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<div class="add">

<b>take </b><font color="Maroon">T</font> =  <a href="msafree2.html#K8" target="_self">Trivial_Algebra</a> <font color="Maroon">S</font>;<br/>

<a NAME="E1:25_1_1"/><i><font color="Green">E29</font></i>: 
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">T</font> <a href="pboole.html#R6">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i/>;<br/>
<b>thus </b><a NAME="E2:25_1_1"/>
<font color="Maroon">T</font> is <a href="msafree2.html#V4" target="_self">locally-finite</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:25_1_1_1"/>
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">T</font> is <a href="finset_1.html#V2">locally-finite</a>
  <i><font color="Red">:: according to </font></i><a class="ref" href="msafree2.html#D11" target="_self">MSAFREE2:def 11</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="pre_circ.html#D3">PRE_CIRC:def 3</a><br/>

<b>assume </b><a NAME="E1:25_1_1_1_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 ;<br/>

<b>hence </b><a NAME="E2:25_1_1_1_1"/>
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">T</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/>


</div><b>end;</b></div>


</div><b>end;</b></div>

<b>thus </b><a NAME="E3:25_1_1"/>
<font color="Maroon">T</font> is <a href="msualg_1.html#V5">non-empty</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:25_1_1_2"/>
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">T</font> is <a href="relat_1.html#V2">non-empty</a>
  <i><font color="Red">:: according to </font></i><a class="ref" href="msualg_1.html#D8">MSUALG_1:def 8</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="pboole.html#D16">PBOOLE:def 16</a><br/>

<b>assume </b><a NAME="E1:25_1_1_2_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 ;<br/>

<b>hence </b><a NAME="E2:25_1_1_2_1"/>
not the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">T</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/>


</div><b>end;</b></div>


</div><b>end;</b></div>

<b>thus </b><a NAME="E4:25_1_1"/>
<font color="Maroon">T</font> is <a href="msualg_1.html#V4">strict</a>
 ;<br/>


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