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<div class="add">

<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="seq_2.html#V3" title="SEQ_2:attr.3">bounded</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#V7" title="MEASURE6:attr.7">with_max</a>  )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c2">cT</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>;<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#V7" title="MEASURE6:attr.7">with_max</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="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a>  )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:73_1"/><span class="lab"><font color="Green" title="E15">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="seq_2.html#V3" title="SEQ_2:attr.3">bounded</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="c3">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="c3">f</font> is  <a href="pscomp_1.html#V1" title="PSCOMP_1:attr.1">continuous</a>  implies <font color="Maroon" title="c3">f</font> is  <a href="measure6.html#V7" title="MEASURE6:attr.7">with_max</a>  )</span><br/><span class="kw">assume </span><a NAME="E2:73_1"/><span class="lab"><font color="Green" title="E16">A2</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="c3">f</font> is  <a href="measure6.html#V7" title="MEASURE6:attr.7">with_max</a> </span><br/><span class="kw">set </span><font color="Maroon" title="c4">fcT</font> = <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>;<br/><a NAME="E3:73_1"/>
<font color="Maroon" title="c3">f</font> is  <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:73_1"><span class="lab"><font color="Green" title="E15">A1</font></span></a>, <a class="txt" href="#E2:73_1"><span class="lab"><font color="Green" title="E16">A2</font></span></a></span>;<br/><a NAME="E4:73_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E17">A3</font></span>: 
<font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> is  <a href="xxreal_2.html#V4" title="XXREAL_2:attr.4">bounded_above</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="measure6.html#D18" title="MEASURE6:def.18">MEASURE6:def 18</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c5">b</font> =  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span>)</span>;<br/><span class="kw">set </span><font color="Maroon" title="c6">bf</font> = <span class="p1">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span>)</span></span>)</span> <a href="valued_1.html#K9" title="VALUED_1:func.9">+</a> <span class="p1">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <font color="Maroon" title="c3">f</font></span>)</span>;<br/><a NAME="E5:73_1"/>
 <a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <font color="Maroon" title="c3">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="txt" href="#E2:73_1"><span class="lab"><font color="Green" title="E16">A2</font></span></a>, <a class="ref" href="pscomp_1.html#T55" target="_self" title="PSCOMP_1:th.55">Th55</a></span>;<br/><a NAME="E6:73_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E18">A4</font></span>: 
<span class="p1">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span>)</span></span>)</span> <a href="valued_1.html#K9" title="VALUED_1:func.9">+</a> <span class="p1">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <font color="Maroon" title="c3">f</font></span>)</span> 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#T56" target="_self" title="PSCOMP_1:th.56">Th56</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c7">bf9</font> = <span class="p1">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span>)</span></span>)</span> <a href="valued_1.html#K9" title="VALUED_1:func.9">+</a> <span class="p1">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <font color="Maroon" title="c3">f</font></span>)</span> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c8">f9</font> = <font color="Maroon" title="c3">f</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  ;<br/><span class="kw">set </span><font color="Maroon" title="c9">g</font> =  <a href="measure6.html#K7" title="MEASURE6:func.7">Inv</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span>)</span></span>)</span> <a href="valued_1.html#K9" title="VALUED_1:func.9">+</a> <span class="p2">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span>;<br/><span class="kw">set </span><font color="Maroon" title="c10">gcT</font> = <span class="p1">(<span class="default"><a href="measure6.html#K7" title="MEASURE6:func.7">Inv</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span>)</span></span>)</span> <a href="valued_1.html#K9" title="VALUED_1:func.9">+</a> <span class="p3">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>;<br/><span class="kw">assume </span><a NAME="E9:73_1"/>
not <font color="Maroon" title="c3">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"> contradiction</span><br/><a NAME="E10:73_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E19">A5</font></span>: 
not <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> is  <a href="measure6.html#V1" title="MEASURE6:attr.1">with_max</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="measure6.html#D19" title="MEASURE6:def.19">MEASURE6:def 19</a></span>;<br/><a NAME="E11:73_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A6</font></span>: 
not  <a href="measure6.html#NK4" title="MEASURE6:NK.4">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:73_1"><span class="lab"><font color="Green" title="E17">A3</font></span></a>, <a class="ref" href="measure6.html#D9" title="MEASURE6:def.9">MEASURE6:def 9</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:73_1_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span>)</span></span>)</span> <a href="valued_1.html#K9" title="VALUED_1:func.9">+</a> <span class="p2">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:73_1_1"/><span class="lab"><font color="Green" title="E21">A7</font></span>: 
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span>)</span></span>)</span> <a href="valued_1.html#K9" title="VALUED_1:func.9">+</a> <span class="p2">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:73_1_1"/><span class="lab"><font color="Green" title="E22">A8</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span>)</span></span>)</span> <a href="valued_1.html#K9" title="VALUED_1:func.9">+</a> <span class="p2">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c12">x</font> = <font color="Maroon" title="c11">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:73_1_1"><span class="lab"><font color="Green" title="E21">A7</font></span></a></span>;<br/><font color="Maroon" title="c7">bf9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">x</font> = 
<span class="p1">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <font color="Maroon" title="c8">f9</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#T2" title="VALUED_1:th.2">VALUED_1:2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">x</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#T8" title="VALUED_1:th.8">VALUED_1:8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">x</font></span>)</span>

;<br/><span class="kw">hence </span><a NAME="E7:73_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:73_1"><span class="lab"><font color="Green" title="E20">A6</font></span></a>, <a class="txt" href="#E4:73_1_1"><span class="lab"><font color="Green" title="E22">A8</font></span></a>, <a class="ref" href="funct_2.html#T43" title="FUNCT_2:th.43">FUNCT_2:43</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><a NAME="E13:73_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E21">A9</font></span>: 
 <a href="measure6.html#K7" title="MEASURE6:func.7">Inv</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span>)</span></span>)</span> <a href="valued_1.html#K9" title="VALUED_1:func.9">+</a> <span class="p2">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> is  <a href="pscomp_1.html#V1" title="PSCOMP_1:attr.1">continuous</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:73_1"><span class="lab"><font color="Green" title="E18">A4</font></span></a>, <a class="ref" href="pscomp_1.html#T57" target="_self" title="PSCOMP_1:th.57">Th57</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_2"><span class="kw">now </span></a><div class="add"><a NAME="E1:73_1_2"/>
 <a href="measure6.html#K7" title="MEASURE6:func.7">Inv</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span>)</span></span>)</span> <a href="valued_1.html#K9" title="VALUED_1:func.9">+</a> <span class="p2">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> is  <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:73_1"><span class="lab"><font color="Green" title="E15">A1</font></span></a>, <a class="txt" href="#E13:73_1"><span class="lab"><font color="Green" title="E21">A9</font></span></a></span>;<br/><a NAME="E2:73_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="measure6.html#K7" title="MEASURE6:func.7">Inv</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span>)</span></span>)</span> <a href="valued_1.html#K9" title="VALUED_1:func.9">+</a> <span class="p3">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> is  <a href="xxreal_2.html#V4" title="XXREAL_2:attr.4">bounded_above</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="measure6.html#D18" title="MEASURE6:def.18">MEASURE6:def 18</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">p</font> being   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <span class="kw"> such that </span><br/><a NAME="E4:73_1_2"/><span class="lab"><font color="Green" title="E22">A10</font></span>: 
for <font color="Olive" title="b1">r</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="measure6.html#K7" title="MEASURE6:func.7">Inv</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span>)</span></span>)</span> <a href="valued_1.html#K9" title="VALUED_1:func.9">+</a> <span class="p3">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> holds <br/><font color="Olive" title="b1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">p</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="seq_4.html#D1" title="SEQ_4:def.1">SEQ_4:def 1</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="73_1_2_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:73_1_2_1"/>
( <font color="Maroon" title="c11">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c11">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="73_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:73_1_2_1_1"/><span class="lab"><font color="Green" title="E23">A11</font></span>: 
<font color="Maroon" title="c11">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">a19</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>( <font color="Olive" title="b1">a19</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; ( for <font color="Olive" title="b2">r</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b2">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="measure6.html#K7" title="MEASURE6:func.7">Inv</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span>)</span></span>)</span> <a href="valued_1.html#K9" title="VALUED_1:func.9">+</a> <span class="p3">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> holds <br/><font color="Olive" title="b2">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">a19</font> ) )</span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c12">a19</font> = 1 as   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  ;<br/><span class="kw">take </span><font color="Maroon" title="c13">a19</font> = <font color="Maroon" title="c12">a19</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="c13">a19</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; ( for <font color="Olive" title="b1">r</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="measure6.html#K7" title="MEASURE6:func.7">Inv</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span>)</span></span>)</span> <a href="valued_1.html#K9" title="VALUED_1:func.9">+</a> <span class="p3">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> holds <br/><font color="Olive" title="b1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">a19</font> ) )</span><br/><span class="kw">thus </span><a NAME="E3:73_1_2_1_1"/>
<font color="Maroon" title="c13">a19</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</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">r</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="measure6.html#K7" title="MEASURE6:func.7">Inv</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span>)</span></span>)</span> <a href="valued_1.html#K9" title="VALUED_1:func.9">+</a> <span class="p3">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> holds <br/><font color="Olive" title="b1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">a19</font></span><br/><span class="kw">let </span><font color="Maroon" title="c14">r</font> be   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</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="c14">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="measure6.html#K7" title="MEASURE6:func.7">Inv</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span>)</span></span>)</span> <a href="valued_1.html#K9" title="VALUED_1:func.9">+</a> <span class="p3">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> implies <font color="Maroon" title="c14">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">a19</font> )</span><br/><span class="kw">assume </span><a NAME="E4:73_1_2_1_1"/>
<font color="Maroon" title="c14">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="measure6.html#K7" title="MEASURE6:func.7">Inv</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span>)</span></span>)</span> <a href="valued_1.html#K9" title="VALUED_1:func.9">+</a> <span class="p3">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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="c14">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">a19</font></span><br/><span class="kw">hence </span><a NAME="E5:73_1_2_1_1"/>
<font color="Maroon" title="c14">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">a19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:73_1_2"><span class="lab"><font color="Green" title="E22">A10</font></span></a>, <a class="txt" href="#E1:73_1_2_1_1"><span class="lab"><font color="Green" title="E23">A11</font></span></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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="73_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:73_1_2_1_2"/><span class="lab"><font color="Green" title="E23">A12</font></span>: 
<font color="Maroon" title="c11">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">p</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>( <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; ( for <font color="Olive" title="b2">r</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b2">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="measure6.html#K7" title="MEASURE6:func.7">Inv</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span>)</span></span>)</span> <a href="valued_1.html#K9" title="VALUED_1:func.9">+</a> <span class="p3">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> holds <br/><font color="Olive" title="b2">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p</font> ) )</span><br/><div class="add"><span class="kw">take </span><font color="Maroon" title="c12">p</font> = <font color="Maroon" title="c11">p</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="c12">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; ( for <font color="Olive" title="b1">r</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="measure6.html#K7" title="MEASURE6:func.7">Inv</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span>)</span></span>)</span> <a href="valued_1.html#K9" title="VALUED_1:func.9">+</a> <span class="p3">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> holds <br/><font color="Olive" title="b1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">p</font> ) )</span><br/><span class="kw">thus </span><a NAME="E2:73_1_2_1_2"/>
<font color="Maroon" title="c12">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:73_1_2_1_2"><span class="lab"><font color="Green" title="E23">A12</font></span></a></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">r</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="measure6.html#K7" title="MEASURE6:func.7">Inv</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span>)</span></span>)</span> <a href="valued_1.html#K9" title="VALUED_1:func.9">+</a> <span class="p3">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> holds <br/><font color="Olive" title="b1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">p</font></span><br/><span class="kw">thus </span><a NAME="E3:73_1_2_1_2"/>
for <font color="Olive" title="b1">r</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="measure6.html#K7" title="MEASURE6:func.7">Inv</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span>)</span></span>)</span> <a href="valued_1.html#K9" title="VALUED_1:func.9">+</a> <span class="p3">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> holds <br/><font color="Olive" title="b1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:73_1_2"><span class="lab"><font color="Green" title="E22">A10</font></span></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></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">p</font> being   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <span class="kw"> such that </span><br/><a NAME="E16:73_1"/><span class="lab"><font color="Green" title="E22">A13</font></span>: 
<font color="Maroon" title="c11">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">and </span><br/><a NAME="E17:73_1"/><span class="lab"><font color="Green" title="E23">A14</font></span>: 
for <font color="Olive" title="b1">r</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="measure6.html#K7" title="MEASURE6:func.7">Inv</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span>)</span></span>)</span> <a href="valued_1.html#K9" title="VALUED_1:func.9">+</a> <span class="p3">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> holds <br/><font color="Olive" title="b1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">p</font>
 ;<br/><span class="kw">consider </span><font color="Maroon" title="c12">r</font> being   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <span class="kw"> such that </span><br/><a NAME="E19:73_1"/><span class="lab"><font color="Green" title="E24">A15</font></span>: 
<font color="Maroon" title="c12">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>
 <span class="kw">and </span><br/><a NAME="E20:73_1"/><span class="lab"><font color="Green" title="E25">A16</font></span>: 
<span class="p1">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c11">p</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:73_1"><span class="lab"><font color="Green" title="E17">A3</font></span></a>, <a class="txt" href="#E16:73_1"><span class="lab"><font color="Green" title="E22">A13</font></span></a>, <a class="ref" href="seq_4.html#D4" title="SEQ_4:def.4">SEQ_4:def 4</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c13">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E22:73_1"/><span class="lab"><font color="Green" title="E26">A17</font></span>: 
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>
 <span class="kw">and </span><br/><a NAME="E23:73_1"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>
 <span class="kw">and </span><br/><a NAME="E24:73_1"/><span class="lab"><font color="Green" title="E27">A18</font></span>: 
<font color="Maroon" title="c12">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:73_1"><span class="lab"><font color="Green" title="E24">A15</font></span></a>, <a class="ref" href="funct_2.html#T115" title="FUNCT_2:th.115">FUNCT_2:115</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c14">x</font> = <font color="Maroon" title="c13">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">T</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:73_1"><span class="lab"><font color="Green" title="E26">A17</font></span></a></span>;<br/><a NAME="E26:73_1"/><span class="lab"><font color="Green" title="E28">A19</font></span>: 
<font color="Maroon" title="c3">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:73_1"><span class="lab"><font color="Green" title="E17">A3</font></span></a>, <a class="txt" href="#E19:73_1"><span class="lab"><font color="Green" title="E24">A15</font></span></a>, <a class="txt" href="#E24:73_1"><span class="lab"><font color="Green" title="E27">A18</font></span></a>, <a class="ref" href="seq_4.html#D4" title="SEQ_4:def.4">SEQ_4:def 4</a></span>;<br/><a NAME="E27:73_1"/>
<font color="Maroon" title="c3">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:73_1"><span class="lab"><font color="Green" title="E17">A3</font></span></a>, <a class="txt" href="#E10:73_1"><span class="lab"><font color="Green" title="E19">A5</font></span></a>, <a class="txt" href="#E19:73_1"><span class="lab"><font color="Green" title="E24">A15</font></span></a>, <a class="txt" href="#E24:73_1"><span class="lab"><font color="Green" title="E27">A18</font></span></a>, <a class="ref" href="measure6.html#D9" title="MEASURE6:def.9">MEASURE6:def 9</a></span>;<br/><a NAME="E28:73_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">x</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:73_1"><span class="lab"><font color="Green" title="E28">A19</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E29:73_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A20</font></span>: 
<span class="p1">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">x</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T22" title="XREAL_1:th.22">XREAL_1:22</a></span>;<br/><span class="p1">(<span class="default"><a href="measure6.html#K7" title="MEASURE6:func.7">Inv</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span>)</span></span>)</span> <a href="valued_1.html#K9" title="VALUED_1:func.9">+</a> <span class="p3">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">bf9</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">x</font></span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#T10" title="VALUED_1:th.10">VALUED_1:10</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <font color="Maroon" title="c8">f9</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">x</font></span>)</span></span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#T2" title="VALUED_1:th.2">VALUED_1:2</a></span>
<br/>.= 
1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <font color="Maroon" title="c8">f9</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">x</font></span>)</span></span>)</span>

<br/>.= 
1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">x</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#T8" title="VALUED_1:th.8">VALUED_1:8</a></span>
<br/>.= 
1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">x</font></span>)</span></span>)</span>

;<br/><a NAME="E31:73_1"/><span class="kw">then </span>
1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">x</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:73_1"><span class="lab"><font color="Green" title="E23">A14</font></span></a>, <a class="ref" href="funct_2.html#T43" title="FUNCT_2:th.43">FUNCT_2:43</a></span>;<br/><a NAME="E32:73_1"/><span class="kw">then </span>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">p</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">x</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:73_1"><span class="lab"><font color="Green" title="E29">A20</font></span></a>, <a class="ref" href="xreal_1.html#T83" title="XREAL_1:th.83">XREAL_1:83</a></span>;<br/><a NAME="E33:73_1"/><span class="kw">then </span>
1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c11">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">x</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:73_1"><span class="lab"><font color="Green" title="E22">A13</font></span></a>, <a class="ref" href="xreal_1.html#T81" title="XREAL_1:th.81">XREAL_1:81</a></span>;<br/><a NAME="E34:73_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">x</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c11">p</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></span>;<br/><span class="kw">hence </span><a NAME="E35:73_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:73_1"><span class="lab"><font color="Green" title="E25">A16</font></span></a>, <a class="txt" href="#E24:73_1"><span class="lab"><font color="Green" title="E27">A18</font></span></a>, <a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</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="E2:73"/><span class="lab"><font color="Green" title="E15">A21</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="seq_2.html#V3" title="SEQ_2:attr.3">bounded</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="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="c3">f</font> is  <a href="pscomp_1.html#V1" title="PSCOMP_1:attr.1">continuous</a>  implies <font color="Maroon" title="c3">f</font> is  <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E3:73"/><span class="lab"><font color="Green" title="E16">A22</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="c3">f</font> is  <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a> </span><br/>

<a NAME="E4:73"/><span class="kw">then </span>
<font color="Maroon" title="c3">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="#E2:73"><span class="lab"><font color="Green" title="E15">A21</font></span></a></span>;<br/>
<a NAME="E5:73"/><span class="kw">then </span>
<font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> is  <a href="measure6.html#V1" title="MEASURE6:attr.1">with_max</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measure6.html#D19" title="MEASURE6:def.19">MEASURE6:def 19</a></span>;<br/>
<a NAME="E6:73"/><span class="kw">then </span>
<font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> is  <a href="xxreal_2.html#V4" title="XXREAL_2:attr.4">bounded_above</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measure6.html#D9" title="MEASURE6:def.9">MEASURE6:def 9</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:73"/>
<font color="Maroon" title="c3">f</font> is  <a href="seq_2.html#V1" title="SEQ_2:attr.1">bounded_above</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="measure6.html#D18" title="MEASURE6:def.18">MEASURE6:def 18</a></span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="seq_2.html#D8" title="SEQ_2:def.8">SEQ_2:def 8</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="c3">f</font> is  <a href="seq_2.html#V2" title="SEQ_2:attr.2">bounded_below</a> </span><br/>

<a NAME="E8:73"/>
<font color="Maroon" title="c3">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="txt" href="#E2:73"><span class="lab"><font color="Green" title="E15">A21</font></span></a>, <a class="txt" href="#E3:73"><span class="lab"><font color="Green" title="E16">A22</font></span></a>, <a class="ref" href="pscomp_1.html#T60" target="_self" title="PSCOMP_1:th.60">Th60</a></span>;<br/>
<a NAME="E9:73"/><span class="kw">then </span>
<font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> is  <a href="measure6.html#V2" title="MEASURE6:attr.2">with_min</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measure6.html#D20" title="MEASURE6:def.20">MEASURE6:def 20</a></span>;<br/>
<a NAME="E10:73"/><span class="kw">then </span>
<font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> is  <a href="xxreal_2.html#V3" title="XXREAL_2:attr.3">bounded_below</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measure6.html#D10" title="MEASURE6:def.10">MEASURE6:def 10</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:73"/>
<font color="Maroon" title="c3">f</font> is  <a href="seq_2.html#V2" title="SEQ_2:attr.2">bounded_below</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="measure6.html#D17" title="MEASURE6:def.17">MEASURE6:def 17</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>
