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<span class="kw">let </span><font color="Maroon" title="c4">P1</font>, <font color="Maroon" title="c5">P2</font> be    <a href="prob_1.html#M2" title="PROB_1:mode.2">Probability</a> of  <a href="prob_4.html#K7" title="PROB_4:func.7">COM</a> <font color="Maroon" title="c2">Sigma</font>,<font color="Maroon" title="c3">P</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( for <font color="Olive" title="b1">B</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Sigma</font> holds <br/>for <font color="Olive" title="b2">C</font> being   <a href="prob_4.html#M2" title="PROB_4:mode.2">thin</a> of <font color="Maroon" title="c3">P</font> holds  <font color="Maroon" title="c4">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Olive" title="b2">C</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">B</font> ) &amp; ( for <font color="Olive" title="b1">B</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Sigma</font> holds <br/>for <font color="Olive" title="b2">C</font> being   <a href="prob_4.html#M2" title="PROB_4:mode.2">thin</a> of <font color="Maroon" title="c3">P</font> holds  <font color="Maroon" title="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Olive" title="b2">C</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">B</font> ) implies <font color="Maroon" title="c4">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P2</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:50_1_2"/><span class="lab"><font color="Green" title="E35">A82</font></span>: 
for <font color="Olive" title="b1">B</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Sigma</font> holds <br/>for <font color="Olive" title="b2">C</font> being   <a href="prob_4.html#M2" title="PROB_4:mode.2">thin</a> of <font color="Maroon" title="c3">P</font> holds  <font color="Maroon" title="c4">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Olive" title="b2">C</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">B</font>
 <span class="kw">and </span><br/><a NAME="E2:50_1_2"/><span class="lab"><font color="Green" title="E36">A83</font></span>: 
for <font color="Olive" title="b1">B</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Sigma</font> holds <br/>for <font color="Olive" title="b2">C</font> being   <a href="prob_4.html#M2" title="PROB_4:mode.2">thin</a> of <font color="Maroon" title="c3">P</font> holds  <font color="Maroon" title="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Olive" title="b2">C</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P2</font></span><br/>

<a NAME="E3:50_1_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="prob_4.html#K7" title="PROB_4:func.7">COM</a> <font color="Maroon" title="c2">Sigma</font>,<font color="Maroon" title="c3">P</font> holds <br/><font color="Maroon" title="c4">P1</font> <a href="funct_1.html#K1" title="FUNCT_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">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_4.html#K7" title="PROB_4:func.7">COM</a> <font color="Maroon" title="c2">Sigma</font>,<font color="Maroon" title="c3">P</font> implies <font color="Maroon" title="c4">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:50_1_2_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_4.html#K7" title="PROB_4:func.7">COM</a> <font color="Maroon" title="c2">Sigma</font>,<font color="Maroon" title="c3">P</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">B</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:50_1_2_1"/><span class="lab"><font color="Green" title="E37">A84</font></span>: 
( <font color="Maroon" title="c7">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Sigma</font> &amp;  ex <font color="Olive" title="b1">C</font> being   <a href="prob_4.html#M2" title="PROB_4:mode.2">thin</a> of <font color="Maroon" title="c3">P</font> st <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Olive" title="b1">C</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="prob_4.html#D5" target="_self" title="PROB_4:def.5">Def5</a></span>;<br/>
<font color="Maroon" title="c4">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> = 
<font color="Maroon" title="c3">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">B</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:50_1_2"><span class="lab"><font color="Green" title="E35">A82</font></span></a>, <a class="txt" href="#E3:50_1_2_1"><span class="lab"><font color="Green" title="E37">A84</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:50_1_2"><span class="lab"><font color="Green" title="E36">A83</font></span></a>, <a class="txt" href="#E3:50_1_2_1"><span class="lab"><font color="Green" title="E37">A84</font></span></a></span>
;<br/>
<span class="kw">hence </span><a NAME="E5:50_1_2_1"/>
<font color="Maroon" title="c4">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:50_1_2"/>
<font color="Maroon" title="c4">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T18" title="FUNCT_2:th.18">FUNCT_2:18</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
