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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>, <font color="Olive" title="b2">U2</font> being   <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">F</font> being  <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Olive" title="b1">U1</font>,<font color="Olive" title="b2">U2</font> holds <br/> ( <font color="Olive" title="b3">F</font> <a href="msualg_3.html#R4" title="MSUALG_3:pred.4">is_isomorphism</a> <font color="Olive" title="b1">U1</font>,<font color="Olive" title="b2">U2</font> iff ( <font color="Olive" title="b3">F</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Olive" title="b1">U1</font>,<font color="Olive" title="b2">U2</font> &amp; <font color="Olive" title="b3">F</font> is <a href="msualg_3.html#V2" title="MSUALG_3:attr.2">"onto"</a> &amp; <font color="Olive" title="b3">F</font> is <a href="msualg_3.html#V1" title="MSUALG_3:attr.1">"1-1"</a> ) )</span><br/><b>let </b><font color="Maroon" title="c2">U1</font>, <font color="Maroon" title="c3">U2</font> be    <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"> 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">U1</font>,<font color="Maroon" title="c3">U2</font> holds <br/> ( <font color="Olive" title="b1">F</font> <a href="msualg_3.html#R4" title="MSUALG_3:pred.4">is_isomorphism</a> <font color="Maroon" title="c2">U1</font>,<font color="Maroon" title="c3">U2</font> iff ( <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">U1</font>,<font color="Maroon" title="c3">U2</font> &amp; <font color="Olive" title="b1">F</font> is <a href="msualg_3.html#V2" title="MSUALG_3:attr.2">"onto"</a> &amp; <font color="Olive" title="b1">F</font> is <a href="msualg_3.html#V1" title="MSUALG_3:attr.1">"1-1"</a> ) )</span><br/><b>let </b><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">U1</font>,<font color="Maroon" title="c3">U2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">F</font> <a href="msualg_3.html#R4" title="MSUALG_3:pred.4">is_isomorphism</a> <font color="Maroon" title="c2">U1</font>,<font color="Maroon" title="c3">U2</font> iff ( <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">U1</font>,<font color="Maroon" title="c3">U2</font> &amp; <font color="Maroon" title="c4">F</font> is <a href="msualg_3.html#V2" title="MSUALG_3:attr.2">"onto"</a> &amp; <font color="Maroon" title="c4">F</font> is <a href="msualg_3.html#V1" title="MSUALG_3:attr.1">"1-1"</a> ) )</span><br/>







<b>thus </b><a NAME="E1:29"/>
( <font color="Maroon" title="c4">F</font> <a href="msualg_3.html#R4" title="MSUALG_3:pred.4">is_isomorphism</a> <font color="Maroon" title="c2">U1</font>,<font color="Maroon" title="c3">U2</font> implies ( <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">U1</font>,<font color="Maroon" title="c3">U2</font> &amp; <font color="Maroon" title="c4">F</font> is <a href="msualg_3.html#V2" title="MSUALG_3:attr.2">"onto"</a> &amp; <font color="Maroon" title="c4">F</font> is <a href="msualg_3.html#V1" title="MSUALG_3:attr.1">"1-1"</a> ) )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <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">U1</font>,<font color="Maroon" title="c3">U2</font> &amp; <font color="Maroon" title="c4">F</font> is <a href="msualg_3.html#V2" title="MSUALG_3:attr.2">"onto"</a> &amp; <font color="Maroon" title="c4">F</font> is <a href="msualg_3.html#V1" title="MSUALG_3:attr.1">"1-1"</a> implies <font color="Maroon" title="c4">F</font> <a href="msualg_3.html#R4" title="MSUALG_3:pred.4">is_isomorphism</a> <font color="Maroon" title="c2">U1</font>,<font color="Maroon" title="c3">U2</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:29_1"/>
<font color="Maroon" title="c4">F</font> <a href="msualg_3.html#R4" title="MSUALG_3:pred.4">is_isomorphism</a> <font color="Maroon" title="c2">U1</font>,<font color="Maroon" title="c3">U2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <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">U1</font>,<font color="Maroon" title="c3">U2</font> &amp; <font color="Maroon" title="c4">F</font> is <a href="msualg_3.html#V2" title="MSUALG_3:attr.2">"onto"</a> &amp; <font color="Maroon" title="c4">F</font> is <a href="msualg_3.html#V1" title="MSUALG_3:attr.1">"1-1"</a> )</span><br/>

<a NAME="E2:29_1"/><b>then </b>
( <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="c2">U1</font>,<font color="Maroon" title="c3">U2</font> &amp; <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">U1</font>,<font color="Maroon" title="c3">U2</font> )
 
<b>by </b><i><a class="ref" href="msualg_3.html#D12" target="_self" title="MSUALG_3:def.12">Def12</a></i>;<br/>
<b>hence </b><a NAME="E3:29_1"/>
( <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">U1</font>,<font color="Maroon" title="c3">U2</font> &amp; <font color="Maroon" title="c4">F</font> is <a href="msualg_3.html#V2" title="MSUALG_3:attr.2">"onto"</a> &amp; <font color="Maroon" title="c4">F</font> is <a href="msualg_3.html#V1" title="MSUALG_3:attr.1">"1-1"</a> )
 <b>by </b><i><a class="ref" href="msualg_3.html#D10" target="_self" title="MSUALG_3:def.10">Def10</a>, <a class="ref" href="msualg_3.html#D11" target="_self" title="MSUALG_3:def.11">Def11</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:29"/><i><font color="Green" title="E24">A1</font></i>: 
( <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">U1</font>,<font color="Maroon" title="c3">U2</font> &amp; <font color="Maroon" title="c4">F</font> is <a href="msualg_3.html#V2" title="MSUALG_3:attr.2">"onto"</a> &amp; <font color="Maroon" title="c4">F</font> is <a href="msualg_3.html#V1" title="MSUALG_3:attr.1">"1-1"</a> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">F</font> <a href="msualg_3.html#R4" title="MSUALG_3:pred.4">is_isomorphism</a> <font color="Maroon" title="c2">U1</font>,<font color="Maroon" title="c3">U2</font></span><br/>

<a NAME="E3:29"/><b>then </b><i><font color="Green" title="E25">A2</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="c2">U1</font>,<font color="Maroon" title="c3">U2</font>
 
<b>by </b><i><a class="ref" href="msualg_3.html#D10" target="_self" title="MSUALG_3:def.10">Def10</a></i>;<br/>
<a NAME="E4:29"/>
<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">U1</font>,<font color="Maroon" title="c3">U2</font>
 
<b>by </b><i><a class="txt" href="#E2:29"><i><font color="Green" title="E24">A1</font></i></a>, <a class="ref" href="msualg_3.html#D11" target="_self" title="MSUALG_3:def.11">Def11</a></i>;<br/>
<b>hence </b><a NAME="E5:29"/>
<font color="Maroon" title="c4">F</font> <a href="msualg_3.html#R4" title="MSUALG_3:pred.4">is_isomorphism</a> <font color="Maroon" title="c2">U1</font>,<font color="Maroon" title="c3">U2</font>
 <b>by </b><i><a class="txt" href="#E3:29"><i><font color="Green" title="E25">A2</font></i></a>, <a class="ref" href="msualg_3.html#D12" target="_self" title="MSUALG_3:def.12">Def12</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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