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

<span class="kw">let </span><font color="Maroon" title="c1">P</font> be   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="proofs_1.html#L1" title="PROOFS_1:struct.1">ProofSystem</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">B</font>, <font color="Olive" title="b2">B1</font>, <font color="Olive" title="b3">B2</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">P</font>  st <font color="Olive" title="b2">B1</font> is <font color="Olive" title="b1">B</font> <a href="proofs_1.html#V13" title="PROOFS_1:attr.13">-omitting</a>  &amp; <font color="Olive" title="b3">B2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">B1</font> holds <br/><font color="Olive" title="b3">B2</font> is <font color="Olive" title="b1">B</font> <a href="proofs_1.html#V13" title="PROOFS_1:attr.13">-omitting</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">B</font>, <font color="Maroon" title="c3">B1</font>, <font color="Maroon" title="c4">B2</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">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="c3">B1</font> is <font color="Maroon" title="c2">B</font> <a href="proofs_1.html#V13" title="PROOFS_1:attr.13">-omitting</a>  &amp; <font color="Maroon" title="c4">B2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">B1</font> implies <font color="Maroon" title="c4">B2</font> is <font color="Maroon" title="c2">B</font> <a href="proofs_1.html#V13" title="PROOFS_1:attr.13">-omitting</a>  )</span><br/>







<span class="kw">set </span><font color="Maroon" title="c5">A</font> =  the <a href="proofs_1.html#U1" title="PROOFS_1:sel.1">Axioms</a> of <font color="Maroon" title="c1">P</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">R</font> =  the <a href="proofs_1.html#U2" title="PROOFS_1:sel.2">Rules</a> of <font color="Maroon" title="c1">P</font>;<br/>
<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:79"/><span class="lab"><font color="Green" title="E30">A1</font></span>: 
<font color="Maroon" title="c3">B1</font> is <font color="Maroon" title="c2">B</font> <a href="proofs_1.html#V13" title="PROOFS_1:attr.13">-omitting</a> 
 <span class="kw">and </span><br/><a NAME="E2:79"/><span class="lab"><font color="Green" title="E31">A2</font></span>: 
<font color="Maroon" title="c4">B2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">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">B2</font> is <font color="Maroon" title="c2">B</font> <a href="proofs_1.html#V13" title="PROOFS_1:attr.13">-omitting</a> </span><br/>

<span class="kw">consider </span><font color="Maroon" title="c7">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E4:79"/><span class="lab"><font color="Green" title="E32">A3</font></span>: 
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font>
 <span class="kw">and </span><br/><a NAME="E5:79"/><span class="lab"><font color="Green" title="E33">A4</font></span>: 
 not <font color="Maroon" title="c1">P</font> <a href="proofs_1.html#K9" title="PROOFS_1:func.9">\/</a> <font color="Maroon" title="c3">B1</font> <a href="proofs_1.html#R3" title="PROOFS_1:pred.3">|-</a> <font color="Maroon" title="c7">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:79"><span class="lab"><font color="Green" title="E30">A1</font></span></a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c7">a</font>
; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="proofs_1.html#D23" target="_self" onmouseover="rs('proofs_1/D23')" onmouseout="rh()">PROOFS_1:def 23</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="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> &amp;  not <font color="Maroon" title="c1">P</font> <a href="proofs_1.html#K9" title="PROOFS_1:func.9">\/</a> <font color="Maroon" title="c4">B2</font> <a href="proofs_1.html#R3" title="PROOFS_1:pred.3">|-</a> <font color="Maroon" title="c7">a</font> )</span><br/>

<a NAME="E6:79"/>
(  the <a href="proofs_1.html#U1" title="PROOFS_1:sel.1">Axioms</a> of <font color="Maroon" title="c1">P</font> <a href="proofs_1.html#K8" title="PROOFS_1:func.8">\/</a> <font color="Maroon" title="c3">B1</font> is   <a href="proofs_1.html#NM10" title="PROOFS_1:NM.10">Extension</a> of  the <a href="proofs_1.html#U1" title="PROOFS_1:sel.1">Axioms</a> of <font color="Maroon" title="c1">P</font> <a href="proofs_1.html#K8" title="PROOFS_1:func.8">\/</a> <font color="Maroon" title="c4">B2</font> &amp;  the <a href="proofs_1.html#U2" title="PROOFS_1:sel.2">Rules</a> of <font color="Maroon" title="c1">P</font> is   <a href="proofs_1.html#NM11" title="PROOFS_1:NM.11">Extension</a> of  the <a href="proofs_1.html#U2" title="PROOFS_1:sel.2">Rules</a> of <font color="Maroon" title="c1">P</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:79"><span class="lab"><font color="Green" title="E31">A2</font></span></a>, <a class="ref" href="proofs_1.html#D12" target="_self" onmouseover="rs('proofs_1/D12')" onmouseout="rh()">Def11</a>, <a class="ref" href="proofs_1.html#D13" target="_self" onmouseover="rs('proofs_1/D13')" onmouseout="rh()">Def12</a>, <a class="ref" href="xboole_1.html#T9" onmouseover="rs('xboole_1/T9')" onmouseout="rh()">XBOOLE_1:9</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:79"/>
( <font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> &amp;  not <font color="Maroon" title="c1">P</font> <a href="proofs_1.html#K9" title="PROOFS_1:func.9">\/</a> <font color="Maroon" title="c4">B2</font> <a href="proofs_1.html#R3" title="PROOFS_1:pred.3">|-</a> <font color="Maroon" title="c7">a</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:79"><span class="lab"><font color="Green" title="E32">A3</font></span></a>, <a class="txt" href="#E5:79"><span class="lab"><font color="Green" title="E33">A4</font></span></a>, <a class="ref" href="proofs_1.html#T15" target="_self" onmouseover="rs('proofs_1/T15')" onmouseout="rh()">Th54</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>
