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<b>let </b><font color="Maroon" title="c2">A</font> be    <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon" title="c1">S</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> is <a href="circcomb.html#V6" title="CIRCCOMB:attr.6">Boolean</a> implies ( <font color="Maroon" title="c2">A</font> is <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a> &amp; <font color="Maroon" title="c2">A</font> is <a href="msafree2.html#V4" title="MSAFREE2:attr.4">locally-finite</a> ) )</span><br/>

<b>assume </b><a NAME="E1:103_1_1"/><i><font color="Green" title="E58">A1</font></i>: 
<font color="Maroon" title="c2">A</font> is <a href="circcomb.html#V6" title="CIRCCOMB:attr.6">Boolean</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">A</font> is <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a> &amp; <font color="Maroon" title="c2">A</font> is <a href="msafree2.html#V4" title="MSAFREE2:attr.4">locally-finite</a> )</span><br/>

<a NAME="E2:103_1_1"/><b>then </b>
the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">A</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <a href="circcomb.html#K10" title="CIRCCOMB:func.10">BOOLEAN</a> 
 
<b>by </b><i><a class="ref" href="circcomb.html#T65" target="_self" title="CIRCCOMB:th.65">Th65</a></i>;<br/>
<b>hence </b><a NAME="E3:103_1_1"/>
<font color="Maroon" title="c2">A</font> is <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a>
 <b>by </b><i><a class="ref" href="msualg_1.html#D8" title="MSUALG_1:def.8">MSUALG_1:def 8</a></i>; <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> is <a href="msafree2.html#V4" title="MSAFREE2:attr.4">locally-finite</a></span><br/>

<b>let </b><font color="Maroon" title="c3">v</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="pre_circ.html#D3" title="PRE_CIRC:def.3">PRE_CIRC:def 3</a>,<a class="ref" href="msafree2.html#D11" title="MSAFREE2:def.11">MSAFREE2:def 11</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">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> or the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> )</span><br/>

<b>assume </b><a NAME="E4:103_1_1"/>
<font color="Maroon" title="c3">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a></span><br/>

<b>hence </b><a NAME="E5:103_1_1"/>
the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>by </b><i><a class="txt" href="#E1:103_1_1"><i><font color="Green" title="E58">A1</font></i></a>, <a class="ref" href="circcomb.html#D15" target="_self" title="CIRCCOMB:def.15">Def15</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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