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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a>  <a href="pre_topc.html#L1">TopStruct</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<a NAME="E1:129"/><i><font color="Green">E66</font></i>: 
( the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="tex_4.html#K6" target="_self">MaxADSspace</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="tex_4.html#K3" target="_self">MaxADSet</a> <font color="Maroon">c<sub>2</sub></font> &amp; the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="tex_4.html#K6" target="_self">MaxADSspace</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="tex_4.html#K3" target="_self">MaxADSet</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="ref" href="tex_4.html#D18" target="_self">Def18</a></i>;<br/>
<b>thus </b><a NAME="E2:129"/>
( <font color="Maroon">c<sub>3</sub></font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="tex_4.html#K6" target="_self">MaxADSspace</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>2</sub></font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="tex_4.html#K6" target="_self">MaxADSspace</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> implies  <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="tex_4.html#K6" target="_self">MaxADSspace</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="tex_4.html#K6" target="_self">MaxADSspace</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> #) <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="tex_4.html#K6" target="_self">MaxADSspace</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="tex_4.html#K6" target="_self">MaxADSspace</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> #) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="129_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:129_1"/>
( <font color="Maroon">c<sub>3</sub></font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="tex_4.html#K6" target="_self">MaxADSspace</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>2</sub></font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="tex_4.html#K6" target="_self">MaxADSspace</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> )
 ;<br/>

<a NAME="E2:129_1"/><b>then </b>
 <a href="tex_4.html#K3" target="_self">MaxADSet</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="tex_4.html#K3" target="_self">MaxADSet</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="tex_4.html#E1:129"><i><font color="Green">E66</font></i></a>, <a class="ref" href="tex_4.html#T37" target="_self">Th37</a></i>;<br/>
<b>hence </b><a NAME="E3:129_1"/>
 <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="tex_4.html#K6" target="_self">MaxADSspace</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="tex_4.html#K6" target="_self">MaxADSspace</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> #) <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="tex_4.html#K6" target="_self">MaxADSspace</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="tex_4.html#K6" target="_self">MaxADSspace</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> #)
 <b>by </b><i><a class="txt" href="tex_4.html#E1:129"><i><font color="Green">E66</font></i></a>, <a class="ref" href="tsep_1.html#T5">TSEP_1:5</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E3:129"/>
 <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="tex_4.html#K6" target="_self">MaxADSspace</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="tex_4.html#K6" target="_self">MaxADSspace</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> #) <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="tex_4.html#K6" target="_self">MaxADSspace</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="tex_4.html#K6" target="_self">MaxADSspace</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> #)
 ;<br/>

<b>hence </b><a NAME="E4:129"/>
( <font color="Maroon">c<sub>3</sub></font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="tex_4.html#K6" target="_self">MaxADSspace</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>2</sub></font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="tex_4.html#K6" target="_self">MaxADSspace</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="tex_4.html#E1:129"><i><font color="Green">E66</font></i></a>, <a class="ref" href="tex_4.html#T34" target="_self">Th34</a></i>;<br/>


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