<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><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" 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">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#V6" title="DECOMP_1:attr.6">p-continuous</a>  iff ( <font color="Olive" title="b1">f</font> is  <a href="decomp_1.html#V9" title="DECOMP_1:attr.9">sp-continuous</a>  &amp; <font color="Olive" title="b1">f</font> is  <a href="decomp_1.html#V18" title="DECOMP_1:attr.18">(p,sp)-continuous</a>  ) )</span><br/><span class="kw">let </span><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" 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">f</font> is  <a href="decomp_1.html#V6" title="DECOMP_1:attr.6">p-continuous</a>  iff ( <font color="Maroon" title="c3">f</font> is  <a href="decomp_1.html#V9" title="DECOMP_1:attr.9">sp-continuous</a>  &amp; <font color="Maroon" title="c3">f</font> is  <a href="decomp_1.html#V18" title="DECOMP_1:attr.18">(p,sp)-continuous</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="c3">f</font> is  <a href="decomp_1.html#V9" title="DECOMP_1:attr.9">sp-continuous</a>  &amp; <font color="Maroon" title="c3">f</font> is  <a href="decomp_1.html#V18" title="DECOMP_1:attr.18">(p,sp)-continuous</a>  implies <font color="Maroon" title="c3">f</font> is  <a href="decomp_1.html#V6" title="DECOMP_1:attr.6">p-continuous</a>  )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:22_1"/><span class="lab"><font color="Green" title="E19">A1</font></span>: 
<font color="Maroon" title="c3">f</font> is  <a href="decomp_1.html#V6" title="DECOMP_1:attr.6">p-continuous</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"> ( <font color="Maroon" title="c3">f</font> is  <a href="decomp_1.html#V9" title="DECOMP_1:attr.9">sp-continuous</a>  &amp; <font color="Maroon" title="c3">f</font> is  <a href="decomp_1.html#V18" title="DECOMP_1:attr.18">(p,sp)-continuous</a>  )</span><br/><span class="kw">thus </span><a NAME="E2:22_1"/>
<font color="Maroon" title="c3">f</font> is  <a href="decomp_1.html#V9" title="DECOMP_1:attr.9">sp-continuous</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"> <font color="Maroon" title="c3">f</font> is  <a href="decomp_1.html#V18" title="DECOMP_1:attr.18">(p,sp)-continuous</a> </span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="22_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="decomp_1.html#D30" target="_self" onmouseover="rs('decomp_1/D30')" onmouseout="rh()">DECOMP_1:def 30</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"> ( <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#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c4">V</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="decomp_1.html#K9" title="DECOMP_1:func.9">SPO</a> <font color="Maroon" title="c1">X</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:22_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" 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">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c4">V</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="decomp_1.html#K9" title="DECOMP_1:func.9">SPO</a> <font color="Maroon" title="c1">X</font></span><br/>

<a NAME="E2:22_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c4">V</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="decomp_1.html#K8" title="DECOMP_1:func.8">PO</a> <font color="Maroon" title="c1">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22_1"><span class="lab"><font color="Green" title="E19">A1</font></span></a></span>;<br/>
<a NAME="E3:22_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c4">V</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="decomp_1.html#K9" title="DECOMP_1:func.9">SPO</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#K18" title="DECOMP_1:func.18">D(p,sp)</a> <font color="Maroon" title="c1">X</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="decomp_1.html#T14" target="_self" onmouseover="rs('decomp_1/T14')" onmouseout="rh()">Th14</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:22_1_1"/>
<font color="Maroon" title="c3">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c4">V</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="decomp_1.html#K9" title="DECOMP_1:func.9">SPO</a> <font color="Maroon" title="c1">X</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</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">thus </span><a NAME="E3:22_1"/>
<font color="Maroon" title="c3">f</font> is  <a href="decomp_1.html#V18" title="DECOMP_1:attr.18">(p,sp)-continuous</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"> verum</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="22_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="decomp_1.html#D39" target="_self" onmouseover="rs('decomp_1/D39')" onmouseout="rh()">DECOMP_1:def 39</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"> ( <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#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c4">G</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="decomp_1.html#K18" title="DECOMP_1:func.18">D(p,sp)</a> <font color="Maroon" title="c1">X</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:22_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" 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">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c4">G</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="decomp_1.html#K18" title="DECOMP_1:func.18">D(p,sp)</a> <font color="Maroon" title="c1">X</font></span><br/>

<a NAME="E2:22_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c4">G</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="decomp_1.html#K8" title="DECOMP_1:func.8">PO</a> <font color="Maroon" title="c1">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22_1"><span class="lab"><font color="Green" title="E19">A1</font></span></a></span>;<br/>
<a NAME="E3:22_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c4">G</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="decomp_1.html#K9" title="DECOMP_1:func.9">SPO</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#K18" title="DECOMP_1:func.18">D(p,sp)</a> <font color="Maroon" title="c1">X</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="decomp_1.html#T14" target="_self" onmouseover="rs('decomp_1/T14')" onmouseout="rh()">Th14</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:22_1_2"/>
<font color="Maroon" title="c3">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c4">G</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="decomp_1.html#K18" title="DECOMP_1:func.18">D(p,sp)</a> <font color="Maroon" title="c1">X</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</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></div>
<span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E2:22"/><span class="lab"><font color="Green" title="E19">A2</font></span>: 
( <font color="Maroon" title="c3">f</font> is  <a href="decomp_1.html#V9" title="DECOMP_1:attr.9">sp-continuous</a>  &amp; <font color="Maroon" title="c3">f</font> is  <a href="decomp_1.html#V18" title="DECOMP_1:attr.18">(p,sp)-continuous</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"> <font color="Maroon" title="c3">f</font> is  <a href="decomp_1.html#V6" title="DECOMP_1:attr.6">p-continuous</a> </span><br/>

<span class="kw">let </span><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>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="decomp_1.html#D27" target="_self" onmouseover="rs('decomp_1/D27')" onmouseout="rh()">DECOMP_1:def 27</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"> ( <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#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c4">V</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="decomp_1.html#K8" title="DECOMP_1:func.8">PO</a> <font color="Maroon" title="c1">X</font> )</span><br/>

<span class="kw">assume </span><a NAME="E3:22"/>
<font color="Maroon" title="c4">V</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</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"> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c4">V</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="decomp_1.html#K8" title="DECOMP_1:func.8">PO</a> <font color="Maroon" title="c1">X</font></span><br/>

<a NAME="E4:22"/><span class="kw">then </span>
( <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c4">V</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="decomp_1.html#K9" title="DECOMP_1:func.9">SPO</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c4">V</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="decomp_1.html#K18" title="DECOMP_1:func.18">D(p,sp)</a> <font color="Maroon" title="c1">X</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:22"><span class="lab"><font color="Green" title="E19">A2</font></span></a></span>;<br/>
<a NAME="E5:22"/><span class="kw">then </span>
<font color="Maroon" title="c3">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c4">V</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="decomp_1.html#K9" title="DECOMP_1:func.9">SPO</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#K18" title="DECOMP_1:func.18">D(p,sp)</a> <font color="Maroon" title="c1">X</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:22"/>
<font color="Maroon" title="c3">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c4">V</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="decomp_1.html#K8" title="DECOMP_1:func.8">PO</a> <font color="Maroon" title="c1">X</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="decomp_1.html#T14" target="_self" onmouseover="rs('decomp_1/T14')" onmouseout="rh()">Th14</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>
