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<b>consider </b><font color="Maroon" title="c1">A</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E2:3_1"/><i><font color="Green" title="E2">A2</font></i>: 
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/> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">x</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Olive" title="b2">y</font></span>)</span>,<font color="Maroon">F<sub>1</sub></font>(<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>)</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">x</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Olive" title="b2">y</font></span>)</span>,<font color="Maroon">F<sub>2</sub></font>(<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>)</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> )
 <b>and </b><br/><a NAME="E3:3_1"/><i><font color="Green" title="E3">A3</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon">F<sub>6</sub></font>(),<font color="Maroon">F<sub>7</sub></font>(<font color="Maroon">F<sub>6</sub></font>())</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 <b>and </b><br/><a NAME="E4:3_1"/><i><font color="Green" title="E4">A4</font></i>: 
for <font color="Olive" title="b1">H</font> being  <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a><br/> for <font color="Olive" title="b2">a</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">H</font>,<font color="Olive" title="b2">a</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> holds <br/>( ( <font color="Olive" title="b1">H</font> is <a href="zf_lang.html#V2" title="ZF_LANG:attr.2">being_equality</a> implies <font color="Olive" title="b2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><a href="zf_lang.html#K18" title="ZF_LANG:func.18">Var1</a> <font color="Olive" title="b1">H</font></span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K19" title="ZF_LANG:func.19">Var2</a> <font color="Olive" title="b1">H</font></span>)</span>) ) &amp; ( <font color="Olive" title="b1">H</font> is <a href="zf_lang.html#V3" title="ZF_LANG:attr.3">being_membership</a> implies <font color="Olive" title="b2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>2</sub></font>(<span class="p1">(<span class="default"><a href="zf_lang.html#K18" title="ZF_LANG:func.18">Var1</a> <font color="Olive" title="b1">H</font></span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K19" title="ZF_LANG:func.19">Var2</a> <font color="Olive" title="b1">H</font></span>)</span>) ) &amp; ( <font color="Olive" title="b1">H</font> is <a href="zf_lang.html#V4" title="ZF_LANG:attr.4">negative</a> implies  ex <font color="Olive" title="b3">b</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive" title="b3">b</font>) &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K20" title="ZF_LANG:func.20">the_argument_of</a> <font color="Olive" title="b1">H</font></span>)</span>,<font color="Olive" title="b3">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> ) ) &amp; ( <font color="Olive" title="b1">H</font> is <a href="zf_lang.html#V5" title="ZF_LANG:attr.5">conjunctive</a> implies  ex <font color="Olive" title="b3">b</font>, <font color="Olive" title="b4">c</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive" title="b3">b</font>,<font color="Olive" title="b4">c</font>) &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K21" title="ZF_LANG:func.21">the_left_argument_of</a> <font color="Olive" title="b1">H</font></span>)</span>,<font color="Olive" title="b3">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K22" title="ZF_LANG:func.22">the_right_argument_of</a> <font color="Olive" title="b1">H</font></span>)</span>,<font color="Olive" title="b4">c</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> ) ) &amp; ( <font color="Olive" title="b1">H</font> is <a href="zf_lang.html#V6" title="ZF_LANG:attr.6">universal</a> implies  ex <font color="Olive" title="b3">b</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Olive" title="b1">H</font></span>)</span>,<font color="Olive" title="b3">b</font>) &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Olive" title="b1">H</font></span>)</span>,<font color="Olive" title="b3">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> ) ) )
 <b>by </b><i><a class="txt" href="#E1:3"><i><font color="Green" title="E1">A1</font></i></a></i>;<br/>
<b>thus </b><a NAME="E5:3_1"/>
( <font color="Maroon">F<sub>6</sub></font>() is <a href="zf_lang.html#V2" title="ZF_LANG:attr.2">being_equality</a> implies <font color="Maroon">F<sub>7</sub></font>(<font color="Maroon">F<sub>6</sub></font>()) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><a href="zf_lang.html#K18" title="ZF_LANG:func.18">Var1</a> <font color="Maroon">F<sub>6</sub></font>()</span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K19" title="ZF_LANG:func.19">Var2</a> <font color="Maroon">F<sub>6</sub></font>()</span>)</span>) )
 <b>by </b><i><a class="txt" href="#E3:3_1"><i><font color="Green" title="E3">A3</font></i></a>, <a class="txt" href="#E4:3_1"><i><font color="Green" title="E4">A4</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( <font color="Maroon">F<sub>6</sub></font>() is <a href="zf_lang.html#V3" title="ZF_LANG:attr.3">being_membership</a> implies <font color="Maroon">F<sub>7</sub></font>(<font color="Maroon">F<sub>6</sub></font>()) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>2</sub></font>(<span class="p1">(<span class="default"><a href="zf_lang.html#K18" title="ZF_LANG:func.18">Var1</a> <font color="Maroon">F<sub>6</sub></font>()</span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K19" title="ZF_LANG:func.19">Var2</a> <font color="Maroon">F<sub>6</sub></font>()</span>)</span>) ) &amp; ( <font color="Maroon">F<sub>6</sub></font>() is <a href="zf_lang.html#V4" title="ZF_LANG:attr.4">negative</a> implies <font color="Maroon">F<sub>7</sub></font>(<font color="Maroon">F<sub>6</sub></font>()) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">F<sub>7</sub></font>(<span class="p1">(<span class="default"><a href="zf_lang.html#K20" title="ZF_LANG:func.20">the_argument_of</a> <font color="Maroon">F<sub>6</sub></font>()</span>)</span>)) ) &amp; ( <font color="Maroon">F<sub>6</sub></font>() is <a href="zf_lang.html#V5" title="ZF_LANG:attr.5">conjunctive</a> implies for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>7</sub></font>(<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">F<sub>6</sub></font>()</span>)</span>) &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>7</sub></font>(<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">F<sub>6</sub></font>()</span>)</span>) holds <br/><font color="Maroon">F<sub>7</sub></font>(<font color="Maroon">F<sub>6</sub></font>()) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>) ) &amp; ( <font color="Maroon">F<sub>6</sub></font>() is <a href="zf_lang.html#V6" title="ZF_LANG:attr.6">universal</a> implies <font color="Maroon">F<sub>7</sub></font>(<font color="Maroon">F<sub>6</sub></font>()) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon">F<sub>6</sub></font>()</span>)</span>,<font color="Maroon">F<sub>7</sub></font>(<span class="p1">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon">F<sub>6</sub></font>()</span>)</span>)) ) )</span><br/>

<b>thus </b><a NAME="E6:3_1"/>
( <font color="Maroon">F<sub>6</sub></font>() is <a href="zf_lang.html#V3" title="ZF_LANG:attr.3">being_membership</a> implies <font color="Maroon">F<sub>7</sub></font>(<font color="Maroon">F<sub>6</sub></font>()) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>2</sub></font>(<span class="p1">(<span class="default"><a href="zf_lang.html#K18" title="ZF_LANG:func.18">Var1</a> <font color="Maroon">F<sub>6</sub></font>()</span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K19" title="ZF_LANG:func.19">Var2</a> <font color="Maroon">F<sub>6</sub></font>()</span>)</span>) )
 <b>by </b><i><a class="txt" href="#E3:3_1"><i><font color="Green" title="E3">A3</font></i></a>, <a class="txt" href="#E4:3_1"><i><font color="Green" title="E4">A4</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( <font color="Maroon">F<sub>6</sub></font>() is <a href="zf_lang.html#V4" title="ZF_LANG:attr.4">negative</a> implies <font color="Maroon">F<sub>7</sub></font>(<font color="Maroon">F<sub>6</sub></font>()) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">F<sub>7</sub></font>(<span class="p1">(<span class="default"><a href="zf_lang.html#K20" title="ZF_LANG:func.20">the_argument_of</a> <font color="Maroon">F<sub>6</sub></font>()</span>)</span>)) ) &amp; ( <font color="Maroon">F<sub>6</sub></font>() is <a href="zf_lang.html#V5" title="ZF_LANG:attr.5">conjunctive</a> implies for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>7</sub></font>(<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">F<sub>6</sub></font>()</span>)</span>) &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>7</sub></font>(<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">F<sub>6</sub></font>()</span>)</span>) holds <br/><font color="Maroon">F<sub>7</sub></font>(<font color="Maroon">F<sub>6</sub></font>()) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>) ) &amp; ( <font color="Maroon">F<sub>6</sub></font>() is <a href="zf_lang.html#V6" title="ZF_LANG:attr.6">universal</a> implies <font color="Maroon">F<sub>7</sub></font>(<font color="Maroon">F<sub>6</sub></font>()) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon">F<sub>6</sub></font>()</span>)</span>,<font color="Maroon">F<sub>7</sub></font>(<span class="p1">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon">F<sub>6</sub></font>()</span>)</span>)) ) )</span><br/>

<b>thus </b><a NAME="E7:3_1"/>
( <font color="Maroon">F<sub>6</sub></font>() is <a href="zf_lang.html#V4" title="ZF_LANG:attr.4">negative</a> implies <font color="Maroon">F<sub>7</sub></font>(<font color="Maroon">F<sub>6</sub></font>()) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">F<sub>7</sub></font>(<span class="p1">(<span class="default"><a href="zf_lang.html#K20" title="ZF_LANG:func.20">the_argument_of</a> <font color="Maroon">F<sub>6</sub></font>()</span>)</span>)) )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( <font color="Maroon">F<sub>6</sub></font>() is <a href="zf_lang.html#V5" title="ZF_LANG:attr.5">conjunctive</a> implies for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>7</sub></font>(<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">F<sub>6</sub></font>()</span>)</span>) &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>7</sub></font>(<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">F<sub>6</sub></font>()</span>)</span>) holds <br/><font color="Maroon">F<sub>7</sub></font>(<font color="Maroon">F<sub>6</sub></font>()) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>) ) &amp; ( <font color="Maroon">F<sub>6</sub></font>() is <a href="zf_lang.html#V6" title="ZF_LANG:attr.6">universal</a> implies <font color="Maroon">F<sub>7</sub></font>(<font color="Maroon">F<sub>6</sub></font>()) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon">F<sub>6</sub></font>()</span>)</span>,<font color="Maroon">F<sub>7</sub></font>(<span class="p1">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon">F<sub>6</sub></font>()</span>)</span>)) ) )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:3_1_1"/>
<font color="Maroon">F<sub>6</sub></font>() is <a href="zf_lang.html#V4" title="ZF_LANG:attr.4">negative</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">F<sub>7</sub></font>(<font color="Maroon">F<sub>6</sub></font>()) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">F<sub>7</sub></font>(<span class="p1">(<span class="default"><a href="zf_lang.html#K20" title="ZF_LANG:func.20">the_argument_of</a> <font color="Maroon">F<sub>6</sub></font>()</span>)</span>))</span><br/>

<a NAME="E2:3_1_1"/><b>then </b>
 ex <font color="Olive" title="b1">b</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Maroon">F<sub>7</sub></font>(<font color="Maroon">F<sub>6</sub></font>()) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive" title="b1">b</font>) &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K20" title="ZF_LANG:func.20">the_argument_of</a> <font color="Maroon">F<sub>6</sub></font>()</span>)</span>,<font color="Olive" title="b1">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> )
 
<b>by </b><i><a class="txt" href="#E3:3_1"><i><font color="Green" title="E3">A3</font></i></a>, <a class="txt" href="#E4:3_1"><i><font color="Green" title="E4">A4</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:3_1_1"/>
<font color="Maroon">F<sub>7</sub></font>(<font color="Maroon">F<sub>6</sub></font>()) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">F<sub>7</sub></font>(<span class="p1">(<span class="default"><a href="zf_lang.html#K20" title="ZF_LANG:func.20">the_argument_of</a> <font color="Maroon">F<sub>6</sub></font>()</span>)</span>))
 <b>by </b><i><a class="txt" href="#E1:3"><i><font color="Green" title="E1">A1</font></i></a>, <a class="txt" href="#E2:3_1"><i><font color="Green" title="E2">A2</font></i></a>, <a class="txt" href="#E4:3_1"><i><font color="Green" title="E4">A4</font></i></a></i>; <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>thus </b><a NAME="E8:3_1"/>
( <font color="Maroon">F<sub>6</sub></font>() is <a href="zf_lang.html#V5" title="ZF_LANG:attr.5">conjunctive</a> implies for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>7</sub></font>(<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">F<sub>6</sub></font>()</span>)</span>) &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>7</sub></font>(<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">F<sub>6</sub></font>()</span>)</span>) holds <br/><font color="Maroon">F<sub>7</sub></font>(<font color="Maroon">F<sub>6</sub></font>()) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>) )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">F<sub>6</sub></font>() is <a href="zf_lang.html#V6" title="ZF_LANG:attr.6">universal</a> implies <font color="Maroon">F<sub>7</sub></font>(<font color="Maroon">F<sub>6</sub></font>()) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon">F<sub>6</sub></font>()</span>)</span>,<font color="Maroon">F<sub>7</sub></font>(<span class="p1">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon">F<sub>6</sub></font>()</span>)</span>)) )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:3_1_2"/>
<font color="Maroon">F<sub>6</sub></font>() is <a href="zf_lang.html#V5" title="ZF_LANG:attr.5">conjunctive</a>
 ; <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>, <font color="Olive" title="b2">b</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>7</sub></font>(<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">F<sub>6</sub></font>()</span>)</span>) &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>7</sub></font>(<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">F<sub>6</sub></font>()</span>)</span>) holds <br/><font color="Maroon">F<sub>7</sub></font>(<font color="Maroon">F<sub>6</sub></font>()) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>)</span><br/>

<b>then consider </b><font color="Maroon" title="c2">b</font>, <font color="Maroon" title="c3">c</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:3_1_2"/><i><font color="Green" title="E5">A5</font></i>: 
( <font color="Maroon">F<sub>7</sub></font>(<font color="Maroon">F<sub>6</sub></font>()) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>) &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K21" title="ZF_LANG:func.21">the_left_argument_of</a> <font color="Maroon">F<sub>6</sub></font>()</span>)</span>,<font color="Maroon" title="c2">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K22" title="ZF_LANG:func.22">the_right_argument_of</a> <font color="Maroon">F<sub>6</sub></font>()</span>)</span>,<font color="Maroon" title="c3">c</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> )
 <b>by </b><i><a class="txt" href="#E3:3_1"><i><font color="Green" title="E3">A3</font></i></a>, <a class="txt" href="#E4:3_1"><i><font color="Green" title="E4">A4</font></i></a></i>;<br/>
<b>let </b><font color="Maroon" title="c4">a1</font>, <font color="Maroon" title="c5">a2</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" title="c4">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>7</sub></font>(<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">F<sub>6</sub></font>()</span>)</span>) &amp; <font color="Maroon" title="c5">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>7</sub></font>(<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">F<sub>6</sub></font>()</span>)</span>) implies <font color="Maroon">F<sub>7</sub></font>(<font color="Maroon">F<sub>6</sub></font>()) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c4">a1</font>,<font color="Maroon" title="c5">a2</font>) )</span><br/>

<b>assume </b><a NAME="E4:3_1_2"/><i><font color="Green" title="E6">A6</font></i>: 
( <font color="Maroon" title="c4">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>7</sub></font>(<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">F<sub>6</sub></font>()</span>)</span>) &amp; <font color="Maroon" title="c5">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>7</sub></font>(<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">F<sub>6</sub></font>()</span>)</span>) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">F<sub>7</sub></font>(<font color="Maroon">F<sub>6</sub></font>()) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c4">a1</font>,<font color="Maroon" title="c5">a2</font>)</span><br/>

<a NAME="E5:3_1_2"/>
( <font color="Maroon">F<sub>7</sub></font>(<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">F<sub>6</sub></font>()</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b</font> &amp; <font color="Maroon">F<sub>7</sub></font>(<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">F<sub>6</sub></font>()</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c</font> )
 
<b>by </b><i><a class="txt" href="#E1:3"><i><font color="Green" title="E1">A1</font></i></a>, <a class="txt" href="#E2:3_1"><i><font color="Green" title="E2">A2</font></i></a>, <a class="txt" href="#E4:3_1"><i><font color="Green" title="E4">A4</font></i></a>, <a class="txt" href="#E3:3_1_2"><i><font color="Green" title="E5">A5</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:3_1_2"/>
<font color="Maroon">F<sub>7</sub></font>(<font color="Maroon">F<sub>6</sub></font>()) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c4">a1</font>,<font color="Maroon" title="c5">a2</font>)
 <b>by </b><i><a class="txt" href="#E3:3_1_2"><i><font color="Green" title="E5">A5</font></i></a>, <a class="txt" href="#E4:3_1_2"><i><font color="Green" title="E6">A6</font></i></a></i>; <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>thus </b><a NAME="E9:3_1"/>
( <font color="Maroon">F<sub>6</sub></font>() is <a href="zf_lang.html#V6" title="ZF_LANG:attr.6">universal</a> implies <font color="Maroon">F<sub>7</sub></font>(<font color="Maroon">F<sub>6</sub></font>()) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon">F<sub>6</sub></font>()</span>)</span>,<font color="Maroon">F<sub>7</sub></font>(<span class="p1">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon">F<sub>6</sub></font>()</span>)</span>)) )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:3_1_3"/>
<font color="Maroon">F<sub>6</sub></font>() is <a href="zf_lang.html#V6" title="ZF_LANG:attr.6">universal</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">F<sub>7</sub></font>(<font color="Maroon">F<sub>6</sub></font>()) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon">F<sub>6</sub></font>()</span>)</span>,<font color="Maroon">F<sub>7</sub></font>(<span class="p1">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon">F<sub>6</sub></font>()</span>)</span>))</span><br/>

<a NAME="E2:3_1_3"/><b>then </b>
 ex <font color="Olive" title="b1">b</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Maroon">F<sub>7</sub></font>(<font color="Maroon">F<sub>6</sub></font>()) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon">F<sub>6</sub></font>()</span>)</span>,<font color="Olive" title="b1">b</font>) &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon">F<sub>6</sub></font>()</span>)</span>,<font color="Olive" title="b1">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> )
 
<b>by </b><i><a class="txt" href="#E3:3_1"><i><font color="Green" title="E3">A3</font></i></a>, <a class="txt" href="#E4:3_1"><i><font color="Green" title="E4">A4</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:3_1_3"/>
<font color="Maroon">F<sub>7</sub></font>(<font color="Maroon">F<sub>6</sub></font>()) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon">F<sub>6</sub></font>()</span>)</span>,<font color="Maroon">F<sub>7</sub></font>(<span class="p1">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon">F<sub>6</sub></font>()</span>)</span>))
 <b>by </b><i><a class="txt" href="#E1:3"><i><font color="Green" title="E1">A1</font></i></a>, <a class="txt" href="#E2:3_1"><i><font color="Green" title="E2">A2</font></i></a>, <a class="txt" href="#E4:3_1"><i><font color="Green" title="E4">A4</font></i></a></i>; <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>


</div>
