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<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">U1</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>  ex <font color="Olive" title="b2">U0</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="msafree.html#V2" title="MSAFREE:attr.2">free</a>  <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon" title="c1">S</font> ex <font color="Olive" title="b3">F</font> being  <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Olive" title="b2">U0</font>,<font color="Olive" title="b1">U1</font> st <br/>( <font color="Olive" title="b3">F</font> <a href="msualg_3.html#R2" title="MSUALG_3:pred.2">is_epimorphism</a> <font color="Olive" title="b2">U0</font>,<font color="Olive" title="b1">U1</font> &amp;  <a href="msualg_3.html#K7" title="MSUALG_3:func.7">Image</a> <font color="Olive" title="b3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">U1</font> )</span><br/><b>let </b><font color="Maroon" title="c2">U1</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">U0</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="msafree.html#V2" title="MSAFREE:attr.2">free</a>  <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon" title="c1">S</font> ex <font color="Olive" title="b2">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="Maroon" title="c2">U1</font> st <br/>( <font color="Olive" title="b2">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="Maroon" title="c2">U1</font> &amp;  <a href="msualg_3.html#K7" title="MSUALG_3:func.7">Image</a> <font color="Olive" title="b2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">U1</font> )</span><br/>



<b>consider </b><font color="Maroon" title="c3">U0</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="msafree.html#V2" title="MSAFREE:attr.2">free</a>  <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon" title="c1">S</font>, <font color="Maroon" title="c4">F</font> being   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon" title="c3">U0</font>,<font color="Maroon" title="c2">U1</font><b> such that </b><br/><a NAME="E2:51"/><i><font color="Green" title="E32">A1</font></i>: 
<font color="Maroon" title="c4">F</font> <a href="msualg_3.html#R2" title="MSUALG_3:pred.2">is_epimorphism</a> <font color="Maroon" title="c3">U0</font>,<font color="Maroon" title="c2">U1</font>
 <b>by </b><i><a class="ref" href="msafree.html#T19" target="_self" title="MSAFREE:th.19">Th19</a></i>;<br/>
<a NAME="E3:51"/>
<font color="Maroon" title="c4">F</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon" title="c3">U0</font>,<font color="Maroon" title="c2">U1</font>
 
<b>by </b><i><a class="txt" href="#E2:51"><i><font color="Green" title="E32">A1</font></i></a>, <a class="ref" href="msualg_3.html#D10" title="MSUALG_3:def.10">MSUALG_3:def 10</a></i>;<br/>
<a NAME="E4:51"/><b>then </b>
 <a href="msualg_3.html#K7" title="MSUALG_3:func.7">Image</a> <font color="Maroon" title="c4">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">U1</font>
 
<b>by </b><i><a class="txt" href="#E2:51"><i><font color="Green" title="E32">A1</font></i></a>, <a class="ref" href="msualg_3.html#T19" title="MSUALG_3:th.19">MSUALG_3:19</a></i>;<br/>
<b>hence </b><a NAME="E5:51"/>
 ex <font color="Olive" title="b1">U0</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="msafree.html#V2" title="MSAFREE:attr.2">free</a>  <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon" title="c1">S</font> ex <font color="Olive" title="b2">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="Maroon" title="c2">U1</font> st <br/>( <font color="Olive" title="b2">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="Maroon" title="c2">U1</font> &amp;  <a href="msualg_3.html#K7" title="MSUALG_3:func.7">Image</a> <font color="Olive" title="b2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">U1</font> )
 <b>by </b><i><a class="txt" href="#E2:51"><i><font color="Green" title="E32">A1</font></i></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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