<?xml version="1.0"?>
<div class="add">

<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> holds <br/> (  <a href="random_2.html#K3" title="RANDOM_2:func.3">Real-Valued-Random-Variables-Set</a> <font color="Olive" title="b1">Sigma</font> is  <a href="c0sp1.html#V4" title="C0SP1:attr.4">additively-linearly-closed</a>  &amp;  <a href="random_2.html#K3" title="RANDOM_2:func.3">Real-Valued-Random-Variables-Set</a> <font color="Olive" title="b1">Sigma</font> is  <a href="c0sp1.html#V3" title="C0SP1:attr.3">multiplicatively-closed</a>  )</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"> (  <a href="random_2.html#K3" title="RANDOM_2:func.3">Real-Valued-Random-Variables-Set</a> <font color="Maroon" title="c2">Sigma</font> is  <a href="c0sp1.html#V4" title="C0SP1:attr.4">additively-linearly-closed</a>  &amp;  <a href="random_2.html#K3" title="RANDOM_2:func.3">Real-Valued-Random-Variables-Set</a> <font color="Maroon" title="c2">Sigma</font> is  <a href="c0sp1.html#V3" title="C0SP1:attr.3">multiplicatively-closed</a>  )</span><br/>



<span class="kw">set </span><font color="Maroon" title="c3">W</font> =  <a href="random_2.html#K3" title="RANDOM_2:func.3">Real-Valued-Random-Variables-Set</a> <font color="Maroon" title="c2">Sigma</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c4">V</font> =  <a href="funcsdom.html#K12" title="FUNCSDOM:func.12">RAlgebra</a> <font color="Maroon" title="c1">Omega</font>;<br/>
<a NAME="E1:32"/><span class="lab"><font color="Green" title="E28">A1</font></span>: 
 for <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">u</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="funcsdom.html#K12" title="FUNCSDOM:func.12">RAlgebra</a> <font color="Maroon" title="c1">Omega</font></span>)</span>  st <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="random_2.html#K3" title="RANDOM_2:func.3">Real-Valued-Random-Variables-Set</a> <font color="Maroon" title="c2">Sigma</font> &amp; <font color="Olive" title="b2">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="random_2.html#K3" title="RANDOM_2:func.3">Real-Valued-Random-Variables-Set</a> <font color="Maroon" title="c2">Sigma</font> holds <br/><font color="Olive" title="b1">v</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Olive" title="b2">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="random_2.html#K3" title="RANDOM_2:func.3">Real-Valued-Random-Variables-Set</a> <font color="Maroon" title="c2">Sigma</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">v</font>, <font color="Maroon" title="c6">u</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="funcsdom.html#K12" title="FUNCSDOM:func.12">RAlgebra</a> <font color="Maroon" title="c1">Omega</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="random_2.html#K3" title="RANDOM_2:func.3">Real-Valued-Random-Variables-Set</a> <font color="Maroon" title="c2">Sigma</font> &amp; <font color="Maroon" title="c6">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="random_2.html#K3" title="RANDOM_2:func.3">Real-Valued-Random-Variables-Set</a> <font color="Maroon" title="c2">Sigma</font> implies <font color="Maroon" title="c5">v</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c6">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="random_2.html#K3" title="RANDOM_2:func.3">Real-Valued-Random-Variables-Set</a> <font color="Maroon" title="c2">Sigma</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:32_1"/><span class="lab"><font color="Green" title="E29">A2</font></span>: 
<font color="Maroon" title="c5">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="random_2.html#K3" title="RANDOM_2:func.3">Real-Valued-Random-Variables-Set</a> <font color="Maroon" title="c2">Sigma</font>
 <span class="kw">and </span><br/><a NAME="E2:32_1"/><span class="lab"><font color="Green" title="E30">A3</font></span>: 
<font color="Maroon" title="c6">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="random_2.html#K3" title="RANDOM_2:func.3">Real-Valued-Random-Variables-Set</a> <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="c5">v</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c6">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="random_2.html#K3" title="RANDOM_2:func.3">Real-Valued-Random-Variables-Set</a> <font color="Maroon" title="c2">Sigma</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c7">u1</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><span class="kw"> such that </span><br/><a NAME="E4:32_1"/><span class="lab"><font color="Green" title="E31">A4</font></span>: 
<font color="Maroon" title="c7">u1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">u</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:32_1"><span class="lab"><font color="Green" title="E30">A3</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">h</font> = <font color="Maroon" title="c5">v</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c6">u</font> as    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<font color="Maroon" title="c1">Omega</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>) ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c9">v1</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><span class="kw"> such that </span><br/><a NAME="E7:32_1"/><span class="lab"><font color="Green" title="E32">A5</font></span>: 
<font color="Maroon" title="c9">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">v</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32_1"><span class="lab"><font color="Green" title="E29">A2</font></span></a></span>;<br/>
<a NAME="E8:32_1"/>
<span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c9">v1</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="c7">u1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="c7">u1</font></span>)</span>
 
<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="E9:32_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A6</font></span>: 
(  ex <font color="Olive" title="b1">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Maroon" title="c8">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">Omega</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  ) &amp; <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c9">v1</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="c7">u1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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 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>, <a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></span>;<br/>
<a NAME="E10:32_1"/>
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   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> <font color="Maroon" title="c8">h</font> holds <br/><font color="Maroon" title="c8">h</font> <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"><font color="Maroon" title="c9">v1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</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="c7">u1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:32_1"><span class="lab"><font color="Green" title="E32">A5</font></span></a>, <a class="txt" href="#E4:32_1"><span class="lab"><font color="Green" title="E31">A4</font></span></a>, <a class="ref" href="funcsdom.html#T2" title="FUNCSDOM:th.2">FUNCSDOM:2</a></span>;<br/>
<a NAME="E11:32_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">v</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c6">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">v1</font> <a href="random_1.html#K6" title="RANDOM_1:func.6">(#)</a> <font color="Maroon" title="c7">u1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:32_1"><span class="lab"><font color="Green" title="E33">A6</font></span></a>, <a class="ref" href="valued_1.html#D4" title="VALUED_1:def.4">VALUED_1:def 4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:32_1"/>
<font color="Maroon" title="c5">v</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c6">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="random_2.html#K3" title="RANDOM_2:func.3">Real-Valued-Random-Variables-Set</a> <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"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:32"/>
 for <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">u</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="funcsdom.html#K12" title="FUNCSDOM:func.12">RAlgebra</a> <font color="Maroon" title="c1">Omega</font></span>)</span>  st <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="random_2.html#K3" title="RANDOM_2:func.3">Real-Valued-Random-Variables-Set</a> <font color="Maroon" title="c2">Sigma</font> &amp; <font color="Olive" title="b2">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="random_2.html#K3" title="RANDOM_2:func.3">Real-Valued-Random-Variables-Set</a> <font color="Maroon" title="c2">Sigma</font> holds <br/><font color="Olive" title="b1">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b2">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="random_2.html#K3" title="RANDOM_2:func.3">Real-Valued-Random-Variables-Set</a> <font color="Maroon" title="c2">Sigma</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">v</font>, <font color="Maroon" title="c6">u</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="funcsdom.html#K12" title="FUNCSDOM:func.12">RAlgebra</a> <font color="Maroon" title="c1">Omega</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="random_2.html#K3" title="RANDOM_2:func.3">Real-Valued-Random-Variables-Set</a> <font color="Maroon" title="c2">Sigma</font> &amp; <font color="Maroon" title="c6">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="random_2.html#K3" title="RANDOM_2:func.3">Real-Valued-Random-Variables-Set</a> <font color="Maroon" title="c2">Sigma</font> implies <font color="Maroon" title="c5">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c6">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="random_2.html#K3" title="RANDOM_2:func.3">Real-Valued-Random-Variables-Set</a> <font color="Maroon" title="c2">Sigma</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:32_2"/><span class="lab"><font color="Green" title="E29">A7</font></span>: 
<font color="Maroon" title="c5">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="random_2.html#K3" title="RANDOM_2:func.3">Real-Valued-Random-Variables-Set</a> <font color="Maroon" title="c2">Sigma</font>
 <span class="kw">and </span><br/><a NAME="E2:32_2"/><span class="lab"><font color="Green" title="E30">A8</font></span>: 
<font color="Maroon" title="c6">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="random_2.html#K3" title="RANDOM_2:func.3">Real-Valued-Random-Variables-Set</a> <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="c5">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c6">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="random_2.html#K3" title="RANDOM_2:func.3">Real-Valued-Random-Variables-Set</a> <font color="Maroon" title="c2">Sigma</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c7">u1</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><span class="kw"> such that </span><br/><a NAME="E4:32_2"/><span class="lab"><font color="Green" title="E31">A9</font></span>: 
<font color="Maroon" title="c7">u1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">u</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:32_2"><span class="lab"><font color="Green" title="E30">A8</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">h</font> = <font color="Maroon" title="c5">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c6">u</font> as    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<font color="Maroon" title="c1">Omega</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>) ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c9">v1</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><span class="kw"> such that </span><br/><a NAME="E7:32_2"/><span class="lab"><font color="Green" title="E32">A10</font></span>: 
<font color="Maroon" title="c9">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">v</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32_2"><span class="lab"><font color="Green" title="E29">A7</font></span></a></span>;<br/>
<a NAME="E8:32_2"/>
<span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c9">v1</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="c7">u1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="c7">u1</font></span>)</span>
 
<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="E9:32_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A11</font></span>: 
(  ex <font color="Olive" title="b1">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Maroon" title="c8">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">Omega</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  ) &amp; <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c9">v1</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="c7">u1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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 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>, <a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></span>;<br/>
<a NAME="E10:32_2"/>
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   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> <font color="Maroon" title="c8">h</font> holds <br/><font color="Maroon" title="c8">h</font> <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"><font color="Maroon" title="c9">v1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">u1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:32_2"><span class="lab"><font color="Green" title="E32">A10</font></span></a>, <a class="txt" href="#E4:32_2"><span class="lab"><font color="Green" title="E31">A9</font></span></a>, <a class="ref" href="funcsdom.html#T1" title="FUNCSDOM:th.1">FUNCSDOM:1</a></span>;<br/>
<a NAME="E11:32_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c6">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">v1</font> <a href="random_1.html#K3" title="RANDOM_1:func.3">+</a> <font color="Maroon" title="c7">u1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:32_2"><span class="lab"><font color="Green" title="E33">A11</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="kw">hence </span><a NAME="E12:32_2"/>
<font color="Maroon" title="c5">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c6">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="random_2.html#K3" title="RANDOM_2:func.3">Real-Valued-Random-Variables-Set</a> <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"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E3:32"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A12</font></span>: 
 <a href="random_2.html#K3" title="RANDOM_2:func.3">Real-Valued-Random-Variables-Set</a> <font color="Maroon" title="c2">Sigma</font> is  <a href="ideal_1.html#V1" title="IDEAL_1:attr.1">add-closed</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ideal_1.html#D1" title="IDEAL_1:def.1">IDEAL_1:def 1</a></span>;<br/>
<a NAME="E4:32"/><span class="lab"><font color="Green" title="E30">A13</font></span>: 
 <a href="funcsdom.html#K12" title="FUNCSDOM:func.12">RAlgebra</a> <font color="Maroon" title="c1">Omega</font> is   <a href="rlvect_1.html#NM2" title="RLVECT_1:NM.2">RealLinearSpace</a>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="c0sp1.html#T7" title="C0SP1:th.7">C0SP1:7</a></span>;<br/>
<a NAME="E5:32"/>
 for <font color="Olive" title="b1">v</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="funcsdom.html#K12" title="FUNCSDOM:func.12">RAlgebra</a> <font color="Maroon" title="c1">Omega</font></span>)</span>  st <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="random_2.html#K3" title="RANDOM_2:func.3">Real-Valued-Random-Variables-Set</a> <font color="Maroon" title="c2">Sigma</font> holds <br/> <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="random_2.html#K3" title="RANDOM_2:func.3">Real-Valued-Random-Variables-Set</a> <font color="Maroon" title="c2">Sigma</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">v</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="funcsdom.html#K12" title="FUNCSDOM:func.12">RAlgebra</a> <font color="Maroon" title="c1">Omega</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="random_2.html#K3" title="RANDOM_2:func.3">Real-Valued-Random-Variables-Set</a> <font color="Maroon" title="c2">Sigma</font> implies  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="random_2.html#K3" title="RANDOM_2:func.3">Real-Valued-Random-Variables-Set</a> <font color="Maroon" title="c2">Sigma</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:32_3"/>
<font color="Maroon" title="c5">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="random_2.html#K3" title="RANDOM_2:func.3">Real-Valued-Random-Variables-Set</a> <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">  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="random_2.html#K3" title="RANDOM_2:func.3">Real-Valued-Random-Variables-Set</a> <font color="Maroon" title="c2">Sigma</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">v1</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><span class="kw"> such that </span><br/><a NAME="E3:32_3"/><span class="lab"><font color="Green" title="E31">A14</font></span>: 
<font color="Maroon" title="c6">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">v</font>
 ;<br/>
<a NAME="E4:32_3"/><span class="lab"><font color="Green" title="E32">A15</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="E5:32_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A16</font></span>: 
 <a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <font color="Maroon" title="c6">v1</font> is    <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_1.html#T20" title="RANDOM_1:th.20">RANDOM_1:20</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">h</font> =  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">v</font> as    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<font color="Maroon" title="c1">Omega</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>) ;<br/>
<a NAME="E7:32_3"/><span class="lab"><font color="Green" title="E34">A17</font></span>: 
<font color="Maroon" title="c7">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1</span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">v</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:32"><span class="lab"><font color="Green" title="E30">A13</font></span></a>, <a class="ref" href="rlvect_1.html#T16" title="RLVECT_1:th.16">RLVECT_1:16</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E35">A18</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c8">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="c8">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="c7">h</font> implies <font color="Maroon" title="c7">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">v1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">x</font></span>)</span> )</span><br/><span class="kw">assume </span><a NAME="E1:32_3_1"/>
<font color="Maroon" title="c8">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="c7">h</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="c7">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">v1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">x</font></span>)</span></span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">y</font> = <font color="Maroon" title="c8">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">Omega</font> ;<br/><span class="kw">thus </span><font color="Maroon" title="c7">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">x</font> = 
<span class="p1">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1</span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">v1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">y</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:32_3"><span class="lab"><font color="Green" title="E34">A17</font></span></a>, <a class="txt" href="#E3:32_3"><span class="lab"><font color="Green" title="E31">A14</font></span></a>, <a class="txt" href="#E4:32_3"><span class="lab"><font color="Green" title="E32">A15</font></span></a>, <a class="ref" href="funcsdom.html#T4" title="FUNCSDOM:th.4">FUNCSDOM:4</a></span>
<br/>.= 
 <a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">v1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">x</font></span>)</span>

; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E9:32_3"/>
(  ex <font color="Olive" title="b1">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Maroon" title="c7">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">Omega</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  ) &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c6">v1</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>, <a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></span>;<br/>
<a NAME="E10:32_3"/><span class="kw">then </span>
 <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <font color="Maroon" title="c6">v1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:32_3"><span class="lab"><font color="Green" title="E35">A18</font></span></a>, <a class="ref" href="valued_1.html#T9" title="VALUED_1:th.9">VALUED_1:9</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:32_3"/>
 <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="random_2.html#K3" title="RANDOM_2:func.3">Real-Valued-Random-Variables-Set</a> <font color="Maroon" title="c2">Sigma</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:32_3"><span class="lab"><font color="Green" title="E33">A16</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>
<a NAME="E6:32"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A19</font></span>: 
 <a href="random_2.html#K3" title="RANDOM_2:func.3">Real-Valued-Random-Variables-Set</a> <font color="Maroon" title="c2">Sigma</font> is  <a href="c0sp1.html#V1" title="C0SP1:attr.1">having-inverse</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="c0sp1.html#D1" title="C0SP1:def.1">C0SP1:def 1</a></span>;<br/>
<a NAME="E7:32"/>
 for <font color="Olive" title="b1">a</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><br/>  for <font color="Olive" title="b2">u</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="funcsdom.html#K12" title="FUNCSDOM:func.12">RAlgebra</a> <font color="Maroon" title="c1">Omega</font></span>)</span>  st <font color="Olive" title="b2">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="random_2.html#K3" title="RANDOM_2:func.3">Real-Valued-Random-Variables-Set</a> <font color="Maroon" title="c2">Sigma</font> holds <br/><font color="Olive" title="b1">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b2">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="random_2.html#K3" title="RANDOM_2:func.3">Real-Valued-Random-Variables-Set</a> <font color="Maroon" title="c2">Sigma</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">a</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</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">u</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="funcsdom.html#K12" title="FUNCSDOM:func.12">RAlgebra</a> <font color="Maroon" title="c1">Omega</font></span>)</span>  st <font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="random_2.html#K3" title="RANDOM_2:func.3">Real-Valued-Random-Variables-Set</a> <font color="Maroon" title="c2">Sigma</font> holds <br/><font color="Maroon" title="c5">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="random_2.html#K3" title="RANDOM_2:func.3">Real-Valued-Random-Variables-Set</a> <font color="Maroon" title="c2">Sigma</font></span><br/><span class="kw">let </span><font color="Maroon" title="c6">u</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="funcsdom.html#K12" title="FUNCSDOM:func.12">RAlgebra</a> <font color="Maroon" title="c1">Omega</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="random_2.html#K3" title="RANDOM_2:func.3">Real-Valued-Random-Variables-Set</a> <font color="Maroon" title="c2">Sigma</font> implies <font color="Maroon" title="c5">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="random_2.html#K3" title="RANDOM_2:func.3">Real-Valued-Random-Variables-Set</a> <font color="Maroon" title="c2">Sigma</font> )</span><br/>



<span class="kw">assume </span><a NAME="E1:32_4"/>
<font color="Maroon" title="c6">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="random_2.html#K3" title="RANDOM_2:func.3">Real-Valued-Random-Variables-Set</a> <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="c5">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="random_2.html#K3" title="RANDOM_2:func.3">Real-Valued-Random-Variables-Set</a> <font color="Maroon" title="c2">Sigma</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">u1</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><span class="kw"> such that </span><br/><a NAME="E3:32_4"/><span class="lab"><font color="Green" title="E32">A20</font></span>: 
<font color="Maroon" title="c7">u1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">u</font>
 ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">h</font> = <font color="Maroon" title="c5">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">u</font> as    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<font color="Maroon" title="c1">Omega</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>) ;<br/>
<a NAME="E5:32_4"/><span class="lab"><font color="Green" title="E33">A21</font></span>: 
(  ex <font color="Olive" title="b1">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Maroon" title="c8">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">Omega</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  ) &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c7">u1</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>, <a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></span>;<br/>
<a NAME="E6:32_4"/>
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   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> <font color="Maroon" title="c8">h</font> holds <br/><font color="Maroon" title="c8">h</font> <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> <font color="Maroon" title="c5">a</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">u1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:32_4"><span class="lab"><font color="Green" title="E32">A20</font></span></a>, <a class="ref" href="funcsdom.html#T4" title="FUNCSDOM:th.4">FUNCSDOM:4</a></span>;<br/>
<a NAME="E7:32_4"/><span class="kw">then </span>
<font color="Maroon" title="c5">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> <a href="random_1.html#K5" title="RANDOM_1:func.5">(#)</a> <font color="Maroon" title="c7">u1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:32_4"><span class="lab"><font color="Green" title="E33">A21</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">hence </span><a NAME="E8:32_4"/>
<font color="Maroon" title="c5">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="random_2.html#K3" title="RANDOM_2:func.3">Real-Valued-Random-Variables-Set</a> <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"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E8:32"/>
 <a href="random_2.html#K3" title="RANDOM_2:func.3">Real-Valued-Random-Variables-Set</a> <font color="Maroon" title="c2">Sigma</font> is  <a href="c0sp1.html#V4" title="C0SP1:attr.4">additively-linearly-closed</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:32"><span class="lab"><font color="Green" title="E29">A12</font></span></a>, <a class="txt" href="#E6:32"><span class="lab"><font color="Green" title="E31">A19</font></span></a>, <a class="ref" href="c0sp1.html#D10" title="C0SP1:def.10">C0SP1:def 10</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="random_2.html#K3" title="RANDOM_2:func.3">Real-Valued-Random-Variables-Set</a> <font color="Maroon" title="c2">Sigma</font> is  <a href="c0sp1.html#V3" title="C0SP1:attr.3">multiplicatively-closed</a> </span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c5">g</font> =  <a href="funcsdom.html#K9" title="FUNCSDOM:func.9">RealFuncUnit</a> <font color="Maroon" title="c1">Omega</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">Omega</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/>
<a NAME="E10:32"/>
<font color="Maroon" title="c5">g</font> is    <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="E11:32"/><span class="kw">then </span>
 <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <span class="p1">(<span class="default"><a href="funcsdom.html#K12" title="FUNCSDOM:func.12">RAlgebra</a> <font color="Maroon" title="c1">Omega</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="random_2.html#K3" title="RANDOM_2:func.3">Real-Valued-Random-Variables-Set</a> <font color="Maroon" title="c2">Sigma</font>
 
;<br/>
<span class="kw">hence </span><a NAME="E12:32"/>
 <a href="random_2.html#K3" title="RANDOM_2:func.3">Real-Valued-Random-Variables-Set</a> <font color="Maroon" title="c2">Sigma</font> is  <a href="c0sp1.html#V3" title="C0SP1:attr.3">multiplicatively-closed</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32"><span class="lab"><font color="Green" title="E28">A1</font></span></a>, <a class="ref" href="c0sp1.html#D4" title="C0SP1:def.4">C0SP1:def 4</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>
