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<span class="kw">let </span><font color="Maroon" title="c1">T</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:()"><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="c1">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="Olive" title="b1">f</font> is  <a href="measure6.html#V7" title="MEASURE6:attr.7">with_max</a>  ) iff 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="c1">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="Olive" title="b1">f</font> is  <a href="measure6.html#V8" title="MEASURE6:attr.8">with_min</a>  )</span><br/>

<div><span class="kw">hereby </span> <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="c1">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="Olive" title="b1">f</font> is  <a href="measure6.html#V8" title="MEASURE6:attr.8">with_min</a>  ) implies 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="c1">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="Olive" title="b1">f</font> is  <a href="measure6.html#V7" title="MEASURE6:attr.7">with_max</a>  )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:72_1"/><span class="lab"><font color="Green" title="E14">A1</font></span>: 
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="c1">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="Olive" title="b1">f</font> is  <a href="measure6.html#V7" title="MEASURE6:attr.7">with_max</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="c1">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="Olive" title="b1">f</font> is  <a href="measure6.html#V8" title="MEASURE6:attr.8">with_min</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">f</font> be   <a href="pscomp_1.html#NM1" title="PSCOMP_1:NM.1">RealMap</a> of <font color="Maroon" title="c1">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="c2">f</font> is  <a href="pscomp_1.html#V1" title="PSCOMP_1:attr.1">continuous</a>  implies <font color="Maroon" title="c2">f</font> is  <a href="measure6.html#V8" title="MEASURE6:attr.8">with_min</a>  )</span><br/><span class="kw">assume </span><a NAME="E2:72_1"/>
<font color="Maroon" title="c2">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="c2">f</font> is  <a href="measure6.html#V8" title="MEASURE6:attr.8">with_min</a> </span><br/><a NAME="E3:72_1"/><span class="kw">then </span>
 <a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <font color="Maroon" title="c2">f</font> is  <a href="pscomp_1.html#V1" title="PSCOMP_1:attr.1">continuous</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pscomp_1.html#T55" target="_self" title="PSCOMP_1:th.55">Th55</a></span>;<br/><a NAME="E4:72_1"/><span class="kw">then </span>
 <a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <font color="Maroon" title="c2">f</font> is  <a href="measure6.html#V7" title="MEASURE6:attr.7">with_max</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:72_1"><span class="lab"><font color="Green" title="E14">A1</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E5:72_1"/>
<font color="Maroon" title="c2">f</font> is  <a href="measure6.html#V8" title="MEASURE6:attr.8">with_min</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="measure6.html#T102" title="MEASURE6:th.102">MEASURE6:102</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/></div>
<span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E2:72"/><span class="lab"><font color="Green" title="E14">A2</font></span>: 
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="c1">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="Olive" title="b1">f</font> is  <a href="measure6.html#V8" title="MEASURE6:attr.8">with_min</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="c1">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="Olive" title="b1">f</font> is  <a href="measure6.html#V7" title="MEASURE6:attr.7">with_max</a> </span><br/>

<span class="kw">let </span><font color="Maroon" title="c2">f</font> be   <a href="pscomp_1.html#NM1" title="PSCOMP_1:NM.1">RealMap</a> of <font color="Maroon" title="c1">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="c2">f</font> is  <a href="pscomp_1.html#V1" title="PSCOMP_1:attr.1">continuous</a>  implies <font color="Maroon" title="c2">f</font> is  <a href="measure6.html#V7" title="MEASURE6:attr.7">with_max</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E3:72"/>
<font color="Maroon" title="c2">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="c2">f</font> is  <a href="measure6.html#V7" title="MEASURE6:attr.7">with_max</a> </span><br/>

<a NAME="E4:72"/><span class="kw">then </span>
 <a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <font color="Maroon" title="c2">f</font> is  <a href="pscomp_1.html#V1" title="PSCOMP_1:attr.1">continuous</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pscomp_1.html#T55" target="_self" title="PSCOMP_1:th.55">Th55</a></span>;<br/>
<a NAME="E5:72"/><span class="kw">then </span>
 <a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <span class="p1">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <font color="Maroon" title="c2">f</font></span>)</span> is  <a href="measure6.html#V7" title="MEASURE6:attr.7">with_max</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:72"><span class="lab"><font color="Green" title="E14">A2</font></span></a>, <a class="txt" href="pscomp_1.html#E3"><span class="lab"><font color="Green" title="E3">Lm8</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:72"/>
<font color="Maroon" title="c2">f</font> is  <a href="measure6.html#V7" title="MEASURE6:attr.7">with_max</a> 
 ; <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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