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<b>let </b><font color="Maroon" title="c2">MS</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> non <a href="struct_0.html#V11" title="STRUCT_0:attr.11">void</a> <a href="msualg_1.html#V5" title="MSUALG_1:attr.5">segmental</a>  <a href="msualg_1.html#L1" title="MSUALG_1:struct.1">ManySortedSign</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">A</font> being <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a>  <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon" title="c2">MS</font>  st the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">MS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> holds <br/> <a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K13" title="MSUALG_1:func.13">1-Alg</a> <font color="Olive" title="b1">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_1.html#G3" title="MSUALG_1:aggr.3">MSAlgebra</a>(# the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Olive" title="b1">A</font>,the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Charact</a> of <font color="Olive" title="b1">A</font> #)</span><br/><b>let </b><font color="Maroon" title="c3">A</font> be  <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a>  <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon" title="c2">MS</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">MS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> implies  <a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K13" title="MSUALG_1:func.13">1-Alg</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_1.html#G3" title="MSUALG_1:aggr.3">MSAlgebra</a>(# the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c3">A</font>,the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Charact</a> of <font color="Maroon" title="c3">A</font> #) )</span><br/>



<b>assume </b><a NAME="E1:26"/><i><font color="Green" title="E22">A1</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">MS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K13" title="MSUALG_1:func.13">1-Alg</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_1.html#G3" title="MSUALG_1:aggr.3">MSAlgebra</a>(# the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c3">A</font>,the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Charact</a> of <font color="Maroon" title="c3">A</font> #)</span><br/>

<a NAME="E2:26"/><b>then </b>
 <a href="msualg_1.html#K7" title="MSUALG_1:func.7">MSSign</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K13" title="MSUALG_1:func.13">1-Alg</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_1.html#G1" title="MSUALG_1:aggr.1">ManySortedSign</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">MS</font>,the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c2">MS</font>,the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">Arity</a> of <font color="Maroon" title="c2">MS</font>,the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">ResultSort</a> of <font color="Maroon" title="c2">MS</font> #)
 
<b>by </b><i><a class="ref" href="mssublat.html#T23" target="_self" title="MSSUBLAT:th.23">Th23</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c4">M1A</font> =  <a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K13" title="MSUALG_1:func.13">1-Alg</a> <font color="Maroon" title="c3">A</font></span>)</span> as  <a href="msualg_1.html#V3" title="MSUALG_1:attr.3">strict</a>  <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon" title="c2">MS</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c5">M1A</font> = <font color="Maroon" title="c4">M1A</font> as  <a href="msualg_1.html#V3" title="MSUALG_1:attr.3">strict</a> <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a>  <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon" title="c2">MS</font> <b>by </b><i><a class="ref" href="msualg_1.html#D8" title="MSUALG_1:def.8">MSUALG_1:def 8</a></i>;<br/>
<a NAME="E5:26"/><i><font color="Green" title="E23">A2</font></i>: 
 <a href="msualg_1.html#K13" title="MSUALG_1:func.13">1-Alg</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K13" title="MSUALG_1:func.13">1-Alg</a> <font color="Maroon" title="c3">A</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="unialg_1.html#G1" title="UNIALG_1:aggr.1">UAStr</a>(# <span class="p1">(<span class="default"><a href="msualg_1.html#K11" title="MSUALG_1:func.11">the_sort_of</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <span class="p3">(<span class="default"><a href="msualg_1.html#K13" title="MSUALG_1:func.13">1-Alg</a> <font color="Maroon" title="c3">A</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_1.html#K12" title="MSUALG_1:func.12">the_charact_of</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <span class="p3">(<span class="default"><a href="msualg_1.html#K13" title="MSUALG_1:func.13">1-Alg</a> <font color="Maroon" title="c3">A</font></span>)</span></span>)</span></span>)</span> #)
 
<b>by </b><i><a class="ref" href="msualg_1.html#D19" title="MSUALG_1:def.19">MSUALG_1:def 19</a></i>;<br/>
<a NAME="E6:26"/><i><font color="Green" title="E24">A3</font></i>: 
 <a href="msualg_1.html#K13" title="MSUALG_1:func.13">1-Alg</a> <font color="Maroon" title="c5">M1A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="unialg_1.html#G1" title="UNIALG_1:aggr.1">UAStr</a>(# <span class="p1">(<span class="default"><a href="msualg_1.html#K11" title="MSUALG_1:func.11">the_sort_of</a> <font color="Maroon" title="c5">M1A</font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_1.html#K12" title="MSUALG_1:func.12">the_charact_of</a> <font color="Maroon" title="c5">M1A</font></span>)</span> #)
 
<b>by </b><i><a class="ref" href="msualg_1.html#D19" title="MSUALG_1:def.19">MSUALG_1:def 19</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">c</font> =  <a href="msualg_1.html#K11" title="MSUALG_1:func.11">the_sort_of</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K13" title="MSUALG_1:func.13">1-Alg</a> <font color="Maroon" title="c3">A</font></span>)</span></span>)</span> as   <a href="pboole.html#NM3" title="PBOOLE:NM.3">Component</a> of the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K13" title="MSUALG_1:func.13">1-Alg</a> <font color="Maroon" title="c3">A</font></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="msualg_1.html#D17" title="MSUALG_1:def.17">MSUALG_1:def 17</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">c</font> = <font color="Maroon" title="c6">c</font> as   <a href="pboole.html#NM3" title="PBOOLE:NM.3">Component</a> of the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c5">M1A</font> ;<br/>
<a NAME="E9:26"/>
<font color="Maroon" title="c7">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_1.html#K11" title="MSUALG_1:func.11">the_sort_of</a> <font color="Maroon" title="c5">M1A</font>
 
<b>by </b><i><a class="ref" href="msualg_1.html#D17" title="MSUALG_1:def.17">MSUALG_1:def 17</a></i>;<br/>
<a NAME="E10:26"/><b>then </b><i><font color="Green" title="E25">A4</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="msualg_1.html#K13" title="MSUALG_1:func.13">1-Alg</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="msualg_1.html#K13" title="MSUALG_1:func.13">1-Alg</a> <font color="Maroon" title="c5">M1A</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E5:26"><i><font color="Green" title="E23">A2</font></i></a>, <a class="txt" href="#E6:26"><i><font color="Green" title="E24">A3</font></i></a>, <a class="ref" href="msualg_1.html#T15" title="MSUALG_1:th.15">MSUALG_1:15</a></i>;<br/>
the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <span class="p1">(<span class="default"><a href="msualg_1.html#K13" title="MSUALG_1:func.13">1-Alg</a> <font color="Maroon" title="c3">A</font></span>)</span> = 
 <a href="msualg_1.html#K12" title="MSUALG_1:func.12">the_charact_of</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K13" title="MSUALG_1:func.13">1-Alg</a> <font color="Maroon" title="c3">A</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E5:26"><i><font color="Green" title="E23">A2</font></i></a>, <a class="ref" href="msualg_1.html#T15" title="MSUALG_1:th.15">MSUALG_1:15</a></i>
<br/>.= 
the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Charact</a> of <font color="Maroon" title="c5">M1A</font>
<b>by </b><i><a class="ref" href="msualg_1.html#D18" title="MSUALG_1:def.18">MSUALG_1:def 18</a></i>
<br/>.= 
the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <span class="p1">(<span class="default"><a href="msualg_1.html#K13" title="MSUALG_1:func.13">1-Alg</a> <font color="Maroon" title="c5">M1A</font></span>)</span>
<b>by </b><i><a class="txt" href="#E6:26"><i><font color="Green" title="E24">A3</font></i></a>, <a class="ref" href="msualg_1.html#D18" title="MSUALG_1:def.18">MSUALG_1:def 18</a></i>
;<br/>
<b>hence </b><a NAME="E12:26"/>
 <a href="msualg_1.html#G3" title="MSUALG_1:aggr.3">MSAlgebra</a>(# the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c3">A</font>,the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Charact</a> of <font color="Maroon" title="c3">A</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K13" title="MSUALG_1:func.13">1-Alg</a> <font color="Maroon" title="c3">A</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:26"><i><font color="Green" title="E22">A1</font></i></a>, <a class="txt" href="#E10:26"><i><font color="Green" title="E25">A4</font></i></a>, <a class="ref" href="mssublat.html#T24" target="_self" title="MSSUBLAT:th.24">Th24</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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