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<span class="kw">let </span><font color="Maroon" title="c1">Omega</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">Sigma</font> being   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">Omega</font><br/>  for <font color="Olive" title="b2">P</font> being    <a href="prob_1.html#M2" title="PROB_1:mode.2">Probability</a> of <font color="Olive" title="b1">Sigma</font><br/>  for <font color="Olive" title="b3">f</font>, <font color="Olive" title="b4">g</font> being    <a href="random_1.html#M1" title="RANDOM_1:mode.1">Real-Valued-Random-Variable</a> of <font color="Olive" title="b1">Sigma</font>  st <font color="Olive" title="b3">f</font> <a href="random_1.html#R1" title="RANDOM_1:pred.1">is_integrable_on</a> <font color="Olive" title="b2">P</font> &amp; <font color="Olive" title="b4">g</font> <a href="random_1.html#R1" title="RANDOM_1:pred.1">is_integrable_on</a> <font color="Olive" title="b2">P</font> holds <br/> <a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<span class="p1">(<span class="default"><font color="Olive" title="b3">f</font> <a href="random_1.html#K3" title="RANDOM_1:func.3">+</a> <font color="Olive" title="b4">g</font></span>)</span>,<font color="Olive" title="b2">P</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="Olive" title="b3">f</font>,<font color="Olive" title="b2">P</font>)</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Olive" title="b4">g</font>,<font color="Olive" title="b2">P</font>)</span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">Sigma</font> be   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</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">  for <font color="Olive" title="b1">P</font> being    <a href="prob_1.html#M2" title="PROB_1:mode.2">Probability</a> of <font color="Maroon" title="c2">Sigma</font><br/>  for <font color="Olive" title="b2">f</font>, <font color="Olive" title="b3">g</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 <font color="Olive" title="b2">f</font> <a href="random_1.html#R1" title="RANDOM_1:pred.1">is_integrable_on</a> <font color="Olive" title="b1">P</font> &amp; <font color="Olive" title="b3">g</font> <a href="random_1.html#R1" title="RANDOM_1:pred.1">is_integrable_on</a> <font color="Olive" title="b1">P</font> holds <br/> <a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<span class="p1">(<span class="default"><font color="Olive" title="b2">f</font> <a href="random_1.html#K3" title="RANDOM_1:func.3">+</a> <font color="Olive" title="b3">g</font></span>)</span>,<font color="Olive" title="b1">P</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="Olive" title="b2">f</font>,<font color="Olive" title="b1">P</font>)</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Olive" title="b3">g</font>,<font color="Olive" title="b1">P</font>)</span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">P</font> be    <a href="prob_1.html#M2" title="PROB_1:mode.2">Probability</a> of <font color="Maroon" title="c2">Sigma</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</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 <font color="Olive" title="b1">f</font> <a href="random_1.html#R1" title="RANDOM_1:pred.1">is_integrable_on</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Olive" title="b2">g</font> <a href="random_1.html#R1" title="RANDOM_1:pred.1">is_integrable_on</a> <font color="Maroon" title="c3">P</font> holds <br/> <a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="random_1.html#K3" title="RANDOM_1:func.3">+</a> <font color="Olive" title="b2">g</font></span>)</span>,<font color="Maroon" title="c3">P</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="Olive" title="b1">f</font>,<font color="Maroon" title="c3">P</font>)</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Olive" title="b2">g</font>,<font color="Maroon" title="c3">P</font>)</span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c4">f</font>, <font color="Maroon" title="c5">g</font> be    <a href="random_1.html#M1" title="RANDOM_1:mode.1">Real-Valued-Random-Variable</a> of <font color="Maroon" title="c2">Sigma</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">f</font> <a href="random_1.html#R1" title="RANDOM_1:pred.1">is_integrable_on</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Maroon" title="c5">g</font> <a href="random_1.html#R1" title="RANDOM_1:pred.1">is_integrable_on</a> <font color="Maroon" title="c3">P</font> implies  <a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="random_1.html#K3" title="RANDOM_1:func.3">+</a> <font color="Maroon" title="c5">g</font></span>)</span>,<font color="Maroon" title="c3">P</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">f</font>,<font color="Maroon" title="c3">P</font>)</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c3">P</font>)</span>)</span> )</span><br/>









<span class="kw">set </span><font color="Maroon" title="c6">h</font> = <font color="Maroon" title="c4">f</font> <a href="random_1.html#K3" title="RANDOM_1:func.3">+</a> <font color="Maroon" title="c5">g</font>;<br/>
<span class="kw">assume </span><a NAME="E1:48"/><span class="lab"><font color="Green" title="E37">A1</font></span>: 
( <font color="Maroon" title="c4">f</font> <a href="random_1.html#R1" title="RANDOM_1:pred.1">is_integrable_on</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Maroon" title="c5">g</font> <a href="random_1.html#R1" title="RANDOM_1:pred.1">is_integrable_on</a> <font color="Maroon" title="c3">P</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="random_1.html#K3" title="RANDOM_1:func.3">+</a> <font color="Maroon" title="c5">g</font></span>)</span>,<font color="Maroon" title="c3">P</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">f</font>,<font color="Maroon" title="c3">P</font>)</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c3">P</font>)</span>)</span></span><br/>

<a NAME="E2:48"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A2</font></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>,<font color="Maroon" title="c4">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">P</font>) &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>,<font color="Maroon" title="c5">g</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c3">P</font>) )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="random_1.html#D4" target="_self" title="RANDOM_1:def.4">Def4</a></span>;<br/>
<a NAME="E3:48"/><span class="lab"><font color="Green" title="E39">A3</font></span>: 
( <font color="Maroon" title="c4">f</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> &amp; <font color="Maroon" title="c5">g</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="#E1:48"><span class="lab"><font color="Green" title="E37">A1</font></span></a>, <a class="ref" href="random_1.html#D3" target="_self" title="RANDOM_1:def.3">Def3</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">E</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="E5:48"/><span class="lab"><font color="Green" title="E40">A4</font></span>: 
<font color="Maroon" title="c7">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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">f</font></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="c5">g</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E6:48"/><span class="lab"><font color="Green" title="E41">A5</font></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"><font color="Maroon" title="c4">f</font> <a href="random_1.html#K3" title="RANDOM_1:func.3">+</a> <font color="Maroon" title="c5">g</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> (<span class="p2">(<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="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c7">E</font></span>)</span>)</span>)</span> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</a> <span class="p1">(<span class="default"><a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> (<span class="p2">(<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="p2">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c7">E</font></span>)</span>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc6.html#T101" title="MESFUNC6:th.101">MESFUNC6:101</a></span>;<br/>
<a NAME="E7:48"/><span class="lab"><font color="Green" title="E42">A6</font></span>: 
(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">Omega</font> &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">g</font> <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="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E8:48"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A7</font></span>: 
<font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c7">E</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c4">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:48"><span class="lab"><font color="Green" title="E40">A4</font></span></a>, <a class="ref" href="funct_2.html#T33" title="FUNCT_2:th.33">FUNCT_2:33</a></span>;<br/>
<a NAME="E9:48"/>
<font color="Maroon" title="c4">f</font> <a href="random_1.html#K3" title="RANDOM_1:func.3">+</a> <font color="Maroon" title="c5">g</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="#E3:48"><span class="lab"><font color="Green" title="E39">A3</font></span></a>, <a class="ref" href="mesfunc6.html#T100" title="MESFUNC6:th.100">MESFUNC6:100</a></span>;<br/>
<a NAME="E10:48"/><span class="kw">then </span>
<font color="Maroon" title="c4">f</font> <a href="random_1.html#K3" title="RANDOM_1:func.3">+</a> <font color="Maroon" title="c5">g</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" target="_self" title="RANDOM_1:def.3">Def3</a></span>;<br/>
<span class="kw">hence </span> <a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="random_1.html#K3" title="RANDOM_1:func.3">+</a> <font color="Maroon" title="c5">g</font></span>)</span>,<font color="Maroon" title="c3">P</font>) = 
 <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"><font color="Maroon" title="c4">f</font> <a href="random_1.html#K3" title="RANDOM_1:func.3">+</a> <font color="Maroon" title="c5">g</font></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="random_1.html#D4" target="_self" title="RANDOM_1:def.4">Def4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> (<span class="p2">(<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>,<font color="Maroon" title="c4">f</font>)</span>)</span> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</a> <span class="p1">(<span class="default"><a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> (<span class="p2">(<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>,<font color="Maroon" title="c5">g</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:48"><span class="lab"><font color="Green" title="E40">A4</font></span></a>, <a class="txt" href="#E6:48"><span class="lab"><font color="Green" title="E41">A5</font></span></a>, <a class="txt" href="#E7:48"><span class="lab"><font color="Green" title="E42">A6</font></span></a>, <a class="txt" href="#E8:48"><span class="lab"><font color="Green" title="E43">A7</font></span></a>, <a class="ref" href="funct_2.html#T33" title="FUNCT_2:th.33">FUNCT_2:33</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">f</font>,<font color="Maroon" title="c3">P</font>)</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="random_1.html#K8" title="RANDOM_1:func.8">expect</a> (<font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c3">P</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48"><span class="lab"><font color="Green" title="E38">A2</font></span></a>, <a class="ref" href="supinf_2.html#T1" title="SUPINF_2:th.1">SUPINF_2:1</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/>


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