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<b>let </b><font color="Maroon" title="c1">U1</font>, <font color="Maroon" title="c2">U2</font> be   <a href="unialg_1.html#NM2" title="UNIALG_1:NM.2">Universal_Algebra</a>;<br/><b>let </b><font color="Maroon" title="c3">f</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c2">U2</font>;<br/>





<b>thus </b><a NAME="E1:12"/>
( <font color="Maroon" title="c3">f</font> <a href="alg_1.html#R4" title="ALG_1:pred.4">is_isomorphism</a> <font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c2">U2</font> implies ( <font color="Maroon" title="c3">f</font> <a href="alg_1.html#R1" title="ALG_1:pred.1">is_homomorphism</a> <font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c2">U2</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">U2</font> &amp; <font color="Maroon" title="c3">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:12_1"/>
<font color="Maroon" title="c3">f</font> <a href="alg_1.html#R4" title="ALG_1:pred.4">is_isomorphism</a> <font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c2">U2</font>
 ;<br/>

<a NAME="E2:12_1"/><b>then </b>
( <font color="Maroon" title="c3">f</font> <a href="alg_1.html#R2" title="ALG_1:pred.2">is_monomorphism</a> <font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c2">U2</font> &amp; <font color="Maroon" title="c3">f</font> <a href="alg_1.html#R3" title="ALG_1:pred.3">is_epimorphism</a> <font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c2">U2</font> )
 
<b>by </b><i><a class="ref" href="alg_1.html#D4" target="_self" title="ALG_1:def.4">Def4</a></i>;<br/>
<b>hence </b><a NAME="E3:12_1"/>
( <font color="Maroon" title="c3">f</font> <a href="alg_1.html#R1" title="ALG_1:pred.1">is_homomorphism</a> <font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c2">U2</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">U2</font> &amp; <font color="Maroon" title="c3">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> )
 <b>by </b><i><a class="ref" href="alg_1.html#D2" target="_self" title="ALG_1:def.2">Def2</a>, <a class="ref" href="alg_1.html#D3" target="_self" title="ALG_1:def.3">Def3</a></i>;<br/>


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

<b>assume </b><a NAME="E2:12"/>
( <font color="Maroon" title="c3">f</font> <a href="alg_1.html#R1" title="ALG_1:pred.1">is_homomorphism</a> <font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c2">U2</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">U2</font> &amp; <font color="Maroon" title="c3">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> )
 ;<br/>

<a NAME="E3:12"/><b>then </b>
( <font color="Maroon" title="c3">f</font> <a href="alg_1.html#R2" title="ALG_1:pred.2">is_monomorphism</a> <font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c2">U2</font> &amp; <font color="Maroon" title="c3">f</font> <a href="alg_1.html#R3" title="ALG_1:pred.3">is_epimorphism</a> <font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c2">U2</font> )
 
<b>by </b><i><a class="ref" href="alg_1.html#D2" target="_self" title="ALG_1:def.2">Def2</a>, <a class="ref" href="alg_1.html#D3" target="_self" title="ALG_1:def.3">Def3</a></i>;<br/>
<b>hence </b><a NAME="E4:12"/>
<font color="Maroon" title="c3">f</font> <a href="alg_1.html#R4" title="ALG_1:pred.4">is_isomorphism</a> <font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c2">U2</font>
 <b>by </b><i><a class="ref" href="alg_1.html#D4" target="_self" title="ALG_1:def.4">Def4</a></i>;<br/>


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