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<a NAME="E1:64_1"/><span class="lab"><font color="Green" title="E26">A5</font></span>: 
for <font color="Olive" title="b1">H</font> being  <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a>  st <font color="Olive" title="b1">H</font> is  <a href="zf_lang.html#V5" title="ZF_LANG:attr.5">conjunctive</a>  &amp; <font color="Maroon">P<sub>1</sub></font>[ <a href="zf_lang.html#K21" title="ZF_LANG:func.21">the_left_argument_of</a> <font color="Olive" title="b1">H</font>] &amp; <font color="Maroon">P<sub>1</sub></font>[ <a href="zf_lang.html#K22" title="ZF_LANG:func.22">the_right_argument_of</a> <font color="Olive" title="b1">H</font>] holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">H</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_1"><span class="kw">proof </span></a><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"> ( <font color="Maroon" title="c1">H</font> is  <a href="zf_lang.html#V5" title="ZF_LANG:attr.5">conjunctive</a>  &amp; <font color="Maroon">P<sub>1</sub></font>[ <a href="zf_lang.html#K21" title="ZF_LANG:func.21">the_left_argument_of</a> <font color="Maroon" title="c1">H</font>] &amp; <font color="Maroon">P<sub>1</sub></font>[ <a href="zf_lang.html#K22" title="ZF_LANG:func.22">the_right_argument_of</a> <font color="Maroon" title="c1">H</font>] implies <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c1">H</font>] )</span><br/>

<span class="kw">assume </span><a NAME="E1:64_1_1"/>
<font color="Maroon" title="c1">H</font> is  <a href="zf_lang.html#V5" title="ZF_LANG:attr.5">conjunctive</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon">P<sub>1</sub></font>[ <a href="zf_lang.html#K21" title="ZF_LANG:func.21">the_left_argument_of</a> <font color="Maroon" title="c1">H</font>] or  not <font color="Maroon">P<sub>1</sub></font>[ <a href="zf_lang.html#K22" title="ZF_LANG:func.22">the_right_argument_of</a> <font color="Maroon" title="c1">H</font>] or <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c1">H</font>] )</span><br/>

<a NAME="E2:64_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K21" title="ZF_LANG:func.21">the_left_argument_of</a> <font color="Maroon" title="c1">H</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#K22" title="ZF_LANG:func.22">the_right_argument_of</a> <font color="Maroon" title="c1">H</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang.html#T58" title="ZF_LANG:th.58">ZF_LANG:58</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:64_1_1"/>
(  not <font color="Maroon">P<sub>1</sub></font>[ <a href="zf_lang.html#K21" title="ZF_LANG:func.21">the_left_argument_of</a> <font color="Maroon" title="c1">H</font>] or  not <font color="Maroon">P<sub>1</sub></font>[ <a href="zf_lang.html#K22" title="ZF_LANG:func.22">the_right_argument_of</a> <font color="Maroon" title="c1">H</font>] or <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c1">H</font>] )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:64"><span class="lab"><font color="Green" title="E24">A3</font></span></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>
<a NAME="E2:64_1"/><span class="lab"><font color="Green" title="E27">A6</font></span>: 
for <font color="Olive" title="b1">H</font> being  <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a>  st <font color="Olive" title="b1">H</font> is  <a href="zf_lang.html#V7" title="ZF_LANG:attr.7">atomic</a>  holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">H</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_2"><span class="kw">proof </span></a><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"> ( <font color="Maroon" title="c1">H</font> is  <a href="zf_lang.html#V7" title="ZF_LANG:attr.7">atomic</a>  implies <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c1">H</font>] )</span><br/>

<span class="kw">assume </span><a NAME="E1:64_1_2"/><span class="lab"><font color="Green" title="E28">A7</font></span>: 
( <font color="Maroon" title="c1">H</font> is  <a href="zf_lang.html#V2" title="ZF_LANG:attr.2">being_equality</a>  or <font color="Maroon" title="c1">H</font> is  <a href="zf_lang.html#V3" title="ZF_LANG:attr.3">being_membership</a>  )
 ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="zf_lang.html#D15" title="ZF_LANG:def.15">ZF_LANG:def 15</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">P<sub>1</sub></font>[<font color="Maroon" title="c1">H</font>]</span><br/>

<a NAME="E2:64_1_2"/><span class="lab"><font color="Green" title="E29">A8</font></span>: 
( <font color="Maroon" title="c1">H</font> is  <a href="zf_lang.html#V3" title="ZF_LANG:attr.3">being_membership</a>  implies <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c1">H</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">given </span><font color="Maroon" title="c2">x1</font>, <font color="Maroon" title="c3">x2</font> being   <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a><span class="kw"> such that </span><a NAME="E2:64_1_2_1"/><span class="lab"><font color="Green" title="E30">A9</font></span>: 
<font color="Maroon" title="c1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c3">x2</font>
 ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="zf_lang.html#D11" title="ZF_LANG:def.11">ZF_LANG:def 11</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">P<sub>1</sub></font>[<font color="Maroon" title="c1">H</font>]</span><br/>

<span class="kw">thus </span><a NAME="E3:64_1_2_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c1">H</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="txt" href="#E2:64_1_2_1"><span class="lab"><font color="Green" title="E30">A9</font></span></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>
<a NAME="E3:64_1_2"/>
( <font color="Maroon" title="c1">H</font> is  <a href="zf_lang.html#V2" title="ZF_LANG:attr.2">being_equality</a>  implies <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c1">H</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">given </span><font color="Maroon" title="c2">x1</font>, <font color="Maroon" title="c3">x2</font> being   <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a><span class="kw"> such that </span><a NAME="E2:64_1_2_2"/><span class="lab"><font color="Green" title="E30">A10</font></span>: 
<font color="Maroon" title="c1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Maroon" title="c3">x2</font>
 ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="zf_lang.html#D10" title="ZF_LANG:def.10">ZF_LANG:def 10</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">P<sub>1</sub></font>[<font color="Maroon" title="c1">H</font>]</span><br/>

<span class="kw">thus </span><a NAME="E3:64_1_2_2"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c1">H</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="txt" href="#E2:64_1_2_2"><span class="lab"><font color="Green" title="E30">A10</font></span></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">hence </span><a NAME="E4:64_1_2"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c1">H</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64_1_2"><span class="lab"><font color="Green" title="E28">A7</font></span></a>, <a class="txt" href="#E2:64_1_2"><span class="lab"><font color="Green" title="E29">A8</font></span></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>
<a NAME="E3:64_1"/><span class="lab"><font color="Green" title="E28">A11</font></span>: 
for <font color="Olive" title="b1">H</font> being  <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a>  st <font color="Olive" title="b1">H</font> is  <a href="zf_lang.html#V6" title="ZF_LANG:attr.6">universal</a>  &amp; <font color="Maroon">P<sub>1</sub></font>[ <a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Olive" title="b1">H</font>] holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">H</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_3"><span class="kw">proof </span></a><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"> ( <font color="Maroon" title="c1">H</font> is  <a href="zf_lang.html#V6" title="ZF_LANG:attr.6">universal</a>  &amp; <font color="Maroon">P<sub>1</sub></font>[ <a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c1">H</font>] implies <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c1">H</font>] )</span><br/>

<span class="kw">assume </span><a NAME="E1:64_1_3"/>
<font color="Maroon" title="c1">H</font> is  <a href="zf_lang.html#V6" title="ZF_LANG:attr.6">universal</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon">P<sub>1</sub></font>[ <a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c1">H</font>] or <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c1">H</font>] )</span><br/>

<a NAME="E2:64_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zf_lang.html#K8" title="ZF_LANG:func.8">All</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c1">H</font></span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c1">H</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang.html#T62" title="ZF_LANG:th.62">ZF_LANG:62</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:64_1_3"/>
(  not <font color="Maroon">P<sub>1</sub></font>[ <a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c1">H</font>] or <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c1">H</font>] )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:64"><span class="lab"><font color="Green" title="E25">A4</font></span></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>
<a NAME="E4:64_1"/><span class="lab"><font color="Green" title="E29">A12</font></span>: 
for <font color="Olive" title="b1">H</font> being  <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a>  st <font color="Olive" title="b1">H</font> is  <a href="zf_lang.html#V4" title="ZF_LANG:attr.4">negative</a>  &amp; <font color="Maroon">P<sub>1</sub></font>[ <a href="zf_lang.html#K20" title="ZF_LANG:func.20">the_argument_of</a> <font color="Olive" title="b1">H</font>] holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">H</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_4"><span class="kw">proof </span></a><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"> ( <font color="Maroon" title="c1">H</font> is  <a href="zf_lang.html#V4" title="ZF_LANG:attr.4">negative</a>  &amp; <font color="Maroon">P<sub>1</sub></font>[ <a href="zf_lang.html#K20" title="ZF_LANG:func.20">the_argument_of</a> <font color="Maroon" title="c1">H</font>] implies <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c1">H</font>] )</span><br/>

<span class="kw">assume </span><a NAME="E1:64_1_4"/>
<font color="Maroon" title="c1">H</font> is  <a href="zf_lang.html#V4" title="ZF_LANG:attr.4">negative</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon">P<sub>1</sub></font>[ <a href="zf_lang.html#K20" title="ZF_LANG:func.20">the_argument_of</a> <font color="Maroon" title="c1">H</font>] or <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c1">H</font>] )</span><br/>

<a NAME="E2:64_1_4"/><span class="kw">then </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> <span class="p1">(<span class="default"><a href="zf_lang.html#K20" title="ZF_LANG:func.20">the_argument_of</a> <font color="Maroon" title="c1">H</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang.html#D30" title="ZF_LANG:def.30">ZF_LANG:def 30</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:64_1_4"/>
(  not <font color="Maroon">P<sub>1</sub></font>[ <a href="zf_lang.html#K20" title="ZF_LANG:func.20">the_argument_of</a> <font color="Maroon" title="c1">H</font>] or <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c1">H</font>] )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:64"><span class="lab"><font color="Green" title="E23">A2</font></span></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">thus </span><a NAME="E5:64_1"/>
for <font color="Olive" title="b1">H</font> being  <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a> holds  <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">H</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="zf_lang.html#S1" title="ZF_LANG:sch.1">ZF_LANG:sch 1</a>(<a class="txt" href="#E2:64_1"><span class="lab"><font color="Green" title="E27">A6</font></span></a>, <a class="txt" href="#E4:64_1"><span class="lab"><font color="Green" title="E29">A12</font></span></a>, <a class="txt" href="#E1:64_1"><span class="lab"><font color="Green" title="E26">A5</font></span></a>, <a class="txt" href="#E3:64_1"><span class="lab"><font color="Green" title="E28">A11</font></span></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>
