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<span class="kw">let </span><font color="Maroon" title="c1">r3</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">T</font> being   <a href="pre_topc.html#L1" title="PRE_TOPC:struct.1">TopStruct</a> <br/> for <font color="Olive" title="b2">f</font> being  <a href="pscomp_1.html#NM1" title="PSCOMP_1:NM.1">RealMap</a> of <font color="Olive" title="b1">T</font>  st <font color="Olive" title="b2">f</font> is  <a href="pscomp_1.html#V1" title="PSCOMP_1:attr.1">continuous</a>  holds <br/><font color="Maroon" title="c1">r3</font> <a href="valued_1.html#K9" title="VALUED_1:func.9">+</a> <font color="Olive" title="b2">f</font> is  <a href="pscomp_1.html#V1" title="PSCOMP_1:attr.1">continuous</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">T</font> be    <a href="pre_topc.html#L1" title="PRE_TOPC:struct.1">TopStruct</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">f</font> being  <a href="pscomp_1.html#NM1" title="PSCOMP_1:NM.1">RealMap</a> of <font color="Maroon" title="c2">T</font>  st <font color="Olive" title="b1">f</font> is  <a href="pscomp_1.html#V1" title="PSCOMP_1:attr.1">continuous</a>  holds <br/><font color="Maroon" title="c1">r3</font> <a href="valued_1.html#K9" title="VALUED_1:func.9">+</a> <font color="Olive" title="b1">f</font> is  <a href="pscomp_1.html#V1" title="PSCOMP_1:attr.1">continuous</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c3">f</font> be   <a href="pscomp_1.html#NM1" title="PSCOMP_1:NM.1">RealMap</a> of <font color="Maroon" title="c2">T</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">f</font> is  <a href="pscomp_1.html#V1" title="PSCOMP_1:attr.1">continuous</a>  implies <font color="Maroon" title="c1">r3</font> <a href="valued_1.html#K9" title="VALUED_1:func.9">+</a> <font color="Maroon" title="c3">f</font> is  <a href="pscomp_1.html#V1" title="PSCOMP_1:attr.1">continuous</a>  )</span><br/>





<span class="kw">assume </span><a NAME="E1:65"/><span class="lab"><font color="Green" title="E11">A1</font></span>: 
<font color="Maroon" title="c3">f</font> is  <a href="pscomp_1.html#V1" title="PSCOMP_1:attr.1">continuous</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">r3</font> <a href="valued_1.html#K9" title="VALUED_1:func.9">+</a> <font color="Maroon" title="c3">f</font> is  <a href="pscomp_1.html#V1" title="PSCOMP_1:attr.1">continuous</a> </span><br/>

<span class="kw">let </span><font color="Maroon" title="c4">X</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="pscomp_1.html#D25" target="_self" title="PSCOMP_1:def.25">PSCOMP_1:def 25</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">X</font> is  <a href="rcomp_1.html#V2" title="RCOMP_1:attr.2">closed</a>  implies <span class="p1">(<span class="default"><font color="Maroon" title="c1">r3</font> <a href="valued_1.html#K9" title="VALUED_1:func.9">+</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c4">X</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E2:65"/>
<font color="Maroon" title="c4">X</font> is  <a href="rcomp_1.html#V2" title="RCOMP_1:attr.2">closed</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r3</font> <a href="valued_1.html#K9" title="VALUED_1:func.9">+</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c4">X</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> </span><br/>

<a NAME="E3:65"/><span class="kw">then </span><span class="lab"><font color="Green" title="E12">A2</font></span>: 
<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c1">r3</font></span>)</span> <a href="measure6.html#K2" title="MEASURE6:func.2">++</a> <font color="Maroon" title="c4">X</font> is  <a href="rcomp_1.html#V2" title="RCOMP_1:attr.2">closed</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measure6.html#T89" title="MEASURE6:th.89">MEASURE6:89</a></span>;<br/>
<a NAME="E4:65"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">r3</font> <a href="valued_1.html#K9" title="VALUED_1:func.9">+</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c4">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c1">r3</font></span>)</span> <a href="measure6.html#K2" title="MEASURE6:func.2">++</a> <font color="Maroon" title="c4">X</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measure6.html#T106" title="MEASURE6:th.106">MEASURE6:106</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:65"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">r3</font> <a href="valued_1.html#K9" title="VALUED_1:func.9">+</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c4">X</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:65"><span class="lab"><font color="Green" title="E11">A1</font></span></a>, <a class="txt" href="#E3:65"><span class="lab"><font color="Green" title="E12">A2</font></span></a>, <a class="ref" href="pscomp_1.html#D25" target="_self" title="PSCOMP_1:def.25">Def25</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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