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<b>let </b><font color="Maroon" title="c1">X</font>, <font color="Maroon" title="c2">Y</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">f</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font> holds <br/> ( <font color="Olive" title="b1">f</font> is <a href="decomp_1.html#V7" title="DECOMP_1:attr.7">alpha-continuous</a> iff ( <font color="Olive" title="b1">f</font> is <a href="decomp_1.html#V8" title="DECOMP_1:attr.8">ps-continuous</a> &amp; <font color="Olive" title="b1">f</font> is <a href="decomp_1.html#V16" title="DECOMP_1:attr.16">(alpha,ps)-continuous</a> ) )</span><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 <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">f</font> is <a href="decomp_1.html#V7" title="DECOMP_1:attr.7">alpha-continuous</a> iff ( <font color="Maroon" title="c3">f</font> is <a href="decomp_1.html#V8" title="DECOMP_1:attr.8">ps-continuous</a> &amp; <font color="Maroon" title="c3">f</font> is <a href="decomp_1.html#V16" title="DECOMP_1:attr.16">(alpha,ps)-continuous</a> ) )</span><br/>





<div><b>hereby </b> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">f</font> is <a href="decomp_1.html#V8" title="DECOMP_1:attr.8">ps-continuous</a> &amp; <font color="Maroon" title="c3">f</font> is <a href="decomp_1.html#V16" title="DECOMP_1:attr.16">(alpha,ps)-continuous</a> implies <font color="Maroon" title="c3">f</font> is <a href="decomp_1.html#V7" title="DECOMP_1:attr.7">alpha-continuous</a> )</span>
<div class="add"><b>assume </b><a NAME="E1:25_1"/><i><font color="Green" title="E35">A1</font></i>: 
<font color="Maroon" title="c3">f</font> is <a href="decomp_1.html#V7" title="DECOMP_1:attr.7">alpha-continuous</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">f</font> is <a href="decomp_1.html#V8" title="DECOMP_1:attr.8">ps-continuous</a> &amp; <font color="Maroon" title="c3">f</font> is <a href="decomp_1.html#V16" title="DECOMP_1:attr.16">(alpha,ps)-continuous</a> )</span><br/><b>thus </b><a NAME="E2:25_1"/>
<font color="Maroon" title="c3">f</font> is <a href="decomp_1.html#V8" title="DECOMP_1:attr.8">ps-continuous</a>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">f</font> is <a href="decomp_1.html#V16" title="DECOMP_1:attr.16">(alpha,ps)-continuous</a></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">V</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">Y</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="decomp_1.html#D29" target="_self" title="DECOMP_1:def.29">DECOMP_1:def 29</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">V</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> implies <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c4">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="decomp_1.html#K10" title="DECOMP_1:func.10">PSO</a> <font color="Maroon" title="c1">X</font> )</span><br/>

<b>assume </b><a NAME="E1:25_1_1"/>
<font color="Maroon" title="c4">V</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c4">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="decomp_1.html#K10" title="DECOMP_1:func.10">PSO</a> <font color="Maroon" title="c1">X</font></span><br/>

<a NAME="E2:25_1_1"/><b>then </b>
<font color="Maroon" title="c3">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c4">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> <a href="decomp_1.html#K6" title="DECOMP_1:func.6">^alpha</a> 
 
<b>by </b><i><a class="txt" href="#E1:25_1"><i><font color="Green" title="E35">A1</font></i></a>, <a class="ref" href="decomp_1.html#D28" target="_self" title="DECOMP_1:def.28">Def28</a></i>;<br/>
<a NAME="E3:25_1_1"/><b>then </b>
<font color="Maroon" title="c3">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c4">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="decomp_1.html#K10" title="DECOMP_1:func.10">PSO</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="decomp_1.html#K17" title="DECOMP_1:func.17">D(alpha,ps)</a> <font color="Maroon" title="c1">X</font></span>)</span>
 
<b>by </b><i><a class="ref" href="decomp_1.html#T13" target="_self" title="DECOMP_1:th.13">Th13</a></i>;<br/>
<b>hence </b><a NAME="E4:25_1_1"/>
<font color="Maroon" title="c3">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c4">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="decomp_1.html#K10" title="DECOMP_1:func.10">PSO</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><b>thus </b><a NAME="E3:25_1"/>
<font color="Maroon" title="c3">f</font> is <a href="decomp_1.html#V16" title="DECOMP_1:attr.16">(alpha,ps)-continuous</a>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">G</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">Y</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="decomp_1.html#D37" target="_self" title="DECOMP_1:def.37">DECOMP_1:def 37</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">G</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> implies <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c4">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="decomp_1.html#K17" title="DECOMP_1:func.17">D(alpha,ps)</a> <font color="Maroon" title="c1">X</font> )</span><br/>

<b>assume </b><a NAME="E1:25_1_2"/>
<font color="Maroon" title="c4">G</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c4">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="decomp_1.html#K17" title="DECOMP_1:func.17">D(alpha,ps)</a> <font color="Maroon" title="c1">X</font></span><br/>

<a NAME="E2:25_1_2"/><b>then </b>
<font color="Maroon" title="c3">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c4">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> <a href="decomp_1.html#K6" title="DECOMP_1:func.6">^alpha</a> 
 
<b>by </b><i><a class="txt" href="#E1:25_1"><i><font color="Green" title="E35">A1</font></i></a>, <a class="ref" href="decomp_1.html#D28" target="_self" title="DECOMP_1:def.28">Def28</a></i>;<br/>
<a NAME="E3:25_1_2"/><b>then </b>
<font color="Maroon" title="c3">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c4">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="decomp_1.html#K10" title="DECOMP_1:func.10">PSO</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="decomp_1.html#K17" title="DECOMP_1:func.17">D(alpha,ps)</a> <font color="Maroon" title="c1">X</font></span>)</span>
 
<b>by </b><i><a class="ref" href="decomp_1.html#T13" target="_self" title="DECOMP_1:th.13">Th13</a></i>;<br/>
<b>hence </b><a NAME="E4:25_1_2"/>
<font color="Maroon" title="c3">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c4">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="decomp_1.html#K17" title="DECOMP_1:func.17">D(alpha,ps)</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<b>assume </b><a NAME="E2:25"/><i><font color="Green" title="E35">A2</font></i>: 
( <font color="Maroon" title="c3">f</font> is <a href="decomp_1.html#V8" title="DECOMP_1:attr.8">ps-continuous</a> &amp; <font color="Maroon" title="c3">f</font> is <a href="decomp_1.html#V16" title="DECOMP_1:attr.16">(alpha,ps)-continuous</a> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">f</font> is <a href="decomp_1.html#V7" title="DECOMP_1:attr.7">alpha-continuous</a></span><br/>

<b>let </b><font color="Maroon" title="c4">V</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">Y</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="decomp_1.html#D28" target="_self" title="DECOMP_1:def.28">DECOMP_1:def 28</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">V</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> implies <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c4">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> <a href="decomp_1.html#K6" title="DECOMP_1:func.6">^alpha</a>  )</span><br/>

<b>assume </b><a NAME="E3:25"/><i><font color="Green" title="E36">A3</font></i>: 
<font color="Maroon" title="c4">V</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c4">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> <a href="decomp_1.html#K6" title="DECOMP_1:func.6">^alpha</a> </span><br/>

<a NAME="E4:25"/><b>then </b><i><font color="Green" title="E37">A4</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c4">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="decomp_1.html#K10" title="DECOMP_1:func.10">PSO</a> <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="txt" href="#E2:25"><i><font color="Green" title="E35">A2</font></i></a>, <a class="ref" href="decomp_1.html#D29" target="_self" title="DECOMP_1:def.29">Def29</a></i>;<br/>
<a NAME="E5:25"/>
<font color="Maroon" title="c3">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c4">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="decomp_1.html#K17" title="DECOMP_1:func.17">D(alpha,ps)</a> <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="txt" href="#E2:25"><i><font color="Green" title="E35">A2</font></i></a>, <a class="txt" href="#E3:25"><i><font color="Green" title="E36">A3</font></i></a>, <a class="ref" href="decomp_1.html#D37" target="_self" title="DECOMP_1:def.37">Def37</a></i>;<br/>
<a NAME="E6:25"/><b>then </b>
<font color="Maroon" title="c3">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c4">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="decomp_1.html#K10" title="DECOMP_1:func.10">PSO</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="decomp_1.html#K17" title="DECOMP_1:func.17">D(alpha,ps)</a> <font color="Maroon" title="c1">X</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E4:25"><i><font color="Green" title="E37">A4</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<b>hence </b><a NAME="E7:25"/>
<font color="Maroon" title="c3">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c4">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> <a href="decomp_1.html#K6" title="DECOMP_1:func.6">^alpha</a> 
 <b>by </b><i><a class="ref" href="decomp_1.html#T13" target="_self" title="DECOMP_1:th.13">Th13</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
