<?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> 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="b1">B</font> is  <a href="proofs_1.html#NV14" title="PROOFS_1:NV.14">inconsistent</a>  holds <br/>( <font color="Olive" title="b2">B1</font> is  <a href="proofs_1.html#V11" title="PROOFS_1:attr.11">consistent</a>  iff <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>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">B</font>, <font color="Maroon" title="c3">B1</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="c2">B</font> is  <a href="proofs_1.html#NV14" title="PROOFS_1:NV.14">inconsistent</a>  implies ( <font color="Maroon" title="c3">B1</font> is  <a href="proofs_1.html#V11" title="PROOFS_1:attr.11">consistent</a>  iff <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><br/>





<span class="kw">set </span><font color="Maroon" title="c4">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="c5">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><a NAME="E1:76"/>
<font color="Maroon" title="c2">B</font> is  <a href="proofs_1.html#NV14" title="PROOFS_1:NV.14">inconsistent</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="c3">B1</font> is  <a href="proofs_1.html#V11" title="PROOFS_1:attr.11">consistent</a>  iff <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><br/>

<a NAME="E2:76"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A1</font></span>: 
<font color="Maroon" title="c1">P</font> <a href="proofs_1.html#K9" title="PROOFS_1:func.9">\/</a> <font color="Maroon" title="c2">B</font> is  <a href="proofs_1.html#NV12" title="PROOFS_1:NV.12">inconsistent</a> 
 
;<br/>
<span class="kw">thus </span><a NAME="E3:76"/>
( <font color="Maroon" title="c3">B1</font> is  <a href="proofs_1.html#V11" title="PROOFS_1:attr.11">consistent</a>  implies <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>  )
  <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>  implies <font color="Maroon" title="c3">B1</font> is  <a href="proofs_1.html#V11" title="PROOFS_1:attr.11">consistent</a>  )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="76_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:76_1"/>
<font color="Maroon" title="c3">B1</font> is  <a href="proofs_1.html#V11" title="PROOFS_1:attr.11">consistent</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="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><br/>

<a NAME="E2:76_1"/><span class="kw">then </span>
<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> is  <a href="proofs_1.html#V10" title="PROOFS_1:attr.10">consistent</a> 
 
;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E4:76_1"/><span class="lab"><font color="Green" title="E30">A2</font></span>: 
<font color="Maroon" title="c6">a</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <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>
 <span class="kw">and </span><br/><a NAME="E5:76_1"/><span class="lab"><font color="Green" title="E31">A3</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="c6">a</font>
 ;<br/>
<a NAME="E6:76_1"/>
<font color="Maroon" title="c6">a</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c1">P</font> <a href="proofs_1.html#K9" title="PROOFS_1:func.9">\/</a> <font color="Maroon" title="c2">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:76_1"><span class="lab"><font color="Green" title="E30">A2</font></span></a></span>;<br/>
<a NAME="E7:76_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A5</font></span>: 
<font color="Maroon" title="c1">P</font> <a href="proofs_1.html#K9" title="PROOFS_1:func.9">\/</a> <font color="Maroon" title="c2">B</font> <a href="proofs_1.html#R3" title="PROOFS_1:pred.3">|-</a> <font color="Maroon" title="c6">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:76"><span class="lab"><font color="Green" title="E29">A1</font></span></a></span>;<br/>
<span class="kw">assume </span><a NAME="E8:76_1"/><span class="lab"><font color="Green" title="E33">A7</font></span>: 
 not <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> 
 ; <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"> contradiction</span><br/>

<a NAME="E9:76_1"/>
 for <font color="Olive" title="b1">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  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="c2">B</font> holds <br/> 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>, the <a href="proofs_1.html#U2" title="PROOFS_1:sel.2">Rules</a> of <font color="Maroon" title="c1">P</font> <a href="proofs_1.html#R2" title="PROOFS_1:pred.2">|-</a> <font color="Olive" title="b1">b</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="76_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  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="c2">B</font> implies  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>, the <a href="proofs_1.html#U2" title="PROOFS_1:sel.2">Rules</a> of <font color="Maroon" title="c1">P</font> <a href="proofs_1.html#R2" title="PROOFS_1:pred.2">|-</a> <font color="Maroon" title="c7">b</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:76_1_1"/>
<font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  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="c2">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">  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>, the <a href="proofs_1.html#U2" title="PROOFS_1:sel.2">Rules</a> of <font color="Maroon" title="c1">P</font> <a href="proofs_1.html#R2" title="PROOFS_1:pred.2">|-</a> <font color="Maroon" title="c7">b</font></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_1_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="E1:76_1_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="proofs_1.html#U1" title="PROOFS_1:sel.1">Axioms</a> of <font color="Maroon" title="c1">P</font> or <font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:76_1_1_1_1"/>
<font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="proofs_1.html#U1" title="PROOFS_1:sel.1">Axioms</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">  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>, the <a href="proofs_1.html#U2" title="PROOFS_1:sel.2">Rules</a> of <font color="Maroon" title="c1">P</font> <a href="proofs_1.html#R2" title="PROOFS_1:pred.2">|-</a> <font color="Maroon" title="c7">b</font></span><br/><div class="add"><a NAME="E2:76_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><span class="kw">hence </span><a NAME="E3:76_1_1_1_1"/>
 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>, the <a href="proofs_1.html#U2" title="PROOFS_1:sel.2">Rules</a> of <font color="Maroon" title="c1">P</font> <a href="proofs_1.html#R2" title="PROOFS_1:pred.2">|-</a> <font color="Maroon" title="c7">b</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="proofs_1.html#T8" target="_self" onmouseover="rs('proofs_1/T8')" onmouseout="rh()">Th46</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="76_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:76_1_1_1_2"/>
<font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">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">  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>, the <a href="proofs_1.html#U2" title="PROOFS_1:sel.2">Rules</a> of <font color="Maroon" title="c1">P</font> <a href="proofs_1.html#R2" title="PROOFS_1:pred.2">|-</a> <font color="Maroon" title="c7">b</font></span><br/><div class="add"><a NAME="E2:76_1_1_1_2"/><span class="kw">then </span>
<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">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:76_1"><span class="lab"><font color="Green" title="E33">A7</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:76_1_1_1_2"/>
 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>, the <a href="proofs_1.html#U2" title="PROOFS_1:sel.2">Rules</a> of <font color="Maroon" title="c1">P</font> <a href="proofs_1.html#R2" title="PROOFS_1:pred.2">|-</a> <font color="Maroon" title="c7">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"> 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="E10:76_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:76_1"><span class="lab"><font color="Green" title="E31">A3</font></span></a>, <a class="txt" href="#E7:76_1"><span class="lab"><font color="Green" title="E32">A5</font></span></a>, <a class="ref" href="proofs_1.html#T14" target="_self" onmouseover="rs('proofs_1/T14')" onmouseout="rh()">Th53</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="E4:76"/>
<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> 
 ; <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  <a href="proofs_1.html#V11" title="PROOFS_1:attr.11">consistent</a> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E6:76"/><span class="lab"><font color="Green" title="E30">A11</font></span>: 
<font color="Maroon" title="c6">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="E7:76"/><span class="lab"><font color="Green" title="E31">A12</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="c6">a</font>
 ;<br/>
<a NAME="E8:76"/>
<font color="Maroon" title="c6">a</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:76"><span class="lab"><font color="Green" title="E30">A11</font></span></a></span>;<br/>
<a NAME="E9:76"/><span class="kw">then </span>
<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> is  <a href="proofs_1.html#V10" title="PROOFS_1:attr.10">consistent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:76"><span class="lab"><font color="Green" title="E31">A12</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:76"/>
<font color="Maroon" title="c3">B1</font> is  <a href="proofs_1.html#V11" title="PROOFS_1:attr.11">consistent</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"> verum</span><br/>


</div>
