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<b>thus </b><a NAME="E1:4_1"/>
( ( for <font color="Olive" title="b1">a</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds  <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">a</font>] ) implies for <font color="Olive" title="b1">a</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds  <font color="Maroon">P<sub>2</sub></font>[<font color="Olive" title="b1">a</font>] )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( for <font color="Olive" title="b1">a</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds  <font color="Maroon">P<sub>2</sub></font>[<font color="Olive" title="b1">a</font>] ) implies for <font color="Olive" title="b1">a</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds  <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">a</font>] )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:4_1_1"/><i><font color="Green" title="E2">A2</font></i>: 
for <font color="Olive" title="b1">a</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds  <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">a</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">a</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds  <font color="Maroon">P<sub>2</sub></font>[<font color="Olive" title="b1">a</font>]</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c1">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">P<sub>2</sub></font>[<font color="Maroon" title="c1">a</font>]</span><br/><a NAME="E1:4_1_1_1"/>
( ( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c1">a</font>] implies <font color="Maroon">P<sub>2</sub></font>[<font color="Maroon" title="c1">a</font>] ) &amp; ( <font color="Maroon">P<sub>2</sub></font>[<font color="Maroon" title="c1">a</font>] implies <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c1">a</font>] ) &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c1">a</font>] )
 <b>by </b><i><a class="txt" href="#E1:4"><i><font color="Green" title="E1">A1</font></i></a>, <a class="txt" href="#E1:4_1_1"><i><font color="Green" title="E2">A2</font></i></a></i>;<br/><b>hence </b><a NAME="E2:4_1_1_1"/>
<font color="Maroon">P<sub>2</sub></font>[<font color="Maroon" title="c1">a</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:4_1_1"/>
for <font color="Olive" title="b1">a</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds  <font color="Maroon">P<sub>2</sub></font>[<font color="Olive" title="b1">a</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:4_1"/><i><font color="Green" title="E2">A3</font></i>: 
for <font color="Olive" title="b1">a</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds  <font color="Maroon">P<sub>2</sub></font>[<font color="Olive" title="b1">a</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">a</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds  <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">a</font>]</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c1">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c1">a</font>]</span><br/><a NAME="E1:4_1_2"/>
( ( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c1">a</font>] implies <font color="Maroon">P<sub>2</sub></font>[<font color="Maroon" title="c1">a</font>] ) &amp; ( <font color="Maroon">P<sub>2</sub></font>[<font color="Maroon" title="c1">a</font>] implies <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c1">a</font>] ) &amp; <font color="Maroon">P<sub>2</sub></font>[<font color="Maroon" title="c1">a</font>] )
 <b>by </b><i><a class="txt" href="#E1:4"><i><font color="Green" title="E1">A1</font></i></a>, <a class="txt" href="#E2:4_1"><i><font color="Green" title="E2">A3</font></i></a></i>;<br/><b>hence </b><a NAME="E2:4_1_2"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c1">a</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:4_1"/>
for <font color="Olive" title="b1">a</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds  <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">a</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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