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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>set </b><font color="Maroon" title="c4">qa</font> =  <a href="alg_1.html#K5" title="ALG_1:func.5">QuotUnivAlg</a> <font color="Maroon" title="c1">U1</font>,<span class="p1">(<span class="default"><a href="alg_1.html#K7" title="ALG_1:func.7">Cng</a> <font color="Maroon" title="c3">f</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c5">u1</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">U1</font>;<br/>
<b>set </b><font color="Maroon" title="c6">u2</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">U2</font>;<br/>
<b>set </b><font color="Maroon" title="c7">F</font> =  <a href="alg_1.html#K8" title="ALG_1:func.8">HomQuot</a> <font color="Maroon" title="c3">f</font>;<br/>
<b>assume </b><a NAME="E1:39"/><i><font color="Green" title="E32">A1</font></i>: 
<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>
 ;<br/>

<a NAME="E2:39"/><b>then </b><i><font color="Green" title="E33">A2</font></i>: 
<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>
 
<b>by </b><i><a class="ref" href="alg_1.html#D3" target="_self" title="ALG_1:def.3">Def3</a></i>;<br/>
<a NAME="E3:39"/><b>then </b><i><font color="Green" title="E34">A3</font></i>: 
(  <a href="alg_1.html#K8" title="ALG_1:func.8">HomQuot</a> <font color="Maroon" title="c3">f</font> <a href="alg_1.html#R1" title="ALG_1:pred.1">is_homomorphism</a>  <a href="alg_1.html#K5" title="ALG_1:func.5">QuotUnivAlg</a> <font color="Maroon" title="c1">U1</font>,<span class="p1">(<span class="default"><a href="alg_1.html#K7" title="ALG_1:func.7">Cng</a> <font color="Maroon" title="c3">f</font></span>)</span>,<font color="Maroon" title="c2">U2</font> &amp;  <a href="alg_1.html#K8" title="ALG_1:func.8">HomQuot</a> <font color="Maroon" title="c3">f</font> <a href="alg_1.html#R2" title="ALG_1:pred.2">is_monomorphism</a>  <a href="alg_1.html#K5" title="ALG_1:func.5">QuotUnivAlg</a> <font color="Maroon" title="c1">U1</font>,<span class="p1">(<span class="default"><a href="alg_1.html#K7" title="ALG_1:func.7">Cng</a> <font color="Maroon" title="c3">f</font></span>)</span>,<font color="Maroon" title="c2">U2</font> )
 
<b>by </b><i><a class="ref" href="alg_1.html#T23" target="_self" title="ALG_1:th.23">Th23</a></i>;<br/>
<a NAME="E4:39"/><b>then </b><i><font color="Green" title="E35">A4</font></i>: 
 <a href="alg_1.html#K8" title="ALG_1:func.8">HomQuot</a> <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></i>;<br/>
<a NAME="E5:39"/><i><font color="Green" title="E36">A5</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</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="c1">U1</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> )
 
<b>by </b><i><a class="txt" href="alg_1.html#E1:39"><i><font color="Green" title="E32">A1</font></i></a>, <a class="ref" href="alg_1.html#D3" target="_self" title="ALG_1:def.3">Def3</a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E6:39"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="alg_1.html#K8" title="ALG_1:func.8">HomQuot</a> <font color="Maroon" title="c3">f</font></span>)</span> <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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:39_1"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="alg_1.html#K8" title="ALG_1:func.8">HomQuot</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">U2</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><br/>

<b>let </b><font color="Maroon" title="c8">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:39_1"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">U2</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c9">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:39_1"/><i><font color="Green" title="E37">A6</font></i>: 
( <font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font> &amp; <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">x</font> )
 <b>by </b><i><a class="txt" href="alg_1.html#E5:39"><i><font color="Green" title="E36">A5</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">y</font> = <font color="Maroon" title="c9">y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">U1</font> <b>by </b><i><a class="txt" href="alg_1.html#E4:39_1"><i><font color="Green" title="E37">A6</font></i></a></i>;<br/>
<a NAME="E6:39_1"/><i><font color="Green" title="E38">A7</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="alg_1.html#K8" title="ALG_1:func.8">HomQuot</a> <font color="Maroon" title="c3">f</font></span>)</span> <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 <span class="p1">(<span class="default"><a href="alg_1.html#K5" title="ALG_1:func.5">QuotUnivAlg</a> <font color="Maroon" title="c1">U1</font>,<span class="p2">(<span class="default"><a href="alg_1.html#K7" title="ALG_1:func.7">Cng</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>set </b><font color="Maroon" title="c11">u</font> =  <a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> <span class="p1">(<span class="default"><a href="alg_1.html#K7" title="ALG_1:func.7">Cng</a> <font color="Maroon" title="c3">f</font></span>)</span>,<font color="Maroon" title="c10">y</font>;<br/>
<a NAME="E7:39_1"/>
 <a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> <span class="p1">(<span class="default"><a href="alg_1.html#K7" title="ALG_1:func.7">Cng</a> <font color="Maroon" title="c3">f</font></span>)</span>,<font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <span class="p1">(<span class="default"><a href="alg_1.html#K7" title="ALG_1:func.7">Cng</a> <font color="Maroon" title="c3">f</font></span>)</span>
 
<b>by </b><i><a class="ref" href="eqrel_1.html#D5" title="EQREL_1:def.5">EQREL_1:def 5</a></i>;<br/>
<a NAME="E8:39_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="alg_1.html#K8" title="ALG_1:func.8">HomQuot</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> <span class="p2">(<span class="default"><a href="alg_1.html#K7" title="ALG_1:func.7">Cng</a> <font color="Maroon" title="c3">f</font></span>)</span>,<font color="Maroon" title="c10">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">x</font> &amp; <span class="p1">(<span class="default"><a href="alg_1.html#K8" title="ALG_1:func.8">HomQuot</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> <span class="p2">(<span class="default"><a href="alg_1.html#K7" title="ALG_1:func.7">Cng</a> <font color="Maroon" title="c3">f</font></span>)</span>,<font color="Maroon" title="c10">y</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="alg_1.html#K8" title="ALG_1:func.8">HomQuot</a> <font color="Maroon" title="c3">f</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="alg_1.html#E2:39"><i><font color="Green" title="E33">A2</font></i></a>, <a class="txt" href="alg_1.html#E4:39_1"><i><font color="Green" title="E37">A6</font></i></a>, <a class="txt" href="alg_1.html#E6:39_1"><i><font color="Green" title="E38">A7</font></i></a>, <a class="ref" href="alg_1.html#D17" target="_self" title="ALG_1:def.17">Def17</a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E9:39_1"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="alg_1.html#K8" title="ALG_1:func.8">HomQuot</a> <font color="Maroon" title="c3">f</font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:39"/>
 <a href="alg_1.html#K8" title="ALG_1:func.8">HomQuot</a> <font color="Maroon" title="c3">f</font> <a href="alg_1.html#R4" title="ALG_1:pred.4">is_isomorphism</a>  <a href="alg_1.html#K5" title="ALG_1:func.5">QuotUnivAlg</a> <font color="Maroon" title="c1">U1</font>,<span class="p1">(<span class="default"><a href="alg_1.html#K7" title="ALG_1:func.7">Cng</a> <font color="Maroon" title="c3">f</font></span>)</span>,<font color="Maroon" title="c2">U2</font>
 <b>by </b><i><a class="txt" href="alg_1.html#E3:39"><i><font color="Green" title="E34">A3</font></i></a>, <a class="txt" href="alg_1.html#E4:39"><i><font color="Green" title="E35">A4</font></i></a>, <a class="ref" href="alg_1.html#T8" target="_self" title="ALG_1:th.8">Th8</a></i>;<br/>


</div>
