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<span class="kw">let </span><font color="Maroon" title="c1">S</font> be  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  non  <a href="struct_0.html#V11" title="STRUCT_0:attr.11">void</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">U1</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="c1">S</font><br/> for <font color="Olive" title="b2">f</font> being   <a href="autalg_1.html#M2" title="AUTALG_1:mode.2">Element</a> of  <a href="autalg_1.html#K4" title="AUTALG_1:func.4">MSAAut</a> <font color="Olive" title="b1">U1</font><br/> for <font color="Olive" title="b3">g</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="autalg_1.html#K6" title="AUTALG_1:func.6">MSAAutGroup</a> <font color="Olive" title="b1">U1</font></span>)</span>  st <font color="Olive" title="b2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">g</font> holds <br/><font color="Olive" title="b2">f</font> <a href="msualg_3.html#K4" title="MSUALG_3:func.4">""</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">g</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">U1</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="c1">S</font>; <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">f</font> being   <a href="autalg_1.html#M2" title="AUTALG_1:mode.2">Element</a> of  <a href="autalg_1.html#K4" title="AUTALG_1:func.4">MSAAut</a> <font color="Maroon" title="c2">U1</font><br/> for <font color="Olive" title="b2">g</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="autalg_1.html#K6" title="AUTALG_1:func.6">MSAAutGroup</a> <font color="Maroon" title="c2">U1</font></span>)</span>  st <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g</font> holds <br/><font color="Olive" title="b1">f</font> <a href="msualg_3.html#K4" title="MSUALG_3:func.4">""</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c3">f</font> be    <a href="autalg_1.html#M2" title="AUTALG_1:mode.2">Element</a> of  <a href="autalg_1.html#K4" title="AUTALG_1:func.4">MSAAut</a> <font color="Maroon" title="c2">U1</font>; <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">g</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="autalg_1.html#K6" title="AUTALG_1:func.6">MSAAutGroup</a> <font color="Maroon" title="c2">U1</font></span>)</span>  st <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> holds <br/><font color="Maroon" title="c3">f</font> <a href="msualg_3.html#K4" title="MSUALG_3:func.4">""</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c4">g</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="autalg_1.html#K6" title="AUTALG_1:func.6">MSAAutGroup</a> <font color="Maroon" title="c2">U1</font></span>)</span>; <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="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> implies <font color="Maroon" title="c3">f</font> <a href="msualg_3.html#K4" title="MSUALG_3:func.4">""</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a>  )</span><br/>







<span class="kw">consider </span><font color="Maroon" title="c5">g1</font> being    <a href="autalg_1.html#M2" title="AUTALG_1:mode.2">Element</a> of  <a href="autalg_1.html#K4" title="AUTALG_1:func.4">MSAAut</a> <font color="Maroon" title="c2">U1</font><span class="kw"> such that </span><br/><a NAME="E2:49"/><span class="lab"><font color="Green" title="E27">A1</font></span>: 
<font color="Maroon" title="c5">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> 
 ;<br/>
<span class="kw">assume </span><a NAME="E3:49"/>
<font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font>
 ; <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="c3">f</font> <a href="msualg_3.html#K4" title="MSUALG_3:func.4">""</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span><br/>

<a NAME="E4:49"/><span class="kw">then </span>
<font color="Maroon" title="c5">g1</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:49"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="ref" href="autalg_1.html#D8" target="_self" title="AUTALG_1:def.8">Def8</a></span>;<br/>
<a NAME="E5:49"/><span class="kw">then </span>
<font color="Maroon" title="c5">g1</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <span class="p1">(<span class="default"><a href="autalg_1.html#K6" title="AUTALG_1:func.6">MSAAutGroup</a> <font color="Maroon" title="c2">U1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D6" title="GROUP_1:def.6">GROUP_1:def 6</a></span>;<br/>
<a NAME="E6:49"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A2</font></span>: 
<font color="Maroon" title="c5">g1</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c3">f</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="msualg_3.html#K2" title="MSUALG_3:func.2">id</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">U1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="autalg_1.html#T36" target="_self" title="AUTALG_1:th.36">Th36</a></span>;<br/>
<a NAME="E7:49"/>
( <font color="Maroon" title="c3">f</font> is  <a href="msualg_3.html#V1" title="MSUALG_3:attr.1">"1-1"</a>  &amp; <font color="Maroon" title="c3">f</font> is  <a href="msualg_3.html#V2" title="MSUALG_3:attr.2">"onto"</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="autalg_1.html#E40"><span class="lab"><font color="Green" title="E20">Lm3</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:49"/>
<font color="Maroon" title="c3">f</font> <a href="msualg_3.html#K4" title="MSUALG_3:func.4">""</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:49"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="txt" href="#E6:49"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="ref" href="autalg_1.html#T19" target="_self" title="AUTALG_1:th.19">Th19</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/>


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