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

<span class="kw">let </span><font color="Maroon" title="c1">B</font>, <font color="Maroon" title="c2">B1</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ; <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">R</font> being   <a href="proofs_1.html#NM2" title="PROOFS_1:NM.2">Rule</a><br/>  for <font color="Olive" title="b2">S</font> being   <a href="proofs_1.html#NM3" title="PROOFS_1:NM.3">Formula-finset</a>  st <font color="Olive" title="b2">S</font> is <font color="Maroon" title="c1">B</font>,<font color="Olive" title="b1">R</font> <a href="proofs_1.html#V3" title="PROOFS_1:attr.3">-derivable</a>  &amp; <font color="Maroon" title="c1">B</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Olive" title="b2">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">B1</font> holds <br/><font color="Olive" title="b2">S</font> is <font color="Maroon" title="c2">B1</font>,<font color="Olive" title="b1">R</font> <a href="proofs_1.html#V3" title="PROOFS_1:attr.3">-derivable</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c3">R</font> be   <a href="proofs_1.html#NM2" title="PROOFS_1:NM.2">Rule</a>; <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">S</font> being   <a href="proofs_1.html#NM3" title="PROOFS_1:NM.3">Formula-finset</a>  st <font color="Olive" title="b1">S</font> is <font color="Maroon" title="c1">B</font>,<font color="Maroon" title="c3">R</font> <a href="proofs_1.html#V3" title="PROOFS_1:attr.3">-derivable</a>  &amp; <font color="Maroon" title="c1">B</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Olive" title="b1">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">B1</font> holds <br/><font color="Olive" title="b1">S</font> is <font color="Maroon" title="c2">B1</font>,<font color="Maroon" title="c3">R</font> <a href="proofs_1.html#V3" title="PROOFS_1:attr.3">-derivable</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c4">S</font> be   <a href="proofs_1.html#NM3" title="PROOFS_1:NM.3">Formula-finset</a>; <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" title="c4">S</font> is <font color="Maroon" title="c1">B</font>,<font color="Maroon" title="c3">R</font> <a href="proofs_1.html#V3" title="PROOFS_1:attr.3">-derivable</a>  &amp; <font color="Maroon" title="c1">B</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c4">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">B1</font> implies <font color="Maroon" title="c4">S</font> is <font color="Maroon" title="c2">B1</font>,<font color="Maroon" title="c3">R</font> <a href="proofs_1.html#V3" title="PROOFS_1:attr.3">-derivable</a>  )</span><br/>







<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:45"/><span class="lab"><font color="Green" title="E19">A1</font></span>: 
<font color="Maroon" title="c4">S</font> is <font color="Maroon" title="c1">B</font>,<font color="Maroon" title="c3">R</font> <a href="proofs_1.html#V3" title="PROOFS_1:attr.3">-derivable</a> 
 <span class="kw">and </span><br/><a NAME="E2:45"/><span class="lab"><font color="Green" title="E20">A2</font></span>: 
<font color="Maroon" title="c1">B</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c4">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">B1</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" title="c4">S</font> is <font color="Maroon" title="c2">B1</font>,<font color="Maroon" title="c3">R</font> <a href="proofs_1.html#V3" title="PROOFS_1:attr.3">-derivable</a> </span><br/>

<span class="kw">consider </span><font color="Maroon" title="c5">P</font> being   <a href="proofs_1.html#NM4" title="PROOFS_1:NM.4">Formula-sequence</a><span class="kw"> such that </span><br/><a NAME="E4:45"/><span class="lab"><font color="Green" title="E21">A3</font></span>: 
<font color="Maroon" title="c4">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="proofs_1.html#K1" title="PROOFS_1:func.1">rng</a> <font color="Maroon" title="c5">P</font>
 <span class="kw">and </span><br/><a NAME="E5:45"/><span class="lab"><font color="Green" title="E22">A4</font></span>: 
<font color="Maroon" title="c5">P</font> is <font color="Maroon" title="c1">B</font>,<font color="Maroon" title="c3">R</font> <a href="proofs_1.html#V2" title="PROOFS_1:attr.2">-correct</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45"><span class="lab"><font color="Green" title="E19">A1</font></span></a></span>;<br/>
<a NAME="E6:45"/>
<font color="Maroon" title="c5">P</font> is <font color="Maroon" title="c2">B1</font>,<font color="Maroon" title="c3">R</font> <a href="proofs_1.html#V2" title="PROOFS_1:attr.2">-correct</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="45_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">k</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="proofs_1.html#D7" target="_self" onmouseover="rs('proofs_1/D7')" onmouseout="rh()">PROOFS_1:def 7</a> <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" title="c6">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c5">P</font> implies <font color="Maroon" title="c5">P</font>,<font color="Maroon" title="c6">k</font> <a href="proofs_1.html#R1" title="PROOFS_1:pred.1">is_a_correct_step_wrt</a> <font color="Maroon" title="c2">B1</font>,<font color="Maroon" title="c3">R</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:45_1"/><span class="lab"><font color="Green" title="E23">A6</font></span>: 
<font color="Maroon" title="c6">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c5">P</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" title="c5">P</font>,<font color="Maroon" title="c6">k</font> <a href="proofs_1.html#R1" title="PROOFS_1:pred.1">is_a_correct_step_wrt</a> <font color="Maroon" title="c2">B1</font>,<font color="Maroon" title="c3">R</font></span><br/>

<a NAME="E2:45_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A7</font></span>: 
<font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:45"><span class="lab"><font color="Green" title="E21">A3</font></span></a>, <a class="ref" href="funct_1.html#T3" onmouseover="rs('funct_1/T3')" onmouseout="rh()">FUNCT_1:3</a></span>;<br/>
<a NAME="E3:45_1"/>
<font color="Maroon" title="c5">P</font>,<font color="Maroon" title="c6">k</font> <a href="proofs_1.html#R1" title="PROOFS_1:pred.1">is_a_correct_step_wrt</a> <font color="Maroon" title="c1">B</font>,<font color="Maroon" title="c3">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:45"><span class="lab"><font color="Green" title="E22">A4</font></span></a>, <a class="txt" href="#E1:45_1"><span class="lab"><font color="Green" title="E23">A6</font></span></a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:45_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">B</font> or  ex <font color="Olive" title="b1">Q</font> being   <a href="proofs_1.html#NM3" title="PROOFS_1:NM.3">Formula-finset</a> st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">Q</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font></span>)</span></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="c3">R</font> &amp; (  for <font color="Olive" title="b2">t</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b2">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Q</font> holds <br/> ex <font color="Olive" title="b3">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>( <font color="Olive" title="b3">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c5">P</font> &amp; <font color="Olive" title="b3">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">k</font> &amp; <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">t</font> ) ) ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:45_1_1_1"/>
<font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">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" title="c5">P</font>,<font color="Maroon" title="c6">k</font> <a href="proofs_1.html#R1" title="PROOFS_1:pred.1">is_a_correct_step_wrt</a> <font color="Maroon" title="c2">B1</font>,<font color="Maroon" title="c3">R</font></span><br/><div class="add"><a NAME="E2:45_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">B</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c4">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:45_1"><span class="lab"><font color="Green" title="E24">A7</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/><span class="kw">hence </span><a NAME="E3:45_1_1_1"/>
<font color="Maroon" title="c5">P</font>,<font color="Maroon" title="c6">k</font> <a href="proofs_1.html#R1" title="PROOFS_1:pred.1">is_a_correct_step_wrt</a> <font color="Maroon" title="c2">B1</font>,<font color="Maroon" title="c3">R</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:45"><span class="lab"><font color="Green" title="E20">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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:45_1_1_2"/>
 ex <font color="Olive" title="b1">Q</font> being   <a href="proofs_1.html#NM3" title="PROOFS_1:NM.3">Formula-finset</a> st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">Q</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font></span>)</span></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="c3">R</font> &amp; (  for <font color="Olive" title="b2">t</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b2">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Q</font> holds <br/> ex <font color="Olive" title="b3">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>( <font color="Olive" title="b3">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c5">P</font> &amp; <font color="Olive" title="b3">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">k</font> &amp; <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">t</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" title="c5">P</font>,<font color="Maroon" title="c6">k</font> <a href="proofs_1.html#R1" title="PROOFS_1:pred.1">is_a_correct_step_wrt</a> <font color="Maroon" title="c2">B1</font>,<font color="Maroon" title="c3">R</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">Q</font> being   <a href="proofs_1.html#NM3" title="PROOFS_1:NM.3">Formula-finset</a><span class="kw"> such that </span><br/><a NAME="E3:45_1_1_2"/><span class="lab"><font color="Green" title="E25">A10</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">Q</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font></span>)</span></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="c3">R</font>
 <span class="kw">and </span><br/><a NAME="E4:45_1_1_2"/><span class="lab"><font color="Green" title="E26">A11</font></span>: 
 for <font color="Olive" title="b1">t</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Q</font> holds <br/> ex <font color="Olive" title="b2">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>( <font color="Olive" title="b2">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c5">P</font> &amp; <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">k</font> &amp; <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">t</font> )
 ;<br/><span class="kw">thus </span><a NAME="E5:45_1_1_2"/>
<font color="Maroon" title="c5">P</font>,<font color="Maroon" title="c6">k</font> <a href="proofs_1.html#R1" title="PROOFS_1:pred.1">is_a_correct_step_wrt</a> <font color="Maroon" title="c2">B1</font>,<font color="Maroon" title="c3">R</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:45_1_1_2"><span class="lab"><font color="Green" title="E25">A10</font></span></a>, <a class="txt" href="#E4:45_1_1_2"><span class="lab"><font color="Green" title="E26">A11</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></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E7:45"/>
<font color="Maroon" title="c4">S</font> is <font color="Maroon" title="c2">B1</font>,<font color="Maroon" title="c3">R</font> <a href="proofs_1.html#V3" title="PROOFS_1:attr.3">-derivable</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:45"><span class="lab"><font color="Green" title="E21">A3</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>
