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<div class="add">

<span class="kw">consider </span><font color="Maroon" title="c5">S</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">Sigma</font><span class="kw"> such that </span><br/><a NAME="E2:9_1_1"/><span class="lab"><font color="Green" title="E8">A2</font></span>: 
( <font color="Maroon" title="c5">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">Omega</font> &amp; <font color="Maroon" title="c4">X</font> <a href="mesfunc6.html#R1" title="MESFUNC6:pred.1">is_measurable_on</a> <font color="Maroon" title="c5">S</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="random_1.html#D2" title="RANDOM_1:def.2">RANDOM_1:def 2</a></span>;<br/>
<a NAME="E3:9_1_1"/>
<span class="p1">(<span class="default"><a href="prob_4.html#K2" title="PROB_4:func.2">P2M</a> <font color="Maroon" title="c3">P</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c5">S</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="prob_1.html#T35" title="PROB_1:th.35">PROB_1:35</a></span>;<br/>
<a NAME="E4:9_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E9">A3</font></span>: 
<span class="p1">(<span class="default"><a href="prob_4.html#K2" title="PROB_4:func.2">P2M</a> <font color="Maroon" title="c3">P</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c5">S</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a>, <a class="ref" href="xxreal_0.html#T9" title="XXREAL_0:th.9">XXREAL_0:9</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">r</font> =  <a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c7">E0</font> = <font color="Maroon" title="c1">Omega</font> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> 1;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">E</font> = <font color="Maroon" title="c1">Omega</font> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <span class="p1">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span>;<br/>
<a NAME="E5:9_1_1"/><span class="lab"><font color="Green" title="E10">A4</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="c1">Omega</font> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">Omega</font> &amp;  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">Omega</font> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> 1</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">E0</font> = <font color="Maroon" title="c1">Omega</font> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> 1 as    <a href="random_1.html#M1" title="RANDOM_1:mode.1">Real-Valued-Random-Variable</a> of <font color="Maroon" title="c2">Sigma</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="random_2.html#T4" target="_self" title="RANDOM_2:th.4">Th4</a></span>;<br/>
<a NAME="E7:9_1_1"/><span class="lab"><font color="Green" title="E11">A5</font></span>: 
 <a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <font color="Maroon" title="c9">E0</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> (<font color="Maroon" title="c5">S</font>,<font color="Maroon" title="c1">Omega</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:9_1_1"><span class="lab"><font color="Green" title="E8">A2</font></span></a>, <a class="ref" href="random_2.html#T3" target="_self" title="RANDOM_2:th.3">Th3</a></span>;<br/>
<a NAME="E8:9_1_1"/><span class="lab"><font color="Green" title="E12">A6</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="c1">Omega</font> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <span class="p2">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span></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="c9">E0</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:9_1_1"><span class="lab"><font color="Green" title="E10">A4</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_1"><span class="kw">now </span></a><div class="add"><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="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">Omega</font> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <span class="p2">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c1">Omega</font> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <span class="p2">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">E0</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">x</font></span>)</span> )</span><br/><span class="kw">assume </span><a NAME="E1:9_1_1_1"/><span class="lab"><font color="Green" title="E13">A7</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="c1">Omega</font> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <span class="p2">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</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"> <span class="p1">(<span class="default"><font color="Maroon" title="c1">Omega</font> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <span class="p2">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">E0</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">x</font></span>)</span></span><br/><span class="kw">hence </span><span class="p1">(<span class="default"><font color="Maroon" title="c1">Omega</font> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <span class="p2">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">x</font> = 
<span class="p1">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> 1
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T7" title="FUNCOP_1:th.7">FUNCOP_1:7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">E0</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9_1_1_1"><span class="lab"><font color="Green" title="E13">A7</font></span></a>, <a class="ref" href="funcop_1.html#T7" title="FUNCOP_1:th.7">FUNCOP_1:7</a></span>
;<br/> <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="E10:9_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E13">A8</font></span>: 
<font color="Maroon" title="c1">Omega</font> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <span class="p1">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span> <a href="random_1.html#K5" title="RANDOM_1:func.5">(#)</a> <font color="Maroon" title="c9">E0</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:9_1_1"><span class="lab"><font color="Green" title="E12">A6</font></span></a>, <a class="ref" href="valued_1.html#D5" title="VALUED_1:def.5">VALUED_1:def 5</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">E</font> = <font color="Maroon" title="c1">Omega</font> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <span class="p1">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span> as    <a href="random_1.html#M1" title="RANDOM_1:mode.1">Real-Valued-Random-Variable</a> of <font color="Maroon" title="c2">Sigma</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="random_2.html#T4" target="_self" title="RANDOM_2:th.4">Th4</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c11">Y</font> = <font color="Maroon" title="c4">X</font> <a href="random_1.html#K4" title="RANDOM_1:func.4">-</a> <font color="Maroon" title="c10">E</font>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c12">Y</font> = <font color="Maroon" title="c4">X</font> <a href="random_1.html#K4" title="RANDOM_1:func.4">-</a> <font color="Maroon" title="c10">E</font> as    <a href="random_1.html#M1" title="RANDOM_1:mode.1">Real-Valued-Random-Variable</a> of <font color="Maroon" title="c2">Sigma</font> ;<br/>
<a NAME="E13:9_1_1"/>
 <a href="mesfunc2.html#K3" title="MESFUNC2:func.3">chi</a> (<font color="Maroon" title="c5">S</font>,<font color="Maroon" title="c1">Omega</font>) <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a>  <a href="prob_4.html#K2" title="PROB_4:func.2">P2M</a> <font color="Maroon" title="c3">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:9_1_1"><span class="lab"><font color="Green" title="E9">A3</font></span></a>, <a class="ref" href="mesfunc7.html#T24" title="MESFUNC7:th.24">MESFUNC7:24</a></span>;<br/>
<a NAME="E14:9_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E14">A9</font></span>: 
<font color="Maroon" title="c9">E0</font> <a href="mesfunc6.html#R3" title="MESFUNC6:pred.3">is_integrable_on</a>  <a href="prob_4.html#K2" title="PROB_4:func.2">P2M</a> <font color="Maroon" title="c3">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:9_1_1"><span class="lab"><font color="Green" title="E11">A5</font></span></a>, <a class="ref" href="mesfunc6.html#D4" title="MESFUNC6:def.4">MESFUNC6:def 4</a></span>;<br/>
<a NAME="E15:9_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E15">A10</font></span>: 
<span class="p1">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span> <a href="random_1.html#K5" title="RANDOM_1:func.5">(#)</a> <font color="Maroon" title="c9">E0</font> <a href="mesfunc6.html#R3" title="MESFUNC6:pred.3">is_integrable_on</a>  <a href="prob_4.html#K2" title="PROB_4:func.2">P2M</a> <font color="Maroon" title="c3">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc6.html#T102" title="MESFUNC6:th.102">MESFUNC6:102</a></span>;<br/>
<a NAME="E16:9_1_1"/><span class="lab"><font color="Green" title="E16">A11</font></span>: 
 <a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1 is   <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/>
<a NAME="E17:9_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E17">A12</font></span>: 
<span class="p1">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span> <a href="random_1.html#K5" title="RANDOM_1:func.5">(#)</a> <font color="Maroon" title="c9">E0</font></span>)</span> <a href="mesfunc6.html#R3" title="MESFUNC6:pred.3">is_integrable_on</a>  <a href="prob_4.html#K2" title="PROB_4:func.2">P2M</a> <font color="Maroon" title="c3">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:9_1_1"><span class="lab"><font color="Green" title="E15">A10</font></span></a>, <a class="ref" href="mesfunc6.html#T102" title="MESFUNC6:th.102">MESFUNC6:102</a></span>;<br/>
<a NAME="E18:9_1_1"/><span class="lab"><font color="Green" title="E18">A13</font></span>: 
<font color="Maroon" title="c4">X</font> <a href="mesfunc6.html#R3" title="MESFUNC6:pred.3">is_integrable_on</a>  <a href="prob_4.html#K2" title="PROB_4:func.2">P2M</a> <font color="Maroon" title="c3">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="ref" href="random_1.html#D3" title="RANDOM_1:def.3">RANDOM_1:def 3</a></span>;<br/>
<a NAME="E19:9_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">Y</font> <a href="mesfunc6.html#R3" title="MESFUNC6:pred.3">is_integrable_on</a>  <a href="prob_4.html#K2" title="PROB_4:func.2">P2M</a> <font color="Maroon" title="c3">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:9_1_1"><span class="lab"><font color="Green" title="E13">A8</font></span></a>, <a class="txt" href="#E17:9_1_1"><span class="lab"><font color="Green" title="E17">A12</font></span></a>, <a class="ref" href="mesfunc6.html#T100" title="MESFUNC6:th.100">MESFUNC6:100</a></span>;<br/>
<a NAME="E20:9_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E19">A14</font></span>: 
<font color="Maroon" title="c12">Y</font> <a href="random_1.html#R1" title="RANDOM_1:pred.1">is_integrable_on</a> <font color="Maroon" title="c3">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="random_1.html#D3" title="RANDOM_1:def.3">RANDOM_1:def 3</a></span>;<br/>
<a NAME="E21:9_1_1"/>
<span class="p1">(<span class="default">2 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p2">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span></span>)</span> <a href="random_1.html#K5" title="RANDOM_1:func.5">(#)</a> <font color="Maroon" title="c4">X</font> <a href="mesfunc6.html#R3" title="MESFUNC6:pred.3">is_integrable_on</a>  <a href="prob_4.html#K2" title="PROB_4:func.2">P2M</a> <font color="Maroon" title="c3">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:9_1_1"><span class="lab"><font color="Green" title="E18">A13</font></span></a>, <a class="ref" href="mesfunc6.html#T102" title="MESFUNC6:th.102">MESFUNC6:102</a></span>;<br/>
<a NAME="E22:9_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p3">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span></span>)</span> <a href="random_1.html#K5" title="RANDOM_1:func.5">(#)</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="mesfunc6.html#R3" title="MESFUNC6:pred.3">is_integrable_on</a>  <a href="prob_4.html#K2" title="PROB_4:func.2">P2M</a> <font color="Maroon" title="c3">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:9_1_1"><span class="lab"><font color="Green" title="E16">A11</font></span></a>, <a class="ref" href="mesfunc6.html#T102" title="MESFUNC6:th.102">MESFUNC6:102</a></span>;<br/>
<a NAME="E23:9_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A15</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="random_1.html#K7" title="RANDOM_1:func.7">abs</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> 2</span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p3">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span></span>)</span> <a href="random_1.html#K5" title="RANDOM_1:func.5">(#)</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="mesfunc6.html#R3" title="MESFUNC6:pred.3">is_integrable_on</a>  <a href="prob_4.html#K2" title="PROB_4:func.2">P2M</a> <font color="Maroon" title="c3">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="ref" href="mesfunc6.html#T100" title="MESFUNC6:th.100">MESFUNC6:100</a></span>;<br/>
<a NAME="E24:9_1_1"/><span class="lab"><font color="Green" title="E21">A16</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">X</font> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> 2</span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p2">(<span class="default">2 <a href="random_1.html#K5" title="RANDOM_1:func.5">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span> <a href="random_1.html#K5" title="RANDOM_1:func.5">(#)</a> <font color="Maroon" title="c9">E0</font></span>)</span> <a href="random_1.html#K6" title="RANDOM_1:func.6">(#)</a> <font color="Maroon" title="c4">X</font></span>)</span></span>)</span></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span> <a href="random_1.html#K5" title="RANDOM_1:func.5">(#)</a> <font color="Maroon" title="c9">E0</font></span>)</span> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> 2</span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">X</font> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> 2</span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p4">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span></span>)</span> <a href="random_1.html#K5" title="RANDOM_1:func.5">(#)</a> <font color="Maroon" title="c4">X</font></span>)</span></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p3">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span></span>)</span> <a href="random_1.html#K5" title="RANDOM_1:func.5">(#)</a> <font color="Maroon" title="c9">E0</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_2"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:9_1_1_2"/><span class="lab"><font color="Green" title="E22">A17</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="c4">X</font> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> 2</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="c4">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mesfun6c.html#D4" title="MESFUN6C:def.4">MESFUN6C:def 4</a></span>;<br/>
<span class="lab"><font color="Green" title="E23">A18</font></span>:  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default">2 <a href="random_1.html#K5" title="RANDOM_1:func.5">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span> <a href="random_1.html#K5" title="RANDOM_1:func.5">(#)</a> <font color="Maroon" title="c9">E0</font></span>)</span> <a href="random_1.html#K6" title="RANDOM_1:func.6">(#)</a> <font color="Maroon" title="c4">X</font></span>)</span></span>)</span> = 
 <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"><span class="p3">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span> <a href="random_1.html#K5" title="RANDOM_1:func.5">(#)</a> <font color="Maroon" title="c9">E0</font></span>)</span> <a href="random_1.html#K6" title="RANDOM_1:func.6">(#)</a> <font color="Maroon" title="c4">X</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#D5" title="VALUED_1:def.5">VALUED_1:def 5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span> <a href="random_1.html#K5" title="RANDOM_1:func.5">(#)</a> <font color="Maroon" title="c9">E0</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">X</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#D4" title="VALUED_1:def.4">VALUED_1:def 4</a></span>
<br/>.= 
<font color="Maroon" title="c1">Omega</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">X</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:9_1_1"><span class="lab"><font color="Green" title="E10">A4</font></span></a>, <a class="ref" href="valued_1.html#D5" title="VALUED_1:def.5">VALUED_1:def 5</a></span>
;<br/>
<span class="lab"><font color="Green" title="E24">A19</font></span>:  <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"><span class="p3">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span> <a href="random_1.html#K5" title="RANDOM_1:func.5">(#)</a> <font color="Maroon" title="c9">E0</font></span>)</span> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> 2</span>)</span> = 
 <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="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span> <a href="random_1.html#K5" title="RANDOM_1:func.5">(#)</a> <font color="Maroon" title="c9">E0</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="mesfun6c.html#D4" title="MESFUN6C:def.4">MESFUN6C:def 4</a></span>
<br/>.= 
<font color="Maroon" title="c1">Omega</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:9_1_1"><span class="lab"><font color="Green" title="E10">A4</font></span></a>, <a class="ref" href="valued_1.html#D5" title="VALUED_1:def.5">VALUED_1:def 5</a></span>
;<br/>
<a NAME="E4:9_1_1_2"/><span class="lab"><font color="Green" title="E25">A20</font></span>: 
 <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"><span class="p3">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p3">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span></span>)</span> <a href="random_1.html#K5" title="RANDOM_1:func.5">(#)</a> <font color="Maroon" title="c9">E0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">Omega</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:9_1_1"><span class="lab"><font color="Green" title="E10">A4</font></span></a>, <a class="ref" href="valued_1.html#D5" title="VALUED_1:def.5">VALUED_1:def 5</a></span>;<br/>
<a NAME="E5:9_1_1_2"/><span class="lab"><font color="Green" title="E26">A21</font></span>: 
 <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">2 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p3">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span></span>)</span> <a href="random_1.html#K5" title="RANDOM_1:func.5">(#)</a> <font color="Maroon" title="c4">X</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="c4">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#D5" title="VALUED_1:def.5">VALUED_1:def 5</a></span>;<br/>
<span class="lab"><font color="Green" title="E27">A22</font></span>:  <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"><font color="Maroon" title="c4">X</font> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> 2</span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p2">(<span class="default">2 <a href="random_1.html#K5" title="RANDOM_1:func.5">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span> <a href="random_1.html#K5" title="RANDOM_1:func.5">(#)</a> <font color="Maroon" title="c9">E0</font></span>)</span> <a href="random_1.html#K6" title="RANDOM_1:func.6">(#)</a> <font color="Maroon" title="c4">X</font></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">Omega</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">X</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9_1_1_2"><span class="lab"><font color="Green" title="E22">A17</font></span></a>, <a class="txt" href="#E2:9_1_1_2"><span class="lab"><font color="Green" title="E23">A18</font></span></a>, <a class="ref" href="valued_1.html#T12" title="VALUED_1:th.12">VALUED_1:12</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">X</font></span>)</span></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c1">Omega</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T16" title="XBOOLE_1:th.16">XBOOLE_1:16</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c1">Omega</font>

;<br/>
<span class="lab"><font color="Green" title="E28">A23</font></span>:  <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"><font color="Maroon" title="c4">X</font> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> 2</span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p4">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span></span>)</span> <a href="random_1.html#K5" title="RANDOM_1:func.5">(#)</a> <font color="Maroon" title="c4">X</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">X</font> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> 2</span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p4">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span></span>)</span> <a href="random_1.html#K5" title="RANDOM_1:func.5">(#)</a> <font color="Maroon" title="c4">X</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#T12" title="VALUED_1:th.12">VALUED_1:12</a></span>
<br/>.= 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">X</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9_1_1_2"><span class="lab"><font color="Green" title="E22">A17</font></span></a>, <a class="txt" href="#E5:9_1_1_2"><span class="lab"><font color="Green" title="E26">A21</font></span></a></span>
;<br/>
<span class="lab"><font color="Green" title="E29">A24</font></span>:  <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"><span class="p3">(<span class="default"><font color="Maroon" title="c4">X</font> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> 2</span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p3">(<span class="default">2 <a href="random_1.html#K5" title="RANDOM_1:func.5">(#)</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span> <a href="random_1.html#K5" title="RANDOM_1:func.5">(#)</a> <font color="Maroon" title="c9">E0</font></span>)</span> <a href="random_1.html#K6" title="RANDOM_1:func.6">(#)</a> <font color="Maroon" title="c4">X</font></span>)</span></span>)</span></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span> <a href="random_1.html#K5" title="RANDOM_1:func.5">(#)</a> <font color="Maroon" title="c9">E0</font></span>)</span> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> 2</span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">X</font> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> 2</span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p3">(<span class="default">2 <a href="random_1.html#K5" title="RANDOM_1:func.5">(#)</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span> <a href="random_1.html#K5" title="RANDOM_1:func.5">(#)</a> <font color="Maroon" title="c9">E0</font></span>)</span> <a href="random_1.html#K6" title="RANDOM_1:func.6">(#)</a> <font color="Maroon" title="c4">X</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span> <a href="random_1.html#K5" title="RANDOM_1:func.5">(#)</a> <font color="Maroon" title="c9">E0</font></span>)</span> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> 2</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#D1" title="VALUED_1:def.1">VALUED_1:def 1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">Omega</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c1">Omega</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9_1_1_2"><span class="lab"><font color="Green" title="E24">A19</font></span></a>, <a class="txt" href="#E6:9_1_1_2"><span class="lab"><font color="Green" title="E27">A22</font></span></a>, <a class="ref" href="xboole_1.html#T16" title="XBOOLE_1:th.16">XBOOLE_1:16</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c1">Omega</font>

;<br/>
<a NAME="E9:9_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A25</font></span>: 
 <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"><span class="p3">(<span class="default"><font color="Maroon" title="c4">X</font> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> 2</span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p3">(<span class="default">2 <a href="random_1.html#K5" title="RANDOM_1:func.5">(#)</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span> <a href="random_1.html#K5" title="RANDOM_1:func.5">(#)</a> <font color="Maroon" title="c9">E0</font></span>)</span> <a href="random_1.html#K6" title="RANDOM_1:func.6">(#)</a> <font color="Maroon" title="c4">X</font></span>)</span></span>)</span></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span> <a href="random_1.html#K5" title="RANDOM_1:func.5">(#)</a> <font color="Maroon" title="c9">E0</font></span>)</span> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> 2</span>)</span></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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">X</font> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> 2</span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p5">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span></span>)</span> <a href="random_1.html#K5" title="RANDOM_1:func.5">(#)</a> <font color="Maroon" title="c4">X</font></span>)</span></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p4">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span></span>)</span> <a href="random_1.html#K5" title="RANDOM_1:func.5">(#)</a> <font color="Maroon" title="c9">E0</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:9_1_1_2"><span class="lab"><font color="Green" title="E25">A20</font></span></a>, <a class="txt" href="#E7:9_1_1_2"><span class="lab"><font color="Green" title="E28">A23</font></span></a>, <a class="ref" href="valued_1.html#D1" title="VALUED_1:def.1">VALUED_1:def 1</a></span>;<br/>
<a NAME="E10:9_1_1_2"/>
 for <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">Omega</font>  st <font color="Olive" title="b1">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"><span class="p3">(<span class="default"><font color="Maroon" title="c4">X</font> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> 2</span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p3">(<span class="default">2 <a href="random_1.html#K5" title="RANDOM_1:func.5">(#)</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span> <a href="random_1.html#K5" title="RANDOM_1:func.5">(#)</a> <font color="Maroon" title="c9">E0</font></span>)</span> <a href="random_1.html#K6" title="RANDOM_1:func.6">(#)</a> <font color="Maroon" title="c4">X</font></span>)</span></span>)</span></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span> <a href="random_1.html#K5" title="RANDOM_1:func.5">(#)</a> <font color="Maroon" title="c9">E0</font></span>)</span> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> 2</span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">X</font> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> 2</span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p3">(<span class="default">2 <a href="random_1.html#K5" title="RANDOM_1:func.5">(#)</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span> <a href="random_1.html#K5" title="RANDOM_1:func.5">(#)</a> <font color="Maroon" title="c9">E0</font></span>)</span> <a href="random_1.html#K6" title="RANDOM_1:func.6">(#)</a> <font color="Maroon" title="c4">X</font></span>)</span></span>)</span></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span> <a href="random_1.html#K5" title="RANDOM_1:func.5">(#)</a> <font color="Maroon" title="c9">E0</font></span>)</span> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> 2</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">X</font> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> 2</span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p5">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span></span>)</span> <a href="random_1.html#K5" title="RANDOM_1:func.5">(#)</a> <font color="Maroon" title="c4">X</font></span>)</span></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p4">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span></span>)</span> <a href="random_1.html#K5" title="RANDOM_1:func.5">(#)</a> <font color="Maroon" title="c9">E0</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_2_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c13">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">Omega</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="c13">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"><span class="p3">(<span class="default"><font color="Maroon" title="c4">X</font> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> 2</span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p3">(<span class="default">2 <a href="random_1.html#K5" title="RANDOM_1:func.5">(#)</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span> <a href="random_1.html#K5" title="RANDOM_1:func.5">(#)</a> <font color="Maroon" title="c9">E0</font></span>)</span> <a href="random_1.html#K6" title="RANDOM_1:func.6">(#)</a> <font color="Maroon" title="c4">X</font></span>)</span></span>)</span></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span> <a href="random_1.html#K5" title="RANDOM_1:func.5">(#)</a> <font color="Maroon" title="c9">E0</font></span>)</span> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> 2</span>)</span></span>)</span> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">X</font> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> 2</span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p3">(<span class="default">2 <a href="random_1.html#K5" title="RANDOM_1:func.5">(#)</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span> <a href="random_1.html#K5" title="RANDOM_1:func.5">(#)</a> <font color="Maroon" title="c9">E0</font></span>)</span> <a href="random_1.html#K6" title="RANDOM_1:func.6">(#)</a> <font color="Maroon" title="c4">X</font></span>)</span></span>)</span></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span> <a href="random_1.html#K5" title="RANDOM_1:func.5">(#)</a> <font color="Maroon" title="c9">E0</font></span>)</span> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> 2</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">X</font> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> 2</span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p5">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span></span>)</span> <a href="random_1.html#K5" title="RANDOM_1:func.5">(#)</a> <font color="Maroon" title="c4">X</font></span>)</span></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p4">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span></span>)</span> <a href="random_1.html#K5" title="RANDOM_1:func.5">(#)</a> <font color="Maroon" title="c9">E0</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">x</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:9_1_1_2_6"/><span class="lab"><font color="Green" title="E31">A26</font></span>: 
<font color="Maroon" title="c13">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"><span class="p3">(<span class="default"><font color="Maroon" title="c4">X</font> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> 2</span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p3">(<span class="default">2 <a href="random_1.html#K5" title="RANDOM_1:func.5">(#)</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span> <a href="random_1.html#K5" title="RANDOM_1:func.5">(#)</a> <font color="Maroon" title="c9">E0</font></span>)</span> <a href="random_1.html#K6" title="RANDOM_1:func.6">(#)</a> <font color="Maroon" title="c4">X</font></span>)</span></span>)</span></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span> <a href="random_1.html#K5" title="RANDOM_1:func.5">(#)</a> <font color="Maroon" title="c9">E0</font></span>)</span> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> 2</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"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">X</font> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> 2</span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p3">(<span class="default">2 <a href="random_1.html#K5" title="RANDOM_1:func.5">(#)</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span> <a href="random_1.html#K5" title="RANDOM_1:func.5">(#)</a> <font color="Maroon" title="c9">E0</font></span>)</span> <a href="random_1.html#K6" title="RANDOM_1:func.6">(#)</a> <font color="Maroon" title="c4">X</font></span>)</span></span>)</span></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span> <a href="random_1.html#K5" title="RANDOM_1:func.5">(#)</a> <font color="Maroon" title="c9">E0</font></span>)</span> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> 2</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">X</font> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> 2</span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p5">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span></span>)</span> <a href="random_1.html#K5" title="RANDOM_1:func.5">(#)</a> <font color="Maroon" title="c4">X</font></span>)</span></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p4">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span></span>)</span> <a href="random_1.html#K5" title="RANDOM_1:func.5">(#)</a> <font color="Maroon" title="c9">E0</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">x</font></span><br/>

<a NAME="E2:9_1_1_2_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A27</font></span>: 
( <font color="Maroon" title="c13">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="c4">X</font> &amp; <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Omega</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:9_1_1_2"><span class="lab"><font color="Green" title="E29">A24</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<span class="lab"><font color="Green" title="E33">A28</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">X</font> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> 2</span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p3">(<span class="default">2 <a href="random_1.html#K5" title="RANDOM_1:func.5">(#)</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span> <a href="random_1.html#K5" title="RANDOM_1:func.5">(#)</a> <font color="Maroon" title="c9">E0</font></span>)</span> <a href="random_1.html#K6" title="RANDOM_1:func.6">(#)</a> <font color="Maroon" title="c4">X</font></span>)</span></span>)</span></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span> <a href="random_1.html#K5" title="RANDOM_1:func.5">(#)</a> <font color="Maroon" title="c9">E0</font></span>)</span> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> 2</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">x</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">X</font> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> 2</span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p3">(<span class="default">2 <a href="random_1.html#K5" title="RANDOM_1:func.5">(#)</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span> <a href="random_1.html#K5" title="RANDOM_1:func.5">(#)</a> <font color="Maroon" title="c9">E0</font></span>)</span> <a href="random_1.html#K6" title="RANDOM_1:func.6">(#)</a> <font color="Maroon" title="c4">X</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">x</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span> <a href="random_1.html#K5" title="RANDOM_1:func.5">(#)</a> <font color="Maroon" title="c9">E0</font></span>)</span> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> 2</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9_1_1_2_6"><span class="lab"><font color="Green" title="E31">A26</font></span></a>, <a class="ref" href="valued_1.html#D1" title="VALUED_1:def.1">VALUED_1:def 1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">X</font> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> 2</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">x</font></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="random_1.html#K5" title="RANDOM_1:func.5">(#)</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span> <a href="random_1.html#K5" title="RANDOM_1:func.5">(#)</a> <font color="Maroon" title="c9">E0</font></span>)</span> <a href="random_1.html#K6" title="RANDOM_1:func.6">(#)</a> <font color="Maroon" title="c4">X</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">x</font></span>)</span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span> <a href="random_1.html#K5" title="RANDOM_1:func.5">(#)</a> <font color="Maroon" title="c9">E0</font></span>)</span> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> 2</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:9_1_1_2"><span class="lab"><font color="Green" title="E27">A22</font></span></a>, <a class="txt" href="#E8:9_1_1_2"><span class="lab"><font color="Green" title="E29">A24</font></span></a>, <a class="txt" href="#E1:9_1_1_2_6"><span class="lab"><font color="Green" title="E31">A26</font></span></a>, <a class="ref" href="valued_1.html#T13" title="VALUED_1:th.13">VALUED_1:13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">X</font> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> 2</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">x</font></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p2">(<span class="default">2 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span> <a href="random_1.html#K5" title="RANDOM_1:func.5">(#)</a> <font color="Maroon" title="c9">E0</font></span>)</span> <a href="random_1.html#K6" title="RANDOM_1:func.6">(#)</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">x</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span> <a href="random_1.html#K5" title="RANDOM_1:func.5">(#)</a> <font color="Maroon" title="c9">E0</font></span>)</span> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> 2</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#T6" title="VALUED_1:th.6">VALUED_1:6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">X</font> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> 2</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">x</font></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p2">(<span class="default">2 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span> <a href="random_1.html#K5" title="RANDOM_1:func.5">(#)</a> <font color="Maroon" title="c9">E0</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">x</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">X</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">x</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span> <a href="random_1.html#K5" title="RANDOM_1:func.5">(#)</a> <font color="Maroon" title="c9">E0</font></span>)</span> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> 2</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#T5" title="VALUED_1:th.5">VALUED_1:5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">X</font> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> 2</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">x</font></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p2">(<span class="default">2 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">E0</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">x</font></span>)</span></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">X</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">x</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span> <a href="random_1.html#K5" title="RANDOM_1:func.5">(#)</a> <font color="Maroon" title="c9">E0</font></span>)</span> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> 2</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#T6" title="VALUED_1:th.6">VALUED_1:6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">X</font> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> 2</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">x</font></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p2">(<span class="default">2 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> 1</span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">X</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">x</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span> <a href="random_1.html#K5" title="RANDOM_1:func.5">(#)</a> <font color="Maroon" title="c9">E0</font></span>)</span> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> 2</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T7" title="FUNCOP_1:th.7">FUNCOP_1:7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">X</font> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> 2</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">x</font></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p4">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">X</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">x</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span> <a href="random_1.html#K5" title="RANDOM_1:func.5">(#)</a> <font color="Maroon" title="c9">E0</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">x</font></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> 2</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9_1_1_2"><span class="lab"><font color="Green" title="E24">A19</font></span></a>, <a class="ref" href="mesfun6c.html#D4" title="MESFUN6C:def.4">MESFUN6C:def 4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">X</font> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> 2</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">x</font></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p4">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">X</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">x</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">E0</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">x</font></span>)</span></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> 2</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#T6" title="VALUED_1:th.6">VALUED_1:6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">X</font> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> 2</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">x</font></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p4">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">X</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">x</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> 1</span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> 2</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T7" title="FUNCOP_1:th.7">FUNCOP_1:7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">X</font> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> 2</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">x</font></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p4">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">X</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">x</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T46" title="POWER:th.46">POWER:46</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">X</font> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> 2</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">x</font></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p4">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">X</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">x</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p2">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span></span>)</span>

;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">X</font> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> 2</span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p5">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span></span>)</span> <a href="random_1.html#K5" title="RANDOM_1:func.5">(#)</a> <font color="Maroon" title="c4">X</font></span>)</span></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p4">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span></span>)</span> <a href="random_1.html#K5" title="RANDOM_1:func.5">(#)</a> <font color="Maroon" title="c9">E0</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">x</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">X</font> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> 2</span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p5">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span></span>)</span> <a href="random_1.html#K5" title="RANDOM_1:func.5">(#)</a> <font color="Maroon" title="c4">X</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">x</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p4">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span></span>)</span> <a href="random_1.html#K5" title="RANDOM_1:func.5">(#)</a> <font color="Maroon" title="c9">E0</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9_1_1_2_6"><span class="lab"><font color="Green" title="E31">A26</font></span></a>, <a class="txt" href="#E9:9_1_1_2"><span class="lab"><font color="Green" title="E30">A25</font></span></a>, <a class="ref" href="valued_1.html#D1" title="VALUED_1:def.1">VALUED_1:def 1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">X</font> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> 2</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">x</font></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p5">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span></span>)</span> <a href="random_1.html#K5" title="RANDOM_1:func.5">(#)</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">x</font></span>)</span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p4">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span></span>)</span> <a href="random_1.html#K5" title="RANDOM_1:func.5">(#)</a> <font color="Maroon" title="c9">E0</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:9_1_1_2"><span class="lab"><font color="Green" title="E28">A23</font></span></a>, <a class="txt" href="#E2:9_1_1_2_6"><span class="lab"><font color="Green" title="E32">A27</font></span></a>, <a class="ref" href="valued_1.html#T13" title="VALUED_1:th.13">VALUED_1:13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">X</font> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> 2</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">x</font></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p4">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">X</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">x</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p4">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span></span>)</span> <a href="random_1.html#K5" title="RANDOM_1:func.5">(#)</a> <font color="Maroon" title="c9">E0</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#T6" title="VALUED_1:th.6">VALUED_1:6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">X</font> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> 2</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">x</font></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p4">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">X</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">x</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p3">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">E0</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">x</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#T6" title="VALUED_1:th.6">VALUED_1:6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">X</font> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> 2</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">x</font></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p4">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">X</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">x</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p3">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T7" title="FUNCOP_1:th.7">FUNCOP_1:7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">X</font> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> 2</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">x</font></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p4">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">X</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">x</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p2">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span></span>)</span>

;<br/>
<span class="kw">hence </span><a NAME="E5:9_1_1_2_6"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">X</font> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> 2</span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p3">(<span class="default">2 <a href="random_1.html#K5" title="RANDOM_1:func.5">(#)</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span> <a href="random_1.html#K5" title="RANDOM_1:func.5">(#)</a> <font color="Maroon" title="c9">E0</font></span>)</span> <a href="random_1.html#K6" title="RANDOM_1:func.6">(#)</a> <font color="Maroon" title="c4">X</font></span>)</span></span>)</span></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span> <a href="random_1.html#K5" title="RANDOM_1:func.5">(#)</a> <font color="Maroon" title="c9">E0</font></span>)</span> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> 2</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">X</font> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> 2</span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p5">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span></span>)</span> <a href="random_1.html#K5" title="RANDOM_1:func.5">(#)</a> <font color="Maroon" title="c4">X</font></span>)</span></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p4">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span></span>)</span> <a href="random_1.html#K5" title="RANDOM_1:func.5">(#)</a> <font color="Maroon" title="c9">E0</font></span>)</span></span>)</span> <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="#E3:9_1_1_2_6"><span class="lab"><font color="Green" title="E33">A28</font></span></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>
<span class="kw">hence </span><a NAME="E11:9_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">X</font> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> 2</span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p2">(<span class="default">2 <a href="random_1.html#K5" title="RANDOM_1:func.5">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span> <a href="random_1.html#K5" title="RANDOM_1:func.5">(#)</a> <font color="Maroon" title="c9">E0</font></span>)</span> <a href="random_1.html#K6" title="RANDOM_1:func.6">(#)</a> <font color="Maroon" title="c4">X</font></span>)</span></span>)</span></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span> <a href="random_1.html#K5" title="RANDOM_1:func.5">(#)</a> <font color="Maroon" title="c9">E0</font></span>)</span> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> 2</span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">X</font> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> 2</span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p4">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span></span>)</span> <a href="random_1.html#K5" title="RANDOM_1:func.5">(#)</a> <font color="Maroon" title="c4">X</font></span>)</span></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p3">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span></span>)</span> <a href="random_1.html#K5" title="RANDOM_1:func.5">(#)</a> <font color="Maroon" title="c9">E0</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:9_1_1_2"><span class="lab"><font color="Green" title="E30">A25</font></span></a>, <a class="ref" href="partfun1.html#T5" title="PARTFUN1:th.5">PARTFUN1:5</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>
<span class="lab"><font color="Green" title="E22">A29</font></span>: <span class="p1">(<span class="default"><a href="random_1.html#K7" title="RANDOM_1:func.7">abs</a> <font color="Maroon" title="c12">Y</font></span>)</span> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> 2 = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">X</font> <a href="random_1.html#K4" title="RANDOM_1:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span> <a href="random_1.html#K5" title="RANDOM_1:func.5">(#)</a> <font color="Maroon" title="c9">E0</font></span>)</span></span>)</span> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> 2
<span class="kw">by</span> <span class="lab"><a class="ref" href="random_2.html#T5" target="_self" title="RANDOM_2:th.5">Th5</a>, <a class="txt" href="#E10:9_1_1"><span class="lab"><font color="Green" title="E13">A8</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">X</font> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> 2</span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p2">(<span class="default">2 <a href="random_1.html#K5" title="RANDOM_1:func.5">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span> <a href="random_1.html#K5" title="RANDOM_1:func.5">(#)</a> <font color="Maroon" title="c9">E0</font></span>)</span> <a href="random_1.html#K6" title="RANDOM_1:func.6">(#)</a> <font color="Maroon" title="c4">X</font></span>)</span></span>)</span></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span> <a href="random_1.html#K5" title="RANDOM_1:func.5">(#)</a> <font color="Maroon" title="c9">E0</font></span>)</span> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> 2</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="random_2.html#T6" target="_self" title="RANDOM_2:th.6">Th6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="random_1.html#K7" title="RANDOM_1:func.7">abs</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> 2</span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p4">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span></span>)</span> <a href="random_1.html#K5" title="RANDOM_1:func.5">(#)</a> <font color="Maroon" title="c4">X</font></span>)</span></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p3">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span></span>)</span> <a href="random_1.html#K5" title="RANDOM_1:func.5">(#)</a> <font color="Maroon" title="c9">E0</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="random_2.html#T5" target="_self" title="RANDOM_2:th.5">Th5</a>, <a class="txt" href="#E24:9_1_1"><span class="lab"><font color="Green" title="E21">A16</font></span></a></span>
;<br/>
<a NAME="E26:9_1_1"/><span class="lab"><font color="Green" title="E23">A30</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p2">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span></span>)</span> <a href="random_1.html#K5" title="RANDOM_1:func.5">(#)</a> <font color="Maroon" title="c9">E0</font> <a href="mesfunc6.html#R3" title="MESFUNC6:pred.3">is_integrable_on</a>  <a href="prob_4.html#K2" title="PROB_4:func.2">P2M</a> <font color="Maroon" title="c3">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:9_1_1"><span class="lab"><font color="Green" title="E14">A9</font></span></a>, <a class="ref" href="mesfunc6.html#T102" title="MESFUNC6:th.102">MESFUNC6:102</a></span>;<br/>
<a NAME="E27:9_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="random_1.html#K7" title="RANDOM_1:func.7">abs</a> <font color="Maroon" title="c12">Y</font></span>)</span> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> 2 <a href="mesfunc6.html#R3" title="MESFUNC6:pred.3">is_integrable_on</a>  <a href="prob_4.html#K2" title="PROB_4:func.2">P2M</a> <font color="Maroon" title="c3">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:9_1_1"><span class="lab"><font color="Green" title="E20">A15</font></span></a>, <a class="txt" href="#E25:9_1_1"><span class="lab"><font color="Green" title="E22">A29</font></span></a>, <a class="ref" href="mesfunc6.html#T100" title="MESFUNC6:th.100">MESFUNC6:100</a></span>;<br/>
<a NAME="E28:9_1_1"/><span class="kw">then </span>
(  <a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> (<span class="p1">(<span class="default"><a href="prob_4.html#K2" title="PROB_4:func.2">P2M</a> <font color="Maroon" title="c3">P</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="random_1.html#K7" title="RANDOM_1:func.7">abs</a> <font color="Maroon" title="c12">Y</font></span>)</span> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> 2</span>)</span>) &amp;  <a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> (<span class="p1">(<span class="default"><a href="prob_4.html#K2" title="PROB_4:func.2">P2M</a> <font color="Maroon" title="c3">P</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="random_1.html#K7" title="RANDOM_1:func.7">abs</a> <font color="Maroon" title="c12">Y</font></span>)</span> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> 2</span>)</span>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc6.html#T90" title="MESFUNC6:th.90">MESFUNC6:90</a></span>;<br/>
<a NAME="E29:9_1_1"/><span class="kw">then </span>
 <a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> (<span class="p1">(<span class="default"><a href="prob_4.html#K2" title="PROB_4:func.2">P2M</a> <font color="Maroon" title="c3">P</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="random_1.html#K7" title="RANDOM_1:func.7">abs</a> <font color="Maroon" title="c12">Y</font></span>)</span> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> 2</span>)</span>) is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T14" title="XXREAL_0:th.14">XXREAL_0:14</a></span>;<br/>
<span class="kw">hence </span><a NAME="E30:9_1_1"/>
(  ex <font color="Olive">b<sub>1</sub></font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  ex <font color="Olive" title="b2">Y</font>, <font color="Olive" title="b3">E</font> being    <a href="random_1.html#M1" title="RANDOM_1:mode.1">Real-Valued-Random-Variable</a> of <font color="Maroon" title="c2">Sigma</font> st <br/>( <font color="Olive" title="b3">E</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c1">Omega</font> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <span class="p1">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span> &amp; <font color="Olive" title="b2">Y</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c4">X</font> <a href="random_1.html#K4" title="RANDOM_1:func.4">-</a> <font color="Olive" title="b3">E</font> &amp; <font color="Olive" title="b2">Y</font> <a href="random_1.html#R1" title="RANDOM_1:pred.1">is_integrable_on</a> <font color="Maroon" title="c3">P</font> &amp; <span class="p1">(<span class="default"><a href="random_1.html#K7" title="RANDOM_1:func.7">abs</a> <font color="Olive" title="b2">Y</font></span>)</span> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> 2 <a href="mesfunc6.html#R3" title="MESFUNC6:pred.3">is_integrable_on</a>  <a href="prob_4.html#K2" title="PROB_4:func.2">P2M</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> (<span class="p1">(<span class="default"><a href="prob_4.html#K2" title="PROB_4:func.2">P2M</a> <font color="Maroon" title="c3">P</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="random_1.html#K7" title="RANDOM_1:func.7">abs</a> <font color="Olive" title="b2">Y</font></span>)</span> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> 2</span>)</span>) ) &amp; (  for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>   st  ex <font color="Olive" title="b3">Y</font>, <font color="Olive" title="b4">E</font> being    <a href="random_1.html#M1" title="RANDOM_1:mode.1">Real-Valued-Random-Variable</a> of <font color="Maroon" title="c2">Sigma</font> st <br/>( <font color="Olive" title="b4">E</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c1">Omega</font> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <span class="p1">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span> &amp; <font color="Olive" title="b3">Y</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c4">X</font> <a href="random_1.html#K4" title="RANDOM_1:func.4">-</a> <font color="Olive" title="b4">E</font> &amp; <font color="Olive" title="b3">Y</font> <a href="random_1.html#R1" title="RANDOM_1:pred.1">is_integrable_on</a> <font color="Maroon" title="c3">P</font> &amp; <span class="p1">(<span class="default"><a href="random_1.html#K7" title="RANDOM_1:func.7">abs</a> <font color="Olive" title="b3">Y</font></span>)</span> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> 2 <a href="mesfunc6.html#R3" title="MESFUNC6:pred.3">is_integrable_on</a>  <a href="prob_4.html#K2" title="PROB_4:func.2">P2M</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> (<span class="p1">(<span class="default"><a href="prob_4.html#K2" title="PROB_4:func.2">P2M</a> <font color="Maroon" title="c3">P</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="random_1.html#K7" title="RANDOM_1:func.7">abs</a> <font color="Olive" title="b3">Y</font></span>)</span> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> 2</span>)</span>) ) &amp;  ex <font color="Olive" title="b3">Y</font>, <font color="Olive" title="b4">E</font> being    <a href="random_1.html#M1" title="RANDOM_1:mode.1">Real-Valued-Random-Variable</a> of <font color="Maroon" title="c2">Sigma</font> st <br/>( <font color="Olive" title="b4">E</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c1">Omega</font> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <span class="p1">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">P</font>)</span>)</span> &amp; <font color="Olive" title="b3">Y</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c4">X</font> <a href="random_1.html#K4" title="RANDOM_1:func.4">-</a> <font color="Olive" title="b4">E</font> &amp; <font color="Olive" title="b3">Y</font> <a href="random_1.html#R1" title="RANDOM_1:pred.1">is_integrable_on</a> <font color="Maroon" title="c3">P</font> &amp; <span class="p1">(<span class="default"><a href="random_1.html#K7" title="RANDOM_1:func.7">abs</a> <font color="Olive" title="b3">Y</font></span>)</span> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> 2 <a href="mesfunc6.html#R3" title="MESFUNC6:pred.3">is_integrable_on</a>  <a href="prob_4.html#K2" title="PROB_4:func.2">P2M</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> (<span class="p1">(<span class="default"><a href="prob_4.html#K2" title="PROB_4:func.2">P2M</a> <font color="Maroon" title="c3">P</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="random_1.html#K7" title="RANDOM_1:func.7">abs</a> <font color="Olive" title="b3">Y</font></span>)</span> <a href="mesfun6c.html#K2" title="MESFUN6C:func.2">to_power</a> 2</span>)</span>) ) holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:9_1_1"><span class="lab"><font color="Green" title="E23">A30</font></span></a>, <a class="txt" href="#E23:9_1_1"><span class="lab"><font color="Green" title="E20">A15</font></span></a>, <a class="txt" href="#E25:9_1_1"><span class="lab"><font color="Green" title="E22">A29</font></span></a>, <a class="txt" href="#E20:9_1_1"><span class="lab"><font color="Green" title="E19">A14</font></span></a>, <a class="ref" href="mesfunc6.html#T100" title="MESFUNC6:th.100">MESFUNC6:100</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>
