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

<span class="kw">let </span><font color="Maroon" title="c1">H</font> be   <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a>; <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">x</font>, <font color="Olive" title="b2">y</font> being  <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a> holds <br/> ( <font color="Maroon" title="c1">H</font> is  <a href="zf_lang.html#V8" title="ZF_LANG:attr.8">disjunctive</a>  iff <font color="Maroon" title="c1">H</font> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font> is  <a href="zf_lang.html#V8" title="ZF_LANG:attr.8">disjunctive</a>  )</span><br/><span class="kw">let </span><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>; <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="c1">H</font> is  <a href="zf_lang.html#V8" title="ZF_LANG:attr.8">disjunctive</a>  iff <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#V8" title="ZF_LANG:attr.8">disjunctive</a>  )</span><br/>





<span class="kw">set </span><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/>
<span class="kw">thus </span><a NAME="E1:185"/>
( <font color="Maroon" title="c1">H</font> is  <a href="zf_lang.html#V8" title="ZF_LANG:attr.8">disjunctive</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#V8" title="ZF_LANG:attr.8">disjunctive</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="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#V8" title="ZF_LANG:attr.8">disjunctive</a>  implies <font color="Maroon" title="c1">H</font> is  <a href="zf_lang.html#V8" title="ZF_LANG:attr.8">disjunctive</a>  )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="185_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">given </span><font color="Maroon" title="c5">H1</font>, <font color="Maroon" title="c6">H2</font> being   <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a><span class="kw"> such that </span><a NAME="E2:185_1"/><span class="lab"><font color="Green" title="E71">A1</font></span>: 
<font color="Maroon" title="c1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">H1</font> <a href="zf_lang.html#K10" title="ZF_LANG:func.10">'or'</a> <font color="Maroon" title="c6">H2</font>
 ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="zf_lang.html#D20" title="ZF_LANG:def.20">ZF_LANG:def 20</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="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#V8" title="ZF_LANG:attr.8">disjunctive</a> </span><br/>

<a NAME="E3:185_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="c5">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#K10" title="ZF_LANG:func.10">'or'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:185_1"><span class="lab"><font color="Green" title="E71">A1</font></span></a>, <a class="ref" href="zf_lang1.html#T175" target="_self" title="ZF_LANG1:th.175">Th175</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:185_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#V8" title="ZF_LANG:attr.8">disjunctive</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang.html#T22" title="ZF_LANG:th.22">ZF_LANG:22</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><span class="kw">end;</span></div>

<span class="kw">given </span><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><span class="kw"> such that </span><a NAME="E3:185"/><span class="lab"><font color="Green" title="E71">A2</font></span>: 
<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#K10" title="ZF_LANG:func.10">'or'</a> <font color="Maroon" title="c6">G2</font>
 ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="zf_lang.html#D20" title="ZF_LANG:def.20">ZF_LANG:def 20</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="c1">H</font> is  <a href="zf_lang.html#V8" title="ZF_LANG:attr.8">disjunctive</a> </span><br/>

<a NAME="E4:185"/>
<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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:185"><span class="lab"><font color="Green" title="E71">A2</font></span></a>, <a class="ref" href="zf_lang.html#T16" title="ZF_LANG:th.16">ZF_LANG:16</a></span>;<br/>
<a NAME="E5:185"/><span class="kw">then </span>
<font color="Maroon" title="c1">H</font> is  <a href="zf_lang.html#V4" title="ZF_LANG:attr.4">negative</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang1.html#T182" target="_self" title="ZF_LANG1:th.182">Th182</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">H9</font> being   <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a><span class="kw"> such that </span><br/><a NAME="E7:185"/><span class="lab"><font color="Green" title="E72">A3</font></span>: 
<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">H9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang.html#T16" title="ZF_LANG:th.16">ZF_LANG:16</a></span>;<br/>
<a NAME="E8:185"/><span class="lab"><font color="Green" title="E73">A4</font></span>: 
<span class="p1">(<span class="default"><a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <font color="Maroon" title="c5">G1</font></span>)</span> <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">H9</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E3:185"><span class="lab"><font color="Green" title="E71">A2</font></span></a>, <a class="txt" href="#E7:185"><span class="lab"><font color="Green" title="E72">A3</font></span></a>, <a class="ref" href="zf_lang1.html#T170" target="_self" title="ZF_LANG1:th.170">Th170</a></span>;<br/>
<a NAME="E9:185"/><span class="kw">then </span>
<font color="Maroon" title="c7">H9</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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang.html#T16" title="ZF_LANG:th.16">ZF_LANG:16</a></span>;<br/>
<a NAME="E10:185"/><span class="kw">then </span>
<font color="Maroon" title="c7">H9</font> is  <a href="zf_lang.html#V5" title="ZF_LANG:attr.5">conjunctive</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang1.html#T183" target="_self" title="ZF_LANG1:th.183">Th183</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E12:185"/><span class="lab"><font color="Green" title="E74">A5</font></span>: 
<font color="Maroon" title="c7">H9</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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang.html#T16" title="ZF_LANG:th.16">ZF_LANG:16</a></span>;<br/>
<a NAME="E13:185"/>
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:185"><span class="lab"><font color="Green" title="E73">A4</font></span></a>, <a class="txt" href="#E12:185"><span class="lab"><font color="Green" title="E74">A5</font></span></a>, <a class="ref" href="zf_lang1.html#T172" target="_self" title="ZF_LANG1:th.172">Th172</a></span>;<br/>
<a NAME="E14:185"/><span class="kw">then </span>
<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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang.html#T16" title="ZF_LANG:th.16">ZF_LANG:16</a></span>;<br/>
<a NAME="E15:185"/><span class="kw">then </span>
<font color="Maroon" title="c9">H2</font> is  <a href="zf_lang.html#V4" title="ZF_LANG:attr.4">negative</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang1.html#T182" target="_self" title="ZF_LANG1:th.182">Th182</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">p2</font> being   <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a><span class="kw"> such that </span><br/><a NAME="E17:185"/><span class="lab"><font color="Green" title="E75">A6</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang.html#T16" title="ZF_LANG:th.16">ZF_LANG:16</a></span>;<br/>
<a NAME="E18:185"/>
 <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:185"><span class="lab"><font color="Green" title="E73">A4</font></span></a>, <a class="txt" href="#E12:185"><span class="lab"><font color="Green" title="E74">A5</font></span></a>, <a class="ref" href="zf_lang1.html#T172" target="_self" title="ZF_LANG1:th.172">Th172</a></span>;<br/>
<a NAME="E19:185"/><span class="kw">then </span>
<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> is  <a href="zf_lang.html#V4" title="ZF_LANG:attr.4">negative</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang.html#T16" title="ZF_LANG:th.16">ZF_LANG:16</a></span>;<br/>
<a NAME="E20:185"/><span class="kw">then </span>
<font color="Maroon" title="c8">H1</font> is  <a href="zf_lang.html#V4" title="ZF_LANG:attr.4">negative</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang1.html#T182" target="_self" title="ZF_LANG1:th.182">Th182</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">p1</font> being   <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a><span class="kw"> such that </span><br/><a NAME="E22:185"/><span class="lab"><font color="Green" title="E76">A7</font></span>: 
<font color="Maroon" title="c8">H1</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="c11">p1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang.html#T16" title="ZF_LANG:th.16">ZF_LANG:16</a></span>;<br/>
<a NAME="E23:185"/>
<font color="Maroon" title="c1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p1</font> <a href="zf_lang.html#K10" title="ZF_LANG:func.10">'or'</a> <font color="Maroon" title="c10">p2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:185"><span class="lab"><font color="Green" title="E72">A3</font></span></a>, <a class="txt" href="#E12:185"><span class="lab"><font color="Green" title="E74">A5</font></span></a>, <a class="txt" href="#E22:185"><span class="lab"><font color="Green" title="E76">A7</font></span></a>, <a class="txt" href="#E17:185"><span class="lab"><font color="Green" title="E75">A6</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E24:185"/>
<font color="Maroon" title="c1">H</font> is  <a href="zf_lang.html#V8" title="ZF_LANG:attr.8">disjunctive</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang.html#T22" title="ZF_LANG:th.22">ZF_LANG:22</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>
