" title="b1">Ut</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>  st <font color="Olive" title="b1">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="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">Ut</font> holds <br/> ex <font color="Olive" title="b2">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> st <br/>( <font color="Olive" title="b2">Vt</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">St</font> &amp; <font color="Olive" title="b2">Vt</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">Ut</font> )</span></a><div class="add"><span class="kw">let </span><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>; <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="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="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">Ut</font> implies  ex <font color="Olive" title="b1">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> st <br/>( <font color="Olive" title="b1">Vt</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">St</font> &amp; <font color="Olive" title="b1">Vt</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">Ut</font> ) )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:35_2"/><span class="lab"><font color="Green" title="E31">A13</font></span>: 
<font color="Maroon" title="c6">Ut</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 <span class="kw">and </span><br/><a NAME="E2:35_2"/><span class="lab"><font color="Green" title="E32">A14</font></span>: 
<font color="Maroon" title="c5">xt</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">Ut</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">  ex <font color="Olive" title="b1">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> st <br/>( <font color="Olive" title="b1">Vt</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">St</font> &amp; <font color="Olive" title="b1">Vt</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">Ut</font> )</span><br/><span class="kw">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_2"><span class="lab"><font color="Green" title="E31">A13</font></span></a>, <a class="ref" href="normsp_2.html#D4" target="_self" onmouseover="rs('normsp_2/D4')" onmouseout="rh()">Def4</a>, <a class="ref" href="normsp_2.html#T20" target="_self" onmouseover="rs('normsp_2/T20')" onmouseout="rh()">Th20</a></span>;<br/><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E5:35_2"/><span class="lab"><font color="Green" title="E33">A15</font></span>: 
( <font color="Maroon" title="c8">V</font> <a href="tarski.html#R2" title="TARSKI: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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35"><span class="lab"><font color="Green" title="E26">A2</font></span></a>, <a class="txt" href="#E5:35"><span class="lab"><font color="Green" title="E28">A10</font></span></a>, <a class="txt" href="#E2:35_2"><span class="lab"><font color="Green" title="E32">A14</font></span></a>, <a class="ref" href="yellow_8.html#D1" onmouseover="rs('yellow_8/D1')" onmouseout="rh()">YELLOW_8:def 1</a></span>;<br/><span class="kw">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="ref" href="normsp_2.html#D4" target="_self" onmouseover="rs('normsp_2/D4')" onmouseout="rh()">Def4</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c10">Vt</font> = <font color="Maroon" title="c9">Vt</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="c10">Vt</font> <a href="tarski.html#R2" title="TARSKI: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> )</span><br/><span class="kw">thus </span><a NAME="E7:35_2"/>
( <font color="Maroon" title="c10">Vt</font> <a href="tarski.html#R2" title="TARSKI: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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35"><span class="lab"><font color="Green" title="E25">A1</font></span></a>, <a class="txt" href="#E5:35_2"><span class="lab"><font color="Green" title="E33">A15</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/></div><span class="kw">end;</span></div>
<a NAME="E9:35"/>
<font color="Maroon" title="c5">xt</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c3">St</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35"><span class="lab"><font color="Green" title="E26">A2</font></span></a>, <a class="txt" href="#E3:35"><span class="lab"><font color="Green" title="E27">A3</font></span></a>, <a class="txt" href="#E5:35"><span class="lab"><font color="Green" title="E28">A10</font></span></a>, <a class="ref" href="yellow_8.html#D1" onmouseover="rs('yellow_8/D1')" onmouseout="rh()">YELLOW_8:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:35"/>
<font color="Maroon" title="c3">St</font> is   <a href="yellow_8.html#NM1" title="YELLOW_8:NM.1">Basis</a> of 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:35"><span class="lab"><font color="Green" title="E29">A11</font></span></a>, <a class="txt" href="#E8:35"><span class="lab"><font color="Green" title="E30">A12</font></span></a>, <a class="ref" href="tops_2.html#T64" onmouseover="rs('tops_2/T64')" onmouseout="rh()">TOPS_2:64</a>, <a class="ref" href="yellow_8.html#D1" onmouseover="rs('yellow_8/D1')" onmouseout="rh()">YELLOW_8:def 1</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/>


</div>
