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

<b>thus </b><a NAME="E1:54_2_1"/>
( <font color="Maroon">c<sub>1</sub></font> is <a href="zf_lang.html#V5" target="_self">conjunctive</a> implies ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="zf_lang.html#NM2">ZF-formula</a> st <font color="Olive">b<sub>2</sub></font> <a href="zf_lang.html#K7" target="_self">'&amp;'</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_2_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:54_2_1_1"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="zf_lang.html#V5" target="_self">conjunctive</a>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> being   <a href="zf_lang.html#NM2">ZF-formula</a><b> such that </b><br/><a NAME="E3:54_2_1_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="zf_lang.html#K7" target="_self">'&amp;'</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="zf_lang.html#D13" target="_self">Def13</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>3</sub></font>
;<br/>

<b>thus </b><a NAME="E4:54_2_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="zf_lang.html#NM2">ZF-formula</a> st <font color="Olive">b<sub>1</sub></font> <a href="zf_lang.html#K7" target="_self">'&amp;'</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="zf_lang.html#E3:54_2_1_1"><i><font color="Green">E42</font></i></a></i>;<br/>


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

<b>thus </b><a NAME="E2:54_2_1"/>
( not <font color="Maroon">c<sub>1</sub></font> is <a href="zf_lang.html#V5" target="_self">conjunctive</a> implies ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="zf_lang.html#NM2">ZF-formula</a> st <font color="Olive">b<sub>2</sub></font> <a href="zf_lang.html#K10" target="_self">'or'</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_2_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:54_2_1_2"/>
not <font color="Maroon">c<sub>1</sub></font> is <a href="zf_lang.html#V5" target="_self">conjunctive</a>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> being   <a href="zf_lang.html#NM2">ZF-formula</a><b> such that </b><br/><a NAME="E3:54_2_1_2"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="zf_lang.html#K10" target="_self">'or'</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="zf_lang.html#E1:54"><i><font color="Green">E41</font></i></a>, <a class="ref" href="zf_lang.html#D20" target="_self">Def20</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>3</sub></font>
;<br/>

<b>thus </b><a NAME="E4:54_2_1_2"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="zf_lang.html#NM2">ZF-formula</a> st <font color="Olive">b<sub>1</sub></font> <a href="zf_lang.html#K10" target="_self">'or'</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="zf_lang.html#E3:54_2_1_2"><i><font color="Green">E42</font></i></a></i>;<br/>


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


</div>
