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<span class="kw">let </span><font color="Maroon" title="c1">k</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="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">X</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/>  for <font color="Olive" title="b2">S</font> being   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Olive" title="b1">X</font><br/>  for <font color="Olive" title="b3">E</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Olive" title="b2">S</font><br/>  for <font color="Olive" title="b4">f</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Olive" title="b1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <font color="Olive" title="b4">f</font> is  <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>  &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">k</font> &amp; <font color="Olive" title="b3">E</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b4">f</font> &amp; <font color="Olive" title="b4">f</font> <a href="mesfunc6.html#R1" title="MESFUNC6:pred.1">is_measurable_on</a> <font color="Olive" title="b3">E</font> holds <br/><font color="Olive" title="b4">f</font> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> <font color="Maroon" title="c1">k</font> <a href="mesfunc6.html#R1" title="MESFUNC6:pred.1">is_measurable_on</a> <font color="Olive" title="b3">E</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">X</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">S</font> being   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c2">X</font><br/>  for <font color="Olive" title="b2">E</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Olive" title="b1">S</font><br/>  for <font color="Olive" title="b3">f</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <font color="Olive" title="b3">f</font> is  <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>  &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">k</font> &amp; <font color="Olive" title="b2">E</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b3">f</font> &amp; <font color="Olive" title="b3">f</font> <a href="mesfunc6.html#R1" title="MESFUNC6:pred.1">is_measurable_on</a> <font color="Olive" title="b2">E</font> holds <br/><font color="Olive" title="b3">f</font> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> <font color="Maroon" title="c1">k</font> <a href="mesfunc6.html#R1" title="MESFUNC6:pred.1">is_measurable_on</a> <font color="Olive" title="b2">E</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">S</font> be   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c2">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">E</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">S</font><br/>  for <font color="Olive" title="b2">f</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <font color="Olive" title="b2">f</font> is  <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>  &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">k</font> &amp; <font color="Olive" title="b1">E</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b2">f</font> &amp; <font color="Olive" title="b2">f</font> <a href="mesfunc6.html#R1" title="MESFUNC6:pred.1">is_measurable_on</a> <font color="Olive" title="b1">E</font> holds <br/><font color="Olive" title="b2">f</font> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> <font color="Maroon" title="c1">k</font> <a href="mesfunc6.html#R1" title="MESFUNC6:pred.1">is_measurable_on</a> <font color="Olive" title="b1">E</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">E</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">S</font>; <a class="txt" 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="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <font color="Olive" title="b1">f</font> is  <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>  &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">k</font> &amp; <font color="Maroon" title="c4">E</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b1">f</font> &amp; <font color="Olive" title="b1">f</font> <a href="mesfunc6.html#R1" title="MESFUNC6:pred.1">is_measurable_on</a> <font color="Maroon" title="c4">E</font> holds <br/><font color="Olive" title="b1">f</font> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> <font color="Maroon" title="c1">k</font> <a href="mesfunc6.html#R1" title="MESFUNC6:pred.1">is_measurable_on</a> <font color="Maroon" title="c4">E</font></span><br/><span class="kw">let </span><font color="Maroon" title="c5">f</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c2">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">f</font> is  <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>  &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">k</font> &amp; <font color="Maroon" title="c4">E</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">f</font> &amp; <font color="Maroon" title="c5">f</font> <a href="mesfunc6.html#R1" title="MESFUNC6:pred.1">is_measurable_on</a> <font color="Maroon" title="c4">E</font> implies <font color="Maroon" title="c5">f</font> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> <font color="Maroon" title="c1">k</font> <a href="mesfunc6.html#R1" title="MESFUNC6:pred.1">is_measurable_on</a> <font color="Maroon" title="c4">E</font> )</span><br/>









<span class="kw">reconsider </span><font color="Maroon" title="c6">k1</font> = <font color="Maroon" title="c1">k</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></span>;<br/>
<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:35"/><span class="lab"><font color="Green" title="E22">A1</font></span>: 
<font color="Maroon" title="c5">f</font> is  <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a> 
 <span class="kw">and </span><br/><a NAME="E3:35"/><span class="lab"><font color="Green" title="E23">A2</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">k</font>
 <span class="kw">and </span><br/><a NAME="E4:35"/><span class="lab"><font color="Green" title="E24">A3</font></span>: 
<font color="Maroon" title="c4">E</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">f</font>
 <span class="kw">and </span><br/><a NAME="E5:35"/><span class="lab"><font color="Green" title="E25">A4</font></span>: 
<font color="Maroon" title="c5">f</font> <a href="mesfunc6.html#R1" title="MESFUNC6:pred.1">is_measurable_on</a> <font color="Maroon" title="c4">E</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">f</font> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> <font color="Maroon" title="c1">k</font> <a href="mesfunc6.html#R1" title="MESFUNC6:pred.1">is_measurable_on</a> <font color="Maroon" title="c4">E</font></span><br/>

<a NAME="E6:35"/><span class="lab"><font color="Green" title="E26">A5</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mesfun6c.html#D4" target="_self" title="MESFUN6C:def.4">Def6</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_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="E1:35_1"/>
( <font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c1">k</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><span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35"><span class="lab"><font color="Green" title="E23">A2</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1"><span class="kw">suppose </span></a><a NAME="E1:35_1_1"/><span class="lab"><font color="Green" title="E27">A6</font></span>: 
<font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">f</font> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> <font color="Maroon" title="c1">k</font> <a href="mesfunc6.html#R1" title="MESFUNC6:pred.1">is_measurable_on</a> <font color="Maroon" title="c4">E</font></span><br/><div class="add"><a NAME="E2:35_1_1"/><span class="lab"><font color="Green" title="E28">A7</font></span>: 
<font color="Maroon" title="c4">E</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> <font color="Maroon" title="c1">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:35"><span class="lab"><font color="Green" title="E24">A3</font></span></a>, <a class="ref" href="mesfun6c.html#D4" target="_self" title="MESFUN6C:def.4">Def6</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">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="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">E</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K12" title="MESFUNC1:func.12">less_dom</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> <font color="Maroon" title="c1">k</font></span>)</span>,<font color="Olive">b<sub>1</sub></font>)</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">S</font></span><br/><span class="kw">reconsider </span><font color="Maroon" title="c8">r1</font> = <font color="Maroon" title="c7">r</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_1_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="E1:35_1_1_1_1"/>
( <font color="Maroon" title="c7">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 or 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">r</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_1_1_1"/><span class="lab"><font color="Green" title="E29">A8</font></span>: 
<font color="Maroon" title="c7">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">E</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K12" title="MESFUNC1:func.12">less_dom</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> <font color="Maroon" title="c1">k</font></span>)</span>,<font color="Olive">b<sub>1</sub></font>)</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">S</font></span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_1_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">E</font> implies <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mesfunc1.html#K15" title="MESFUNC1:func.15">great_eq_dom</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> <font color="Maroon" title="c1">k</font></span>)</span>,<font color="Maroon" title="c8">r1</font>) )</span><br/><span class="kw">assume </span><a NAME="E1:35_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E30">A9</font></span>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">E</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mesfunc1.html#K15" title="MESFUNC1:func.15">great_eq_dom</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> <font color="Maroon" title="c1">k</font></span>)</span>,<font color="Maroon" title="c8">r1</font>)</span><br/><a NAME="E2:35_1_1_1_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">x</font></span>)</span> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c1">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:35"><span class="lab"><font color="Green" title="E24">A3</font></span></a>, <a class="txt" href="#E6:35"><span class="lab"><font color="Green" title="E26">A5</font></span></a>, <a class="ref" href="mesfun6c.html#D4" target="_self" title="MESFUN6C:def.4">Def6</a></span>;<br/><a NAME="E3:35_1_1_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1"><span class="lab"><font color="Green" title="E27">A6</font></span></a>, <a class="txt" href="#E1:35_1_1_1_1_1"><span class="lab"><font color="Green" title="E29">A8</font></span></a>, <a class="ref" href="power.html#T24" title="POWER:th.24">POWER:24</a></span>;<br/><span class="kw">hence </span><a NAME="E4:35_1_1_1_1_1_1"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mesfunc1.html#K15" title="MESFUNC1:func.15">great_eq_dom</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> <font color="Maroon" title="c1">k</font></span>)</span>,<font color="Maroon" title="c8">r1</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:35"><span class="lab"><font color="Green" title="E24">A3</font></span></a>, <a class="txt" href="#E6:35"><span class="lab"><font color="Green" title="E26">A5</font></span></a>, <a class="txt" href="#E1:35_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E30">A9</font></span></a>, <a class="ref" href="mesfunc6.html#T6" title="MESFUNC6:th.6">MESFUNC6:6</a></span>; <a class="txt" 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><a NAME="E3:35_1_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">E</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="mesfunc1.html#K15" title="MESFUNC1:func.15">great_eq_dom</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> <font color="Maroon" title="c1">k</font></span>)</span>,<font color="Maroon" title="c7">r</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/><a NAME="E4:35_1_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">E</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K15" title="MESFUNC1:func.15">great_eq_dom</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> <font color="Maroon" title="c1">k</font></span>)</span>,<font color="Maroon" title="c7">r</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">E</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></span>;<br/><a NAME="E5:35_1_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">E</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K12" title="MESFUNC1:func.12">less_dom</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> <font color="Maroon" title="c1">k</font></span>)</span>,<font color="Maroon" title="c8">r1</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">E</font> <a href="prob_1.html#K7" title="PROB_1:func.7">\</a> <font color="Maroon" title="c4">E</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35_1_1"><span class="lab"><font color="Green" title="E28">A7</font></span></a>, <a class="ref" href="mesfun6c.html#T28" target="_self" title="MESFUN6C:th.28">Th28</a></span>;<br/><span class="kw">hence </span><a NAME="E6:35_1_1_1_1_1"/>
<font color="Maroon" title="c4">E</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K12" title="MESFUNC1:func.12">less_dom</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> <font color="Maroon" title="c1">k</font></span>)</span>,<font color="Maroon" title="c7">r</font>)</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">S</font>
 ; <a class="txt" 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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:35_1_1_1_1_2"/><span class="lab"><font color="Green" title="E29">A10</font></span>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">r</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">E</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K12" title="MESFUNC1:func.12">less_dom</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> <font color="Maroon" title="c1">k</font></span>)</span>,<font color="Olive">b<sub>1</sub></font>)</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">S</font></span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_1_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">E</font> implies <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mesfunc1.html#K12" title="MESFUNC1:func.12">less_dom</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> <font color="Maroon" title="c1">k</font></span>)</span>,<font color="Maroon" title="c7">r</font>) )</span><br/><span class="kw">assume </span><a NAME="E1:35_1_1_1_1_2_1"/><span class="lab"><font color="Green" title="E30">A11</font></span>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">E</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mesfunc1.html#K12" title="MESFUNC1:func.12">less_dom</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> <font color="Maroon" title="c1">k</font></span>)</span>,<font color="Maroon" title="c7">r</font>)</span><br/><a NAME="E2:35_1_1_1_1_2_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">x</font></span>)</span> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c1">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:35"><span class="lab"><font color="Green" title="E24">A3</font></span></a>, <a class="txt" href="#E6:35"><span class="lab"><font color="Green" title="E26">A5</font></span></a>, <a class="ref" href="mesfun6c.html#D4" target="_self" title="MESFUN6C:def.4">Def6</a></span>;<br/><a NAME="E3:35_1_1_1_1_2_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1"><span class="lab"><font color="Green" title="E27">A6</font></span></a>, <a class="txt" href="#E1:35_1_1_1_1_2"><span class="lab"><font color="Green" title="E29">A10</font></span></a>, <a class="ref" href="power.html#T24" title="POWER:th.24">POWER:24</a></span>;<br/><span class="kw">hence </span><a NAME="E4:35_1_1_1_1_2_1"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mesfunc1.html#K12" title="MESFUNC1:func.12">less_dom</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> <font color="Maroon" title="c1">k</font></span>)</span>,<font color="Maroon" title="c7">r</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:35"><span class="lab"><font color="Green" title="E24">A3</font></span></a>, <a class="txt" href="#E6:35"><span class="lab"><font color="Green" title="E26">A5</font></span></a>, <a class="txt" href="#E1:35_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E30">A11</font></span></a>, <a class="ref" href="mesfunc6.html#T3" title="MESFUNC6:th.3">MESFUNC6:3</a></span>; <a class="txt" 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><a NAME="E3:35_1_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c4">E</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="mesfunc1.html#K12" title="MESFUNC1:func.12">less_dom</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> <font color="Maroon" title="c1">k</font></span>)</span>,<font color="Maroon" title="c7">r</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/><a NAME="E4:35_1_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c4">E</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K12" title="MESFUNC1:func.12">less_dom</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> <font color="Maroon" title="c1">k</font></span>)</span>,<font color="Maroon" title="c7">r</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">E</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></span>;<br/><span class="kw">hence </span><a NAME="E5:35_1_1_1_1_2"/>
<font color="Maroon" title="c4">E</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K12" title="MESFUNC1:func.12">less_dom</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> <font color="Maroon" title="c1">k</font></span>)</span>,<font color="Maroon" title="c7">r</font>)</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">S</font>
 ; <a class="txt" 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></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E4:35_1_1"/>
<font color="Maroon" title="c5">f</font> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> <font color="Maroon" title="c1">k</font> <a href="mesfunc6.html#R1" title="MESFUNC6:pred.1">is_measurable_on</a> <font color="Maroon" title="c4">E</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc6.html#T12" title="MESFUNC6:th.12">MESFUNC6:12</a></span>; <a class="txt" 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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_2"><span class="kw">suppose </span></a><a NAME="E1:35_1_2"/><span class="lab"><font color="Green" title="E27">A12</font></span>: 
<font color="Maroon" title="c1">k</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="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">f</font> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> <font color="Maroon" title="c1">k</font> <a href="mesfunc6.html#R1" title="MESFUNC6:pred.1">is_measurable_on</a> <font color="Maroon" title="c4">E</font></span><br/><div class="add"><a NAME="E2:35_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>  holds  <font color="Maroon" title="c4">E</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K15" title="MESFUNC1:func.15">great_eq_dom</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> <font color="Maroon" title="c1">k</font></span>)</span>,<font color="Olive" title="b1">r</font>)</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">S</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">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="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">E</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K15" title="MESFUNC1:func.15">great_eq_dom</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> <font color="Maroon" title="c1">k</font></span>)</span>,<font color="Maroon" title="c7">r</font>)</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">S</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c8">r1</font> = <font color="Maroon" title="c7">r</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_2_1_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="E1:35_1_2_1_1"/>
( <font color="Maroon" title="c8">r1</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  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">r1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:35_1_2_1_1_1"/><span class="lab"><font color="Green" title="E28">A13</font></span>: 
<font color="Maroon" title="c8">r1</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="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">E</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K15" title="MESFUNC1:func.15">great_eq_dom</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> <font color="Maroon" title="c1">k</font></span>)</span>,<font color="Maroon" title="c7">r</font>)</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">S</font></span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_2_1_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">E</font> implies <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mesfunc1.html#K15" title="MESFUNC1:func.15">great_eq_dom</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> <font color="Maroon" title="c1">k</font></span>)</span>,<font color="Maroon" title="c8">r1</font>) )</span><br/><span class="kw">assume </span><a NAME="E1:35_1_2_1_1_1_1"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">E</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mesfunc1.html#K15" title="MESFUNC1:func.15">great_eq_dom</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> <font color="Maroon" title="c1">k</font></span>)</span>,<font color="Maroon" title="c8">r1</font>)</span><br/><a NAME="E2:35_1_2_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">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> <font color="Maroon" title="c5">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:35"><span class="lab"><font color="Green" title="E24">A3</font></span></a></span>;<br/><a NAME="E3:35_1_2_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A14</font></span>: 
<font color="Maroon" title="c9">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"><font color="Maroon" title="c5">f</font> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> <font color="Maroon" title="c1">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="mesfun6c.html#D4" target="_self" title="MESFUN6C:def.4">Def6</a></span>;<br/><a NAME="E4:35_1_2_1_1_1_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="txt" href="#E1:35_1_2"><span class="lab"><font color="Green" title="E27">A12</font></span></a>, <a class="ref" href="mesfunc6.html#T51" title="MESFUNC6:th.51">MESFUNC6:51</a></span>;<br/><span class="kw">hence </span><a NAME="E5:35_1_2_1_1_1_1"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mesfunc1.html#K15" title="MESFUNC1:func.15">great_eq_dom</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> <font color="Maroon" title="c1">k</font></span>)</span>,<font color="Maroon" title="c8">r1</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_2_1_1_1"><span class="lab"><font color="Green" title="E28">A13</font></span></a>, <a class="txt" href="#E3:35_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E29">A14</font></span></a>, <a class="ref" href="mesfunc6.html#T6" title="MESFUNC6:th.6">MESFUNC6:6</a></span>; <a class="txt" 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><a NAME="E3:35_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">E</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="mesfunc1.html#K15" title="MESFUNC1:func.15">great_eq_dom</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> <font color="Maroon" title="c1">k</font></span>)</span>,<font color="Maroon" title="c7">r</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/><a NAME="E4:35_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">E</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K15" title="MESFUNC1:func.15">great_eq_dom</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> <font color="Maroon" title="c1">k</font></span>)</span>,<font color="Maroon" title="c7">r</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">E</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></span>;<br/><span class="kw">hence </span><a NAME="E5:35_1_2_1_1_1"/>
<font color="Maroon" title="c4">E</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K15" title="MESFUNC1:func.15">great_eq_dom</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> <font color="Maroon" title="c1">k</font></span>)</span>,<font color="Maroon" title="c7">r</font>)</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">S</font>
 ; <a class="txt" 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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:35_1_2_1_1_2"/><span class="lab"><font color="Green" title="E28">A15</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">r1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">E</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K15" title="MESFUNC1:func.15">great_eq_dom</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> <font color="Maroon" title="c1">k</font></span>)</span>,<font color="Maroon" title="c7">r</font>)</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">S</font></span><br/><div class="add"><div><span class="lab"><font color="Green" title="E29">A16</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_2_1_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">set </span><font color="Maroon" title="c9">R</font> = <font color="Maroon" title="c7">r</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">k</font></span>)</span>;<br/><span class="kw">let </span><font color="Maroon" title="c10">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mesfunc1.html#K15" title="MESFUNC1:func.15">great_eq_dom</a> (<font color="Maroon" title="c5">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">r1</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">k</font></span>)</span></span>)</span>) implies <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mesfunc1.html#K15" title="MESFUNC1:func.15">great_eq_dom</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> <font color="Maroon" title="c1">k</font></span>)</span>,<font color="Maroon" title="c8">r1</font>) )</span><br/><a NAME="E1:35_1_2_1_1_2_1"/><span class="lab"><font color="Green" title="E30">A17</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">r</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_2_1_1_2"><span class="lab"><font color="Green" title="E28">A15</font></span></a>, <a class="ref" href="power.html#T34" title="POWER:th.34">POWER:34</a></span>;<br/><span class="kw">assume </span><a NAME="E2:35_1_2_1_1_2_1"/><span class="lab"><font color="Green" title="E31">A18</font></span>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mesfunc1.html#K15" title="MESFUNC1:func.15">great_eq_dom</a> (<font color="Maroon" title="c5">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">r1</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">k</font></span>)</span></span>)</span>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mesfunc1.html#K15" title="MESFUNC1:func.15">great_eq_dom</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> <font color="Maroon" title="c1">k</font></span>)</span>,<font color="Maroon" title="c8">r1</font>)</span><br/><a NAME="E3:35_1_2_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A19</font></span>: 
<font color="Maroon" title="c10">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"><font color="Maroon" title="c5">f</font> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> <font color="Maroon" title="c1">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:35"><span class="lab"><font color="Green" title="E26">A5</font></span></a>, <a class="ref" href="mesfunc6.html#T6" title="MESFUNC6:th.6">MESFUNC6:6</a></span>;<br/><a NAME="E4:35_1_2_1_1_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">k</font></span>)</span></span>)</span> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">r</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c1">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_2_1_1_2"><span class="lab"><font color="Green" title="E28">A15</font></span></a>, <a class="ref" href="power.html#T33" title="POWER:th.33">POWER:33</a></span>;<br/><a NAME="E5:35_1_2_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A20</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">k</font></span>)</span></span>)</span> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">r</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_2"><span class="lab"><font color="Green" title="E27">A12</font></span></a>, <a class="ref" href="xcmplx_1.html#T87" title="XCMPLX_1:th.87">XCMPLX_1:87</a></span>;<br/><a NAME="E6:35_1_2_1_1_2_1"/>
 ex <font color="Olive" title="b1">y</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">x</font> &amp; <font color="Maroon" title="c8">r1</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">y</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35_1_2_1_1_2_1"><span class="lab"><font color="Green" title="E31">A18</font></span></a>, <a class="ref" href="mesfunc6.html#T6" title="MESFUNC6:th.6">MESFUNC6:6</a></span>;<br/><a NAME="E7:35_1_2_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">r1</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">x</font></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c6">k1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_2"><span class="lab"><font color="Green" title="E27">A12</font></span></a>, <a class="txt" href="#E1:35_1_2_1_1_2_1"><span class="lab"><font color="Green" title="E30">A17</font></span></a>, <a class="txt" href="#E5:35_1_2_1_1_2_1"><span class="lab"><font color="Green" title="E33">A20</font></span></a>, <a class="ref" href="holder_1.html#T3" title="HOLDER_1:th.3">HOLDER_1:3</a></span>;<br/><a NAME="E8:35_1_2_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">x</font></span>)</span> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c1">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T25" title="POWER:th.25">POWER:25</a></span>;<br/><a NAME="E9:35_1_2_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_2_1_1_2_1"><span class="lab"><font color="Green" title="E32">A19</font></span></a>, <a class="ref" href="mesfun6c.html#D4" target="_self" title="MESFUN6C:def.4">Def6</a></span>;<br/><span class="kw">hence </span><a NAME="E10:35_1_2_1_1_2_1"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mesfunc1.html#K15" title="MESFUNC1:func.15">great_eq_dom</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> <font color="Maroon" title="c1">k</font></span>)</span>,<font color="Maroon" title="c8">r1</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_2_1_1_2_1"><span class="lab"><font color="Green" title="E32">A19</font></span></a>, <a class="ref" href="mesfunc6.html#T6" title="MESFUNC6:th.6">MESFUNC6:6</a></span>; <a class="txt" 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><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_2_1_1_2_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mesfunc1.html#K15" title="MESFUNC1:func.15">great_eq_dom</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> <font color="Maroon" title="c1">k</font></span>)</span>,<font color="Maroon" title="c8">r1</font>) implies <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mesfunc1.html#K15" title="MESFUNC1:func.15">great_eq_dom</a> (<font color="Maroon" title="c5">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">r1</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">k</font></span>)</span></span>)</span>) )</span><br/><span class="kw">assume </span><a NAME="E1:35_1_2_1_1_2_2"/><span class="lab"><font color="Green" title="E30">A21</font></span>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mesfunc1.html#K15" title="MESFUNC1:func.15">great_eq_dom</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> <font color="Maroon" title="c1">k</font></span>)</span>,<font color="Maroon" title="c8">r1</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mesfunc1.html#K15" title="MESFUNC1:func.15">great_eq_dom</a> (<font color="Maroon" title="c5">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">r1</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">k</font></span>)</span></span>)</span>)</span><br/><a NAME="E2:35_1_2_1_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A22</font></span>: 
<font color="Maroon" title="c9">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"><font color="Maroon" title="c5">f</font> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> <font color="Maroon" title="c1">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc6.html#T6" title="MESFUNC6:th.6">MESFUNC6:6</a></span>;<br/><a NAME="E3:35_1_2_1_1_2_2"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="ref" href="mesfunc6.html#T51" title="MESFUNC6:th.51">MESFUNC6:51</a></span>;<br/><a NAME="E4:35_1_2_1_1_2_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">x</font></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c6">k1</font></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c6">k1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">x</font></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">k1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> 1</span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c6">k1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_2"><span class="lab"><font color="Green" title="E27">A12</font></span></a>, <a class="ref" href="holder_1.html#T2" title="HOLDER_1:th.2">HOLDER_1:2</a></span>;<br/><a NAME="E5:35_1_2_1_1_2_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">x</font></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c6">k1</font></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c6">k1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">x</font></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_2"><span class="lab"><font color="Green" title="E27">A12</font></span></a>, <a class="ref" href="xcmplx_1.html#T87" title="XCMPLX_1:th.87">XCMPLX_1:87</a></span>;<br/><a NAME="E6:35_1_2_1_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A23</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">x</font></span>)</span> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T25" title="POWER:th.25">POWER:25</a></span>;<br/><a NAME="E7:35_1_2_1_1_2_2"/>
 ex <font color="Olive" title="b1">y2</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">x</font> &amp; <font color="Maroon" title="c8">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">y2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_2_1_1_2_2"><span class="lab"><font color="Green" title="E30">A21</font></span></a>, <a class="ref" href="mesfunc6.html#T6" title="MESFUNC6:th.6">MESFUNC6:6</a></span>;<br/><a NAME="E8:35_1_2_1_1_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">x</font></span>)</span> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c1">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35_1_2_1_1_2_2"><span class="lab"><font color="Green" title="E31">A22</font></span></a>, <a class="ref" href="mesfun6c.html#D4" target="_self" title="MESFUN6C:def.4">Def6</a></span>;<br/><a NAME="E9:35_1_2_1_1_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">r</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c6">k1</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">x</font></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c6">k1</font></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c6">k1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_2"><span class="lab"><font color="Green" title="E27">A12</font></span></a>, <a class="txt" href="#E1:35_1_2_1_1_2"><span class="lab"><font color="Green" title="E28">A15</font></span></a>, <a class="ref" href="holder_1.html#T3" title="HOLDER_1:th.3">HOLDER_1:3</a></span>;<br/><span class="kw">hence </span><a NAME="E10:35_1_2_1_1_2_2"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mesfunc1.html#K15" title="MESFUNC1:func.15">great_eq_dom</a> (<font color="Maroon" title="c5">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">r1</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">k</font></span>)</span></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:35"><span class="lab"><font color="Green" title="E26">A5</font></span></a>, <a class="txt" href="#E2:35_1_2_1_1_2_2"><span class="lab"><font color="Green" title="E31">A22</font></span></a>, <a class="txt" href="#E6:35_1_2_1_1_2_2"><span class="lab"><font color="Green" title="E32">A23</font></span></a>, <a class="ref" href="mesfunc6.html#T6" title="MESFUNC6:th.6">MESFUNC6:6</a></span>; <a class="txt" 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><a NAME="E4:35_1_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c4">E</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K15" title="MESFUNC1:func.15">great_eq_dom</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> <font color="Maroon" title="c1">k</font></span>)</span>,<font color="Maroon" title="c8">r1</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">E</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K15" title="MESFUNC1:func.15">great_eq_dom</a> (<font color="Maroon" title="c5">f</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">r1</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">k</font></span>)</span></span>)</span>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35_1_2_1_1_2"><span class="lab"><font color="Green" title="E29">A16</font></span></a>, <a class="ref" href="tarski.html#T1" title="TARSKI:th.1">TARSKI:1</a></span>;<br/><span class="kw">hence </span><a NAME="E5:35_1_2_1_1_2"/>
<font color="Maroon" title="c4">E</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K15" title="MESFUNC1:func.15">great_eq_dom</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> <font color="Maroon" title="c1">k</font></span>)</span>,<font color="Maroon" title="c7">r</font>)</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:35"><span class="lab"><font color="Green" title="E24">A3</font></span></a>, <a class="txt" href="#E5:35"><span class="lab"><font color="Green" title="E25">A4</font></span></a>, <a class="ref" href="mesfunc6.html#T13" title="MESFUNC6:th.13">MESFUNC6:13</a></span>; <a class="txt" 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>

</div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:35_1_2"/>
<font color="Maroon" title="c5">f</font> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> <font color="Maroon" title="c1">k</font> <a href="mesfunc6.html#R1" title="MESFUNC6:pred.1">is_measurable_on</a> <font color="Maroon" title="c4">E</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:35"><span class="lab"><font color="Green" title="E24">A3</font></span></a>, <a class="txt" href="#E6:35"><span class="lab"><font color="Green" title="E26">A5</font></span></a>, <a class="ref" href="mesfunc6.html#T13" title="MESFUNC6:th.13">MESFUNC6:13</a></span>; <a class="txt" 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>

</div>
