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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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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> over <font color="Maroon" title="c1">S</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="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font>  st <font color="Olive" title="b3">F</font> <a href="msualg_3.html#R3" title="MSUALG_3:pred.3">is_monomorphism</a> <font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font> holds <br/><font color="Olive" title="b1">A</font>, <a href="msualg_3.html#K6" title="MSUALG_3:func.6">Image</a> <font color="Olive" title="b3">F</font> <a href="msualg_3.html#R6" title="MSUALG_3:pred.6">are_isomorphic</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">A</font>, <font color="Maroon" title="c3">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> over <font color="Maroon" title="c1">S</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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">A</font>,<font color="Maroon" title="c3">B</font>  st <font color="Olive" title="b1">F</font> <a href="msualg_3.html#R3" title="MSUALG_3:pred.3">is_monomorphism</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font> holds <br/><font color="Maroon" title="c2">A</font>, <a href="msualg_3.html#K6" title="MSUALG_3:func.6">Image</a> <font color="Olive" title="b1">F</font> <a href="msualg_3.html#R6" title="MSUALG_3:pred.6">are_isomorphic</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c4">F</font> be   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">F</font> <a href="msualg_3.html#R3" title="MSUALG_3:pred.3">is_monomorphism</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font> implies <font color="Maroon" title="c2">A</font>, <a href="msualg_3.html#K6" title="MSUALG_3:func.6">Image</a> <font color="Maroon" title="c4">F</font> <a href="msualg_3.html#R6" title="MSUALG_3:pred.6">are_isomorphic</a>  )</span><br/>





<span class="kw">assume </span><a NAME="E1:23"/><span class="lab"><font color="Green" title="E7">A1</font></span>: 
<font color="Maroon" title="c4">F</font> <a href="msualg_3.html#R3" title="MSUALG_3:pred.3">is_monomorphism</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">A</font>, <a href="msualg_3.html#K6" title="MSUALG_3:func.6">Image</a> <font color="Maroon" title="c4">F</font> <a href="msualg_3.html#R6" title="MSUALG_3:pred.6">are_isomorphic</a> </span><br/>

<a NAME="E2:23"/><span class="kw">then </span>
<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="c2">A</font>,<font color="Maroon" title="c3">B</font>
 
;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">G</font> being   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon" title="c2">A</font>,<span class="p1">(<span class="default"><a href="msualg_3.html#K6" title="MSUALG_3:func.6">Image</a> <font color="Maroon" title="c4">F</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E4:23"/><span class="lab"><font color="Green" title="E8">A2</font></span>: 
<font color="Maroon" title="c5">G</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a> <font color="Maroon" title="c4">F</font>
 <span class="kw">and </span><br/><a NAME="E5:23"/><span class="lab"><font color="Green" title="E9">A3</font></span>: 
<font color="Maroon" title="c5">G</font> <a href="msualg_3.html#R2" title="MSUALG_3:pred.2">is_epimorphism</a> <font color="Maroon" title="c2">A</font>, <a href="msualg_3.html#K6" title="MSUALG_3:func.6">Image</a> <font color="Maroon" title="c4">F</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_3.html#T21" onmouseover="rs('msualg_3/T21')" onmouseout="rh()">MSUALG_3:21</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c5">G</font>
; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="msualg_3.html#D11" onmouseover="rs('msualg_3/D11')" onmouseout="rh()">MSUALG_3:def 11</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">G</font> <a href="msualg_3.html#R4" title="MSUALG_3:pred.4">is_isomorphism</a> <font color="Maroon" title="c2">A</font>, <a href="msualg_3.html#K6" title="MSUALG_3:func.6">Image</a> <font color="Maroon" title="c4">F</font></span><br/>

<span class="kw">thus </span><a NAME="E6:23"/>
<font color="Maroon" title="c5">G</font> <a href="msualg_3.html#R2" title="MSUALG_3:pred.2">is_epimorphism</a> <font color="Maroon" title="c2">A</font>, <a href="msualg_3.html#K6" title="MSUALG_3:func.6">Image</a> <font color="Maroon" title="c4">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:23"><span class="lab"><font color="Green" title="E9">A3</font></span></a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="msualg_3.html#D10" onmouseover="rs('msualg_3/D10')" onmouseout="rh()">MSUALG_3:def 10</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">G</font> <a href="msualg_3.html#R3" title="MSUALG_3:pred.3">is_monomorphism</a> <font color="Maroon" title="c2">A</font>, <a href="msualg_3.html#K6" title="MSUALG_3:func.6">Image</a> <font color="Maroon" title="c4">F</font></span><br/>

<span class="kw">thus </span><a NAME="E7:23"/>
<font color="Maroon" title="c5">G</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon" title="c2">A</font>, <a href="msualg_3.html#K6" title="MSUALG_3:func.6">Image</a> <font color="Maroon" title="c4">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:23"><span class="lab"><font color="Green" title="E9">A3</font></span></a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="msualg_3.html#D9" onmouseover="rs('msualg_3/D9')" onmouseout="rh()">MSUALG_3:def 9</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">G</font> is  <a href="msualg_3.html#V1" title="MSUALG_3:attr.1">"1-1"</a> </span><br/>

<span class="kw">thus </span><a NAME="E8:23"/>
<font color="Maroon" title="c5">G</font> is  <a href="msualg_3.html#V1" title="MSUALG_3:attr.1">"1-1"</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="txt" href="#E4:23"><span class="lab"><font color="Green" title="E8">A2</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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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