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<div class="add">

<b>let </b><font color="Maroon" title="c1">H</font> be   <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a>;<br/><b>let </b><font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">y</font> be   <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a>;<br/>





<b>thus </b><a NAME="E1:174"/>
( <font color="Maroon" title="c1">H</font> is <a href="zf_lang.html#V9" title="ZF_LANG:attr.9">conditional</a> implies <font color="Maroon" title="c1">H</font> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> is <a href="zf_lang.html#V9" title="ZF_LANG:attr.9">conditional</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="174_1"><b>proof </b></a><div class="add">

<b>given </b><font color="Maroon" title="c4">H1</font>, <font color="Maroon" title="c5">H2</font> being   <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a><b> such that </b><a NAME="E2:174_1"/><i><font color="Green" title="E74">A1</font></i>: 
<font color="Maroon" title="c1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">H1</font> <a href="zf_lang.html#K11" title="ZF_LANG:func.11">=&gt;</a> <font color="Maroon" title="c5">H2</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="zf_lang.html#D21" title="ZF_LANG:def.21">ZF_LANG:def 21</a><br/>

<a NAME="E3:174_1"/>
<font color="Maroon" title="c1">H</font> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">H1</font> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span>)</span> <a href="zf_lang.html#K11" title="ZF_LANG:func.11">=&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">H2</font> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span>)</span>
 
<b>by </b><i><a class="txt" href="zf_lang1.html#E2:174_1"><i><font color="Green" title="E74">A1</font></i></a>, <a class="ref" href="zf_lang1.html#T176" target="_self" title="ZF_LANG1:th.176">Th176</a></i>;<br/>
<b>hence </b><a NAME="E4:174_1"/>
<font color="Maroon" title="c1">H</font> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> is <a href="zf_lang.html#V9" title="ZF_LANG:attr.9">conditional</a>
 <b>by </b><i><a class="ref" href="zf_lang.html#T22" title="ZF_LANG:th.22">ZF_LANG:22</a></i>;<br/>


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

<b>set </b><font color="Maroon" title="c4">G</font> = <font color="Maroon" title="c1">H</font> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font>;<br/>
<b>given </b><font color="Maroon" title="c5">G1</font>, <font color="Maroon" title="c6">G2</font> being   <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a><b> such that </b><a NAME="E3:174"/><i><font color="Green" title="E74">A2</font></i>: 
<font color="Maroon" title="c1">H</font> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="zf_lang.html#K11" title="ZF_LANG:func.11">=&gt;</a> <font color="Maroon" title="c6">G2</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="zf_lang.html#D21" title="ZF_LANG:def.21">ZF_LANG:def 21</a><br/>

<a NAME="E4:174"/>
<font color="Maroon" title="c1">H</font> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> is <a href="zf_lang.html#V4" title="ZF_LANG:attr.4">negative</a>
 
<b>by </b><i><a class="txt" href="zf_lang1.html#E3:174"><i><font color="Green" title="E74">A2</font></i></a>, <a class="ref" href="zf_lang.html#T16" title="ZF_LANG:th.16">ZF_LANG:16</a></i>;<br/>
<a NAME="E5:174"/><b>then </b>
<font color="Maroon" title="c1">H</font> is <a href="zf_lang.html#V4" title="ZF_LANG:attr.4">negative</a>
 
<b>by </b><i><a class="ref" href="zf_lang1.html#T182" target="_self" title="ZF_LANG1:th.182">Th182</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c7">H'</font> being   <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a><b> such that </b><br/><a NAME="E7:174"/><i><font color="Green" title="E75">A3</font></i>: 
<font color="Maroon" title="c1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <font color="Maroon" title="c7">H'</font>
 <b>by </b><i><a class="ref" href="zf_lang.html#T16" title="ZF_LANG:th.16">ZF_LANG:16</a></i>;<br/>
<a NAME="E8:174"/><i><font color="Green" title="E76">A4</font></i>: 
<font color="Maroon" title="c5">G1</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <font color="Maroon" title="c6">G2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">H'</font> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font>
 
<b>by </b><i><a class="txt" href="zf_lang1.html#E3:174"><i><font color="Green" title="E74">A2</font></i></a>, <a class="txt" href="zf_lang1.html#E7:174"><i><font color="Green" title="E75">A3</font></i></a>, <a class="ref" href="zf_lang1.html#T170" target="_self" title="ZF_LANG1:th.170">Th170</a></i>;<br/>
<a NAME="E9:174"/><b>then </b>
<font color="Maroon" title="c7">H'</font> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> is <a href="zf_lang.html#V5" title="ZF_LANG:attr.5">conjunctive</a>
 
<b>by </b><i><a class="ref" href="zf_lang.html#T16" title="ZF_LANG:th.16">ZF_LANG:16</a></i>;<br/>
<a NAME="E10:174"/><b>then </b>
<font color="Maroon" title="c7">H'</font> is <a href="zf_lang.html#V5" title="ZF_LANG:attr.5">conjunctive</a>
 
<b>by </b><i><a class="ref" href="zf_lang1.html#T183" target="_self" title="ZF_LANG1:th.183">Th183</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c8">H1</font>, <font color="Maroon" title="c9">H2</font> being   <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a><b> such that </b><br/><a NAME="E12:174"/><i><font color="Green" title="E77">A5</font></i>: 
<font color="Maroon" title="c7">H'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">H1</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c9">H2</font>
 <b>by </b><i><a class="ref" href="zf_lang.html#T16" title="ZF_LANG:th.16">ZF_LANG:16</a></i>;<br/>
<a NAME="E13:174"/>
( <font color="Maroon" title="c5">G1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">H1</font> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> &amp;  <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <font color="Maroon" title="c6">G2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">H2</font> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> )
 
<b>by </b><i><a class="txt" href="zf_lang1.html#E8:174"><i><font color="Green" title="E76">A4</font></i></a>, <a class="txt" href="zf_lang1.html#E12:174"><i><font color="Green" title="E77">A5</font></i></a>, <a class="ref" href="zf_lang1.html#T172" target="_self" title="ZF_LANG1:th.172">Th172</a></i>;<br/>
<a NAME="E14:174"/><b>then </b>
<font color="Maroon" title="c9">H2</font> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> is <a href="zf_lang.html#V4" title="ZF_LANG:attr.4">negative</a>
 
<b>by </b><i><a class="ref" href="zf_lang.html#T16" title="ZF_LANG:th.16">ZF_LANG:16</a></i>;<br/>
<a NAME="E15:174"/><b>then </b>
<font color="Maroon" title="c9">H2</font> is <a href="zf_lang.html#V4" title="ZF_LANG:attr.4">negative</a>
 
<b>by </b><i><a class="ref" href="zf_lang1.html#T182" target="_self" title="ZF_LANG1:th.182">Th182</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c10">p2</font> being   <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a><b> such that </b><br/><a NAME="E17:174"/><i><font color="Green" title="E78">A6</font></i>: 
<font color="Maroon" title="c9">H2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <font color="Maroon" title="c10">p2</font>
 <b>by </b><i><a class="ref" href="zf_lang.html#T16" title="ZF_LANG:th.16">ZF_LANG:16</a></i>;<br/>
<a NAME="E18:174"/>
<font color="Maroon" title="c1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">H1</font> <a href="zf_lang.html#K11" title="ZF_LANG:func.11">=&gt;</a> <font color="Maroon" title="c10">p2</font>
 
<b>by </b><i><a class="txt" href="zf_lang1.html#E7:174"><i><font color="Green" title="E75">A3</font></i></a>, <a class="txt" href="zf_lang1.html#E12:174"><i><font color="Green" title="E77">A5</font></i></a>, <a class="txt" href="zf_lang1.html#E17:174"><i><font color="Green" title="E78">A6</font></i></a></i>;<br/>
<b>hence </b><a NAME="E19:174"/>
<font color="Maroon" title="c1">H</font> is <a href="zf_lang.html#V9" title="ZF_LANG:attr.9">conditional</a>
 <b>by </b><i><a class="ref" href="zf_lang.html#T22" title="ZF_LANG:th.22">ZF_LANG:22</a></i>;<br/>


</div>
