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<b>take </b><font color="Maroon" title="c2">T</font> =  <a href="msafree2.html#K8" title="MSAFREE2:func.8">Trivial_Algebra</a> <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">T</font> is <a href="msafree2.html#V4" title="MSAFREE2:attr.4">locally-finite</a> &amp; <font color="Maroon" title="c2">T</font> is <a href="msualg_1.html#V5" title="MSUALG_1:attr.5">non-empty</a> &amp; <font color="Maroon" title="c2">T</font> is <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">strict</a> )</span><br/>

<a NAME="E1:25_1_1"/><i><font color="Green" title="E10">A1</font></i>: 
the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">T</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#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="msafree2.html#D12" target="_self" title="MSAFREE2:def.12">Def12</a></i>;<br/>
<b>thus </b><a NAME="E2:25_1_1"/>
<font color="Maroon" title="c2">T</font> is <a href="msafree2.html#V4" title="MSAFREE2:attr.4">locally-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">T</font> is <a href="msualg_1.html#V5" title="MSUALG_1:attr.5">non-empty</a> &amp; <font color="Maroon" title="c2">T</font> is <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">strict</a> )</span><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" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">T</font> is <a href="finset_1.html#V2" title="FINSET_1:attr.2">locally-finite</a>
  <i><font color="Red">:: according to </font></i><a class="ref" href="msafree2.html#D11" target="_self" 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"> verum</span><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" 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="pre_circ.html#D3" title="PRE_CIRC:def.3">PRE_CIRC: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="c3">i</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#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">T</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="E1:25_1_1_1_1"/>
<font color="Maroon" title="c3">i</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#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">T</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>hence </b><a NAME="E2:25_1_1_1_1"/>
the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">T</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="msafree2.html#E1:25_1_1"><i><font color="Green" title="E10">A1</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</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>


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

<b>thus </b><a NAME="E3:25_1_1"/>
<font color="Maroon" title="c2">T</font> is <a href="msualg_1.html#V5" title="MSUALG_1:attr.5">non-empty</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">T</font> is <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">strict</a></span><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" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">T</font> is <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a>
 <b>by </b><i><a class="txt" href="msafree2.html#E1:25_1_1"><i><font color="Green" title="E10">A1</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="msualg_1.html#D8" title="MSUALG_1:def.8">MSUALG_1:def 8</a> <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="E4:25_1_1"/>
<font color="Maroon" title="c2">T</font> is <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">strict</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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