<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</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  <a href="msualg_1.html#K13" title="MSUALG_1:func.13">1-Alg</a> <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_1.html#K13" title="MSUALG_1:func.13">1-Alg</a> <font color="Olive" title="b2">B</font> holds <br/> <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> #) <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="b2">B</font>,the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Charact</a> of <font color="Olive" title="b2">B</font> #)</span><br/><span class="kw">let </span><font color="Maroon" title="c3">A</font>, <font color="Maroon" title="c4">B</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="msualg_1.html#K13" title="MSUALG_1:func.13">1-Alg</a> <font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_1.html#K13" title="MSUALG_1:func.13">1-Alg</a> <font color="Maroon" title="c4">B</font> implies  <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#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="c4">B</font>,the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Charact</a> of <font color="Maroon" title="c4">B</font> #) )</span><br/>



<span class="kw">assume </span><a NAME="E1:24"/><span class="lab"><font color="Green" title="E21">A1</font></span>: 
 <a href="msualg_1.html#K13" title="MSUALG_1:func.13">1-Alg</a> <font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_1.html#K13" title="MSUALG_1:func.13">1-Alg</a> <font color="Maroon" title="c4">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <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#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="c4">B</font>,the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Charact</a> of <font color="Maroon" title="c4">B</font> #)</span><br/>

<a NAME="E2:24"/><span class="lab"><font color="Green" title="E22">A2</font></span>: 
 <a href="msualg_1.html#K13" title="MSUALG_1:func.13">1-Alg</a> <font color="Maroon" title="c4">B</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="c4">B</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="c4">B</font></span>)</span> #)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_1.html#D19" title="MSUALG_1:def.19">MSUALG_1:def 19</a></span>;<br/>
<a NAME="E3:24"/><span class="lab"><font color="Green" title="E23">A3</font></span>: 
 <a href="msualg_1.html#K13" title="MSUALG_1:func.13">1-Alg</a> <font color="Maroon" title="c3">A</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="c3">A</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="c3">A</font></span>)</span> #)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_1.html#D19" title="MSUALG_1:def.19">MSUALG_1:def 19</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E24">A4</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">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="c2">MS</font> implies the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c4">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> )</span><br/><span class="kw">assume </span><a NAME="E1:24_1"/><span class="lab"><font color="Green" title="E25">A5</font></span>: 
<font color="Maroon" title="c5">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="c2">MS</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c4">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font></span><br/><a NAME="E2:24_1"/><span class="lab"><font color="Green" title="E26">A6</font></span>: 
 ex <font color="Olive" title="b1">c</font> being  <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="c4">B</font> st the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c4">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">c</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c6">c</font> = the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c4">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</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="c4">B</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24_1"><span class="lab"><font color="Green" title="E25">A5</font></span></a>, <a class="ref" href="pboole.html#T151" title="PBOOLE:th.151">PBOOLE:151</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c6">c</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c4">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">c</font></span><br/>

<span class="kw">thus </span><a NAME="E2:24_1_1"/>
the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c4">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">c</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E3:24_1"/>
 ex <font color="Olive" title="b1">c</font> being  <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="c3">A</font> st the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">c</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c6">c</font> = the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</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="c3">A</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24_1"><span class="lab"><font color="Green" title="E25">A5</font></span></a>, <a class="ref" href="pboole.html#T151" title="PBOOLE:th.151">PBOOLE:151</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c6">c</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">c</font></span><br/>

<span class="kw">thus </span><a NAME="E2:24_1_2"/>
the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">c</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">then </span>the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> = 
 <a href="msualg_1.html#K11" title="MSUALG_1:func.11">the_sort_of</a> <font color="Maroon" title="c4">B</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24"><span class="lab"><font color="Green" title="E21">A1</font></span></a>, <a class="txt" href="#E3:24"><span class="lab"><font color="Green" title="E23">A3</font></span></a>, <a class="txt" href="#E2:24"><span class="lab"><font color="Green" title="E22">A2</font></span></a>, <a class="ref" href="msualg_1.html#D17" title="MSUALG_1:def.17">MSUALG_1:def 17</a></span>
<br/>.= 
the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c4">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:24_1"><span class="lab"><font color="Green" title="E26">A6</font></span></a>, <a class="ref" href="msualg_1.html#D17" title="MSUALG_1:def.17">MSUALG_1:def 17</a></span>
;<br/><span class="kw">hence </span><a NAME="E5:24_1"/>
the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c4">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Charact</a> of <font color="Maroon" title="c3">A</font> = 
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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:24"><span class="lab"><font color="Green" title="E23">A3</font></span></a>, <a class="ref" href="msualg_1.html#D18" title="MSUALG_1:def.18">MSUALG_1:def 18</a></span>
<br/>.= 
the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Charact</a> of <font color="Maroon" title="c4">B</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24"><span class="lab"><font color="Green" title="E21">A1</font></span></a>, <a class="txt" href="#E2:24"><span class="lab"><font color="Green" title="E22">A2</font></span></a>, <a class="ref" href="msualg_1.html#D18" title="MSUALG_1:def.18">MSUALG_1:def 18</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E6:24"/>
 <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#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="c4">B</font>,the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Charact</a> of <font color="Maroon" title="c4">B</font> #)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:24"><span class="lab"><font color="Green" title="E24">A4</font></span></a>, <a class="ref" href="pboole.html#T3" title="PBOOLE:th.3">PBOOLE:3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
