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<a NAME="E1:12_1_1"/><i><font color="Green" title="E3">A1</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">IT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<div><b>hereby </b> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( for <font color="Olive" title="b1">i</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> holds <br/><font color="Maroon" title="c2">IT</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> ) implies <font color="Maroon" title="c2">IT</font> is <a href="finset_1.html#V2" title="FINSET_1:attr.2">locally-finite</a> )</span>
<div class="add"><b>assume </b><a NAME="E1:12_1_1_1"/><i><font color="Green" title="E4">A2</font></i>: 
<font color="Maroon" title="c2">IT</font> is <a href="finset_1.html#V2" title="FINSET_1:attr.2">locally-finite</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">i</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> holds <br/><font color="Maroon" title="c2">IT</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a></span><br/><b>let </b><font color="Maroon" title="c3">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> implies <font color="Maroon" title="c2">IT</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> )</span><br/><b>assume </b><a NAME="E2:12_1_1_1"/>
<font color="Maroon" title="c3">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</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">IT</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a></span><br/><a NAME="E3:12_1_1_1"/><b>then </b>
<font color="Maroon" title="c2">IT</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">IT</font>
 <b>by </b><i><a class="txt" href="pre_circ.html#E1:12_1_1"><i><font color="Green" title="E3">A1</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/><b>hence </b><a NAME="E4:12_1_1_1"/>
<font color="Maroon" title="c2">IT</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>by </b><i><a class="txt" href="pre_circ.html#E1:12_1_1_1"><i><font color="Green" title="E4">A2</font></i></a>, <a class="ref" href="finset_1.html#D2" title="FINSET_1:def.2">FINSET_1:def 2</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>assume </b><a NAME="E3:12_1_1"/><i><font color="Green" title="E4">A3</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> holds <br/><font color="Maroon" title="c2">IT</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">IT</font> is <a href="finset_1.html#V2" title="FINSET_1:attr.2">locally-finite</a></span><br/>

<b>let </b><font color="Maroon" title="c3">i</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="finset_1.html#D2" title="FINSET_1:def.2">FINSET_1:def 2</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="c3">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">proj2</a> <font color="Maroon" title="c2">IT</font> or <font color="Maroon" title="c3">i</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> )</span><br/>

<b>assume </b><a NAME="E4:12_1_1"/>
<font color="Maroon" title="c3">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">IT</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">i</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a></span><br/>

<a NAME="E5:12_1_1"/><b>then </b>
 ex <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">IT</font> &amp; <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">IT</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> )
 
<b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E6:12_1_1"/>
<font color="Maroon" title="c3">i</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>by </b><i><a class="txt" href="pre_circ.html#E1:12_1_1"><i><font color="Green" title="E3">A1</font></i></a>, <a class="txt" href="pre_circ.html#E3:12_1_1"><i><font color="Green" title="E4">A3</font></i></a></i>; <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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