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<span class="kw">thus </span><a NAME="E1:10_1"/>
( <font color="Maroon">F<sub>4</sub></font>() <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Maroon">F<sub>3</sub></font>()) implies  for <font color="Olive" title="b1">b</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>()  st <font color="Olive" title="b1">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">F<sub>4</sub></font>(),<font color="Olive" title="b1">b</font>] )
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  for <font color="Olive" title="b1">b</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>()  st <font color="Olive" title="b1">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">F<sub>4</sub></font>(),<font color="Olive" title="b1">b</font>] ) implies <font color="Maroon">F<sub>4</sub></font>() <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Maroon">F<sub>3</sub></font>()) )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="10_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ] <span class="kw">means </span><font color="Maroon">P<sub>2</sub></font>[$1,<font color="Maroon">F<sub>3</sub></font>()];<br/>
<span class="kw">assume </span><a NAME="E1:10_1_1"/>
<font color="Maroon">F<sub>4</sub></font>() <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Maroon">F<sub>3</sub></font>())
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">b</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>()  st <font color="Olive" title="b1">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">F<sub>4</sub></font>(),<font color="Olive" title="b1">b</font>]</span><br/>

<a NAME="E2:10_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E3">A3</font></span>: 
<font color="Maroon">F<sub>4</sub></font>() <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">a</font>] </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:10"><span class="lab"><font color="Green" title="E1">A1</font></span></a></span>;<br/>
<a NAME="E3:10_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">F<sub>4</sub></font>()]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="lmod_7.html#S6" target="_self" onmouseover="rs('lmod_7/S6')" onmouseout="rh()">LMOD_7:sch 6</a>(<a class="txt" href="#E2:10_1_1"><span class="lab"><font color="Green" title="E3">A3</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E4:10_1_1"/>
 for <font color="Olive" title="b1">b</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>()  st <font color="Olive" title="b1">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">F<sub>4</sub></font>(),<font color="Olive" title="b1">b</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:10"><span class="lab"><font color="Green" title="E2">A2</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E2:10_1"/>
 for <font color="Olive" title="b1">b</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>()  st <font color="Olive" title="b1">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">F<sub>4</sub></font>(),<font color="Olive" title="b1">b</font>]
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">F<sub>4</sub></font>() <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Maroon">F<sub>3</sub></font>())</span><br/>

<span class="kw">hence </span><a NAME="E3:10_1"/>
<font color="Maroon">F<sub>4</sub></font>() <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Maroon">F<sub>3</sub></font>())
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:10"><span class="lab"><font color="Green" title="E1">A1</font></span></a>, <a class="txt" href="#E2:10"><span class="lab"><font color="Green" title="E2">A2</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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