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<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">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  holds <br/> ( <font color="Maroon" title="c1">P</font> <a href="proofs_1.html#R3" title="PROOFS_1:pred.3">|-</a> <font color="Olive" title="b1">a</font> iff  for <font color="Olive" title="b2">X</font> being  <font color="Maroon" title="c1">P</font> <a href="proofs_1.html#V19" title="PROOFS_1:attr.19">-closed</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  holds  <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">X</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">a</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="c1">P</font> <a href="proofs_1.html#R3" title="PROOFS_1:pred.3">|-</a> <font color="Maroon" title="c2">a</font> iff  for <font color="Olive" title="b1">X</font> being  <font color="Maroon" title="c1">P</font> <a href="proofs_1.html#V19" title="PROOFS_1:attr.19">-closed</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  holds  <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">X</font> )</span><br/>



<span class="kw">set </span><font color="Maroon" title="c3">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">thus </span><a NAME="E1:107"/>
( <font color="Maroon" title="c1">P</font> <a href="proofs_1.html#R3" title="PROOFS_1:pred.3">|-</a> <font color="Maroon" title="c2">a</font> implies  for <font color="Olive" title="b1">X</font> being  <font color="Maroon" title="c1">P</font> <a href="proofs_1.html#V19" title="PROOFS_1:attr.19">-closed</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  holds  <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">X</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="proofs_1.html#T24" target="_self" onmouseover="rs('proofs_1/T24')" onmouseout="rh()">Th65</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"> ( (  for <font color="Olive" title="b1">X</font> being  <font color="Maroon" title="c1">P</font> <a href="proofs_1.html#V19" title="PROOFS_1:attr.19">-closed</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  holds  <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">X</font> ) implies <font color="Maroon" title="c1">P</font> <a href="proofs_1.html#R3" title="PROOFS_1:pred.3">|-</a> <font color="Maroon" title="c2">a</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:107"/>
 for <font color="Olive" title="b1">X</font> being  <font color="Maroon" title="c1">P</font> <a href="proofs_1.html#V19" title="PROOFS_1:attr.19">-closed</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  holds  <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">X</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="c1">P</font> <a href="proofs_1.html#R3" title="PROOFS_1:pred.3">|-</a> <font color="Maroon" title="c2">a</font></span><br/>

<a NAME="E3:107"/><span class="kw">then </span>
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="proofs_1.html#K12" title="PROOFS_1:func.12">Theorems</a> <font color="Maroon" title="c1">P</font>
 
;<br/>
<span class="kw">hence </span><a NAME="E4:107"/>
<font color="Maroon" title="c1">P</font> <a href="proofs_1.html#R3" title="PROOFS_1:pred.3">|-</a> <font color="Maroon" title="c2">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="proofs_1.html#D34" target="_self" onmouseover="rs('proofs_1/D34')" onmouseout="rh()">Def30r</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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