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<b>let </b><font color="Maroon" title="c1">Y</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="pre_topc.html#L1" title="PRE_TOPC:struct.1">TopStruct</a> ;<br/><b>let </b><font color="Maroon" title="c2">Y0</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="tsp_1.html#M2" title="TSP_1:mode.2">SubSpace</a> of <font color="Maroon" title="c1">Y</font>;<br/><b>let </b><font color="Maroon" title="c3">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">Y</font>;<br/>





<b>assume </b><a NAME="E1:37"/><i><font color="Green" title="E25">A1</font></i>: 
<font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y0</font>
 ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:37_1"/>
<font color="Maroon" title="c3">A</font> is <a href="tsp_1.html#V1" title="TSP_1:attr.1">T_0</a>
 ;<br/><a NAME="E2:37_1"/><b>then </b>
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">Y</font>  st <font color="Olive" title="b1">x</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">Y0</font> &amp; <font color="Olive" title="b2">y</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">Y0</font> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">y</font> &amp; ( for <font color="Olive" title="b3">E</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">Y</font>  holds <br/>(  not <font color="Olive" title="b3">E</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> or  not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">E</font> or <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">E</font> ) ) holds <br/> ex <font color="Olive" title="b3">F</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">Y</font> st <br/>( <font color="Olive" title="b3">F</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> &amp; not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">F</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">F</font> )
 <b>by </b><i><a class="txt" href="tsp_1.html#E1:37"><i><font color="Green" title="E25">A1</font></i></a>, <a class="ref" href="tsp_1.html#D9" target="_self" title="TSP_1:def.9">Def9</a></i>;<br/><b>hence </b><a NAME="E3:37_1"/>
<font color="Maroon" title="c2">Y0</font> is <a href="t_0topsp.html#V2" title="T_0TOPSP:attr.2">T_0</a>
 <b>by </b><i><a class="ref" href="tsp_1.html#D14" target="_self" title="TSP_1:def.14">Def14</a></i>;<br/></div>
<b>end;</b></div>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:37_2"/>
<font color="Maroon" title="c2">Y0</font> is <a href="t_0topsp.html#V2" title="T_0TOPSP:attr.2">T_0</a>
 ;<br/><a NAME="E2:37_2"/><b>then </b>
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">Y</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">y</font> &amp; ( for <font color="Olive" title="b3">E</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">Y</font>  holds <br/>(  not <font color="Olive" title="b3">E</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> or  not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">E</font> or <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">E</font> ) ) holds <br/> ex <font color="Olive" title="b3">F</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">Y</font> st <br/>( <font color="Olive" title="b3">F</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> &amp; not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">F</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">F</font> )
 <b>by </b><i><a class="txt" href="tsp_1.html#E1:37"><i><font color="Green" title="E25">A1</font></i></a>, <a class="ref" href="tsp_1.html#D14" target="_self" title="TSP_1:def.14">Def14</a></i>;<br/><b>hence </b><a NAME="E3:37_2"/>
<font color="Maroon" title="c3">A</font> is <a href="tsp_1.html#V1" title="TSP_1:attr.1">T_0</a>
 <b>by </b><i><a class="ref" href="tsp_1.html#D9" target="_self" title="TSP_1:def.9">Def9</a></i>;<br/></div>
<b>end;</b></div>


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