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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="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">U0</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">B</font> being   <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="c1">S</font><br/>  for <font color="Olive" title="b3">G</font> being    <a href="msafree.html#M1" title="MSAFREE:mode.1">GeneratorSet</a> of <font color="Olive" title="b1">U0</font><br/>  for <font color="Olive" title="b4">H</font> being  <a href="relat_1.html#V2" title="RELAT_1:attr.2">V2</a>()  <a href="msafree.html#M1" title="MSAFREE:mode.1">GeneratorSet</a> of <font color="Olive" title="b2">B</font><br/>  for <font color="Olive" title="b5">f</font> being   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Olive" title="b1">U0</font>,<font color="Olive" title="b2">B</font>  st <font color="Olive" title="b4">H</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Olive" title="b5">f</font> <a href="pboole.html#K9" title="PBOOLE:func.9">.:.:</a> <font color="Olive" title="b3">G</font> &amp; <font color="Olive" title="b5">f</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Olive" title="b1">U0</font>,<font color="Olive" title="b2">B</font> holds <br/><font color="Olive" title="b5">f</font> <a href="msualg_3.html#R2" title="MSUALG_3:pred.2">is_epimorphism</a> <font color="Olive" title="b1">U0</font>,<font color="Olive" title="b2">B</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">U0</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="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">B</font> being   <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="c1">S</font><br/>  for <font color="Olive" title="b2">G</font> being    <a href="msafree.html#M1" title="MSAFREE:mode.1">GeneratorSet</a> of <font color="Maroon" title="c2">U0</font><br/>  for <font color="Olive" title="b3">H</font> being  <a href="relat_1.html#V2" title="RELAT_1:attr.2">V2</a>()  <a href="msafree.html#M1" title="MSAFREE:mode.1">GeneratorSet</a> of <font color="Olive" title="b1">B</font><br/>  for <font color="Olive" title="b4">f</font> being   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon" title="c2">U0</font>,<font color="Olive" title="b1">B</font>  st <font color="Olive" title="b3">H</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Olive" title="b4">f</font> <a href="pboole.html#K9" title="PBOOLE:func.9">.:.:</a> <font color="Olive" title="b2">G</font> &amp; <font color="Olive" title="b4">f</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon" title="c2">U0</font>,<font color="Olive" title="b1">B</font> holds <br/><font color="Olive" title="b4">f</font> <a href="msualg_3.html#R2" title="MSUALG_3:pred.2">is_epimorphism</a> <font color="Maroon" title="c2">U0</font>,<font color="Olive" title="b1">B</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">B</font> be   <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="c1">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">G</font> being    <a href="msafree.html#M1" title="MSAFREE:mode.1">GeneratorSet</a> of <font color="Maroon" title="c2">U0</font><br/>  for <font color="Olive" title="b2">H</font> being  <a href="relat_1.html#V2" title="RELAT_1:attr.2">V2</a>()  <a href="msafree.html#M1" title="MSAFREE:mode.1">GeneratorSet</a> of <font color="Maroon" title="c3">B</font><br/>  for <font color="Olive" title="b3">f</font> being   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon" title="c2">U0</font>,<font color="Maroon" title="c3">B</font>  st <font color="Olive" title="b2">H</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Olive" title="b3">f</font> <a href="pboole.html#K9" title="PBOOLE:func.9">.:.:</a> <font color="Olive" title="b1">G</font> &amp; <font color="Olive" title="b3">f</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon" title="c2">U0</font>,<font color="Maroon" title="c3">B</font> holds <br/><font color="Olive" title="b3">f</font> <a href="msualg_3.html#R2" title="MSUALG_3:pred.2">is_epimorphism</a> <font color="Maroon" title="c2">U0</font>,<font color="Maroon" title="c3">B</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">G</font> be    <a href="msafree.html#M1" title="MSAFREE:mode.1">GeneratorSet</a> of <font color="Maroon" title="c2">U0</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">H</font> being  <a href="relat_1.html#V2" title="RELAT_1:attr.2">V2</a>()  <a href="msafree.html#M1" title="MSAFREE:mode.1">GeneratorSet</a> of <font color="Maroon" title="c3">B</font><br/>  for <font color="Olive" title="b2">f</font> being   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon" title="c2">U0</font>,<font color="Maroon" title="c3">B</font>  st <font color="Olive" title="b1">H</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Olive" title="b2">f</font> <a href="pboole.html#K9" title="PBOOLE:func.9">.:.:</a> <font color="Maroon" title="c4">G</font> &amp; <font color="Olive" title="b2">f</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon" title="c2">U0</font>,<font color="Maroon" title="c3">B</font> holds <br/><font color="Olive" title="b2">f</font> <a href="msualg_3.html#R2" title="MSUALG_3:pred.2">is_epimorphism</a> <font color="Maroon" title="c2">U0</font>,<font color="Maroon" title="c3">B</font></span><br/><span class="kw">let </span><font color="Maroon" title="c5">H</font> be  <a href="relat_1.html#V2" title="RELAT_1:attr.2">V2</a>()  <a href="msafree.html#M1" title="MSAFREE:mode.1">GeneratorSet</a> of <font color="Maroon" title="c3">B</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">f</font> being   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon" title="c2">U0</font>,<font color="Maroon" title="c3">B</font>  st <font color="Maroon" title="c5">H</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Olive" title="b1">f</font> <a href="pboole.html#K9" title="PBOOLE:func.9">.:.:</a> <font color="Maroon" title="c4">G</font> &amp; <font color="Olive" title="b1">f</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon" title="c2">U0</font>,<font color="Maroon" title="c3">B</font> holds <br/><font color="Olive" title="b1">f</font> <a href="msualg_3.html#R2" title="MSUALG_3:pred.2">is_epimorphism</a> <font color="Maroon" title="c2">U0</font>,<font color="Maroon" title="c3">B</font></span><br/><span class="kw">let </span><font color="Maroon" title="c6">f</font> be   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon" title="c2">U0</font>,<font color="Maroon" title="c3">B</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">H</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c6">f</font> <a href="pboole.html#K9" title="PBOOLE:func.9">.:.:</a> <font color="Maroon" title="c4">G</font> &amp; <font color="Maroon" title="c6">f</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon" title="c2">U0</font>,<font color="Maroon" title="c3">B</font> implies <font color="Maroon" title="c6">f</font> <a href="msualg_3.html#R2" title="MSUALG_3:pred.2">is_epimorphism</a> <font color="Maroon" title="c2">U0</font>,<font color="Maroon" title="c3">B</font> )</span><br/>











<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:30"/><span class="lab"><font color="Green" title="E15">A1</font></span>: 
<font color="Maroon" title="c5">H</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c6">f</font> <a href="pboole.html#K9" title="PBOOLE:func.9">.:.:</a> <font color="Maroon" title="c4">G</font>
 <span class="kw">and </span><br/><a NAME="E2:30"/><span class="lab"><font color="Green" title="E16">A2</font></span>: 
<font color="Maroon" title="c6">f</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon" title="c2">U0</font>,<font color="Maroon" title="c3">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">f</font> <a href="msualg_3.html#R2" title="MSUALG_3:pred.2">is_epimorphism</a> <font color="Maroon" title="c2">U0</font>,<font color="Maroon" title="c3">B</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c7">I</font> =  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_3.html#K6" title="MSUALG_3:func.6">Image</a> <font color="Maroon" title="c6">f</font></span>)</span> as  <a href="relat_1.html#V2" title="RELAT_1:attr.2">V2</a>() <a href="msualg_2.html#NM1" title="MSUALG_2:NM.1">MSSubset</a> of <font color="Maroon" title="c3">B</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_2.html#D9" title="MSUALG_2:def.9">MSUALG_2:def 9</a></span>;<br/>
<a NAME="E4:30"/>
<font color="Maroon" title="c6">f</font> <a href="pboole.html#K9" title="PBOOLE:func.9">.:.:</a> <font color="Maroon" title="c4">G</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c6">f</font> <a href="pboole.html#K9" title="PBOOLE:func.9">.:.:</a>  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">U0</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="extens_1.html#T2" title="EXTENS_1:th.2">EXTENS_1:2</a></span>;<br/>
<a NAME="E5:30"/><span class="kw">then </span>
<font color="Maroon" title="c5">H</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c6">f</font> <a href="pboole.html#K9" title="PBOOLE:func.9">.:.:</a>  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">U0</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30"><span class="lab"><font color="Green" title="E15">A1</font></span></a>, <a class="ref" href="pboole.html#T13" title="PBOOLE:th.13">PBOOLE:13</a></span>;<br/>
<a NAME="E6:30"/><span class="kw">then </span>
<font color="Maroon" title="c5">H</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a>  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_3.html#K6" title="MSUALG_3:func.6">Image</a> <font color="Maroon" title="c6">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:30"><span class="lab"><font color="Green" title="E16">A2</font></span></a>, <a class="ref" href="msualg_3.html#D12" title="MSUALG_3:def.12">MSUALG_3:def 12</a></span>;<br/>
<a NAME="E7:30"/><span class="kw">then </span>
<font color="Maroon" title="c5">H</font> is    <a href="pboole.html#M3" title="PBOOLE:mode.3">ManySortedSubset</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_3.html#K6" title="MSUALG_3:func.6">Image</a> <font color="Maroon" title="c6">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pboole.html#D18" title="PBOOLE:def.18">PBOOLE:def 18</a></span>;<br/>
<a NAME="E8:30"/><span class="kw">then </span><span class="lab"><font color="Green" title="E17">A3</font></span>: 
 <a href="msualg_2.html#K12" title="MSUALG_2:func.12">GenMSAlg</a> <font color="Maroon" title="c5">H</font> is    <a href="msualg_2.html#M1" title="MSUALG_2:mode.1">MSSubAlgebra</a> of  <a href="msualg_2.html#K12" title="MSUALG_2:func.12">GenMSAlg</a> <font color="Maroon" title="c7">I</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="extens_1.html#T17" title="EXTENS_1:th.17">EXTENS_1:17</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">I1</font> =  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_3.html#K6" title="MSUALG_3:func.6">Image</a> <font color="Maroon" title="c6">f</font></span>)</span> as  <a href="relat_1.html#V2" title="RELAT_1:attr.2">V2</a>() <a href="msualg_2.html#NM1" title="MSUALG_2:NM.1">MSSubset</a> of <span class="p1">(<span class="default"><a href="msualg_3.html#K6" title="MSUALG_3:func.6">Image</a> <font color="Maroon" title="c6">f</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="pboole.html#D18" title="PBOOLE:def.18">PBOOLE:def 18</a></span>;<br/>
<a NAME="E10:30"/>
( <font color="Maroon" title="c7">I</font> is    <a href="msafree.html#M1" title="MSAFREE:mode.1">GeneratorSet</a> of  <a href="msualg_3.html#K6" title="MSUALG_3:func.6">Image</a> <font color="Maroon" title="c6">f</font> &amp;  <a href="msualg_2.html#K12" title="MSUALG_2:func.12">GenMSAlg</a> <font color="Maroon" title="c7">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_2.html#K12" title="MSUALG_2:func.12">GenMSAlg</a> <font color="Maroon" title="c8">I1</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="extens_1.html#T18" title="EXTENS_1:th.18">EXTENS_1:18</a>, <a class="ref" href="msafree2.html#T6" title="MSAFREE2:th.6">MSAFREE2:6</a></span>;<br/>
<a NAME="E11:30"/><span class="kw">then </span>
 <a href="msualg_2.html#K12" title="MSUALG_2:func.12">GenMSAlg</a> <font color="Maroon" title="c7">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_3.html#K6" title="MSUALG_3:func.6">Image</a> <font color="Maroon" title="c6">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="msafree.html#T3" title="MSAFREE:th.3">MSAFREE:3</a></span>;<br/>
<a NAME="E12:30"/><span class="kw">then </span>
<font color="Maroon" title="c3">B</font> is    <a href="msualg_2.html#M1" title="MSUALG_2:mode.1">MSSubAlgebra</a> of  <a href="msualg_3.html#K6" title="MSUALG_3:func.6">Image</a> <font color="Maroon" title="c6">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:30"><span class="lab"><font color="Green" title="E17">A3</font></span></a>, <a class="ref" href="msafree.html#T3" title="MSAFREE:th.3">MSAFREE:3</a></span>;<br/>
<a NAME="E13:30"/><span class="kw">then </span>
 <a href="msualg_3.html#K6" title="MSUALG_3:func.6">Image</a> <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_2.html#T7" title="MSUALG_2:th.7">MSUALG_2:7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E14:30"/>
<font color="Maroon" title="c6">f</font> <a href="msualg_3.html#R2" title="MSUALG_3:pred.2">is_epimorphism</a> <font color="Maroon" title="c2">U0</font>,<font color="Maroon" title="c3">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:30"><span class="lab"><font color="Green" title="E16">A2</font></span></a>, <a class="ref" href="msualg_3.html#T19" title="MSUALG_3:th.19">MSUALG_3:19</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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