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<b>let </b><font color="Maroon" title="c1">T</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="intpro_1.html#K1" title="INTPRO_1:func.1">MC-wff</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">T</font> is <a href="intpro_1.html#V10" title="INTPRO_1:attr.10">S4_theory</a> iff  <a href="intpro_1.html#K13" title="INTPRO_1:func.13">CnS4</a> <font color="Maroon" title="c1">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">T</font> )</span><br/>

<div><b>hereby </b> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="intpro_1.html#K13" title="INTPRO_1:func.13">CnS4</a> <font color="Maroon" title="c1">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">T</font> implies <font color="Maroon" title="c1">T</font> is <a href="intpro_1.html#V10" title="INTPRO_1:attr.10">S4_theory</a> )</span>
<div class="add"><b>assume </b><a NAME="E1:144_1"/><i><font color="Green" title="E108">A1</font></i>: 
<font color="Maroon" title="c1">T</font> is <a href="intpro_1.html#V10" title="INTPRO_1:attr.10">S4_theory</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="intpro_1.html#K13" title="INTPRO_1:func.13">CnS4</a> <font color="Maroon" title="c1">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">T</font></span><br/><a NAME="E2:144_1"/><i><font color="Green" title="E109">A2</font></i>: 
<font color="Maroon" title="c1">T</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="intpro_1.html#K13" title="INTPRO_1:func.13">CnS4</a> <font color="Maroon" title="c1">T</font>
 <b>by </b><i><a class="ref" href="intpro_1.html#T89" target="_self" title="INTPRO_1:th.89">Th89</a></i>;<br/><a NAME="E3:144_1"/>
 <a href="intpro_1.html#K13" title="INTPRO_1:func.13">CnS4</a> <font color="Maroon" title="c1">T</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">T</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="144_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="intpro_1.html#K13" title="INTPRO_1:func.13">CnS4</a> <font color="Maroon" title="c1">T</font> or <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">T</font> )</span><br/>

<b>assume </b><a NAME="E1:144_1_1"/>
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="intpro_1.html#K13" title="INTPRO_1:func.13">CnS4</a> <font color="Maroon" title="c1">T</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">T</font></span><br/>

<b>hence </b><a NAME="E2:144_1_1"/>
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">T</font>
 <b>by </b><i><a class="txt" href="#E1:144_1"><i><font color="Green" title="E108">A1</font></i></a>, <a class="ref" href="intpro_1.html#D23" target="_self" title="INTPRO_1:def.23">Def23</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>hence </b><a NAME="E4:144_1"/>
 <a href="intpro_1.html#K13" title="INTPRO_1:func.13">CnS4</a> <font color="Maroon" title="c1">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">T</font>
 <b>by </b><i><a class="txt" href="#E2:144_1"><i><font color="Green" title="E109">A2</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</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="E2:144"/>
(  <a href="intpro_1.html#K13" title="INTPRO_1:func.13">CnS4</a> <font color="Maroon" title="c1">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">T</font> implies <font color="Maroon" title="c1">T</font> is <a href="intpro_1.html#V10" title="INTPRO_1:attr.10">S4_theory</a> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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