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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">h</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">f</font> =  <a href="msuhom_1.html#K3" title="MSUHOM_1:func.3">MSAlg</a> <font color="Maroon" title="c3">h</font>;<br/>
<b>set </b><font color="Maroon" title="c5">A</font> = <span class="p1">(<span class="default"><a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <font color="Maroon" title="c2">U2</font></span>)</span> <a href="msuhom_1.html#K2" title="MSUHOM_1:func.2">Over</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K7" title="MSUALG_1:func.7">MSSign</a> <font color="Maroon" title="c1">U1</font></span>)</span>;<br/>
<b>assume </b><a NAME="E1:25"/><i><font color="Green" title="E21">A1</font></i>: 
<font color="Maroon" title="c3">h</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="E2:25"/><i><font color="Green" title="E22">A2</font></i>: 
0 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E3:25"/><i><font color="Green" title="E23">A3</font></i>: 
<font color="Maroon" title="c3">h</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="txt" href="msuhom_1.html#E1:25"><i><font color="Green" title="E21">A1</font></i></a>, <a class="ref" href="alg_1.html#D3" title="ALG_1:def.3">ALG_1:def 3</a></i>;<br/>
<a NAME="E4:25"/><b>then </b>
<font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c2">U2</font> <a href="unialg_2.html#R1" title="UNIALG_2:pred.1">are_similar</a> 
 
<b>by </b><i><a class="ref" href="alg_1.html#D1" title="ALG_1:def.1">ALG_1:def 1</a></i>;<br/>
<a NAME="E5:25"/><b>then </b><i><font color="Green" title="E24">A4</font></i>: 
 <a href="msualg_1.html#K7" title="MSUALG_1:func.7">MSSign</a> <font color="Maroon" title="c1">U1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_1.html#K7" title="MSUALG_1:func.7">MSSign</a> <font color="Maroon" title="c2">U2</font>
 
<b>by </b><i><a class="ref" href="msuhom_1.html#T10" target="_self" title="MSUHOM_1:th.10">Th10</a></i>;<br/>
<b>thus </b><a NAME="E6:25"/>
 <a href="msuhom_1.html#K3" title="MSUHOM_1:func.3">MSAlg</a> <font color="Maroon" title="c3">h</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a>  <a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <font color="Maroon" title="c1">U1</font>,<span class="p1">(<span class="default"><a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <font color="Maroon" title="c2">U2</font></span>)</span> <a href="msuhom_1.html#K2" title="MSUHOM_1:func.2">Over</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K7" title="MSUALG_1:func.7">MSSign</a> <font color="Maroon" title="c1">U1</font></span>)</span>
 <b>by </b><i><a class="txt" href="msuhom_1.html#E3:25"><i><font color="Green" title="E23">A3</font></i></a>, <a class="ref" href="msuhom_1.html#T16" target="_self" title="MSUHOM_1:th.16">Th16</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="msualg_3.html#D10" title="MSUALG_3:def.10">MSUALG_3:def 10</a><br/>

<b>let </b><font color="Maroon" title="c6">i</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="msualg_3.html#D3" title="MSUALG_3:def.3">MSUALG_3:def 3</a><br/>

<b>assume </b><a NAME="E7:25"/>
<font color="Maroon" title="c6">i</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 <span class="p1">(<span class="default"><a href="msualg_1.html#K7" title="MSUALG_1:func.7">MSSign</a> <font color="Maroon" title="c1">U1</font></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c7">i'</font> = <font color="Maroon" title="c6">i</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="msualg_1.html#K7" title="MSUALG_1:func.7">MSSign</a> <font color="Maroon" title="c1">U1</font></span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon" title="c8">h'</font> = <span class="p1">(<span class="default"><a href="msuhom_1.html#K3" title="MSUHOM_1:func.3">MSAlg</a> <font color="Maroon" title="c3">h</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <font color="Maroon" title="c1">U1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i'</font>,the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <font color="Maroon" title="c2">U2</font></span>)</span> <a href="msuhom_1.html#K2" title="MSUHOM_1:func.2">Over</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K7" title="MSUALG_1:func.7">MSSign</a> <font color="Maroon" title="c1">U1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i'</font> <b>by </b><i><a class="ref" href="pboole.html#D18" title="PBOOLE:def.18">PBOOLE:def 18</a></i>;<br/>
<a NAME="E10:25"/><i><font color="Green" title="E25">A5</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">h</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="msuhom_1.html#E1:25"><i><font color="Green" title="E21">A1</font></i></a>, <a class="ref" href="alg_1.html#D3" title="ALG_1:def.3">ALG_1:def 3</a></i>;<br/>
<a NAME="E11:25"/>
 <a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <font color="Maroon" title="c2">U2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_1.html#G3" title="MSUALG_1:aggr.3">MSAlgebra</a>(# <span class="p1">(<span class="default"><a href="msualg_1.html#K8" title="MSUALG_1:func.8">MSSorts</a> <font color="Maroon" title="c2">U2</font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_1.html#K9" title="MSUALG_1:func.9">MSCharact</a> <font color="Maroon" title="c2">U2</font></span>)</span> #)
 
<b>by </b><i><a class="ref" href="msualg_1.html#D16" title="MSUALG_1:def.16">MSUALG_1:def 16</a></i>;<br/>
<a NAME="E12:25"/><b>then </b><i><font color="Green" title="E26">A6</font></i>: 
the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <font color="Maroon" title="c2">U2</font></span>)</span> <a href="msuhom_1.html#K2" title="MSUHOM_1:func.2">Over</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K7" title="MSUALG_1:func.7">MSSign</a> <font color="Maroon" title="c1">U1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_1.html#K8" title="MSUALG_1:func.8">MSSorts</a> <font color="Maroon" title="c2">U2</font>
 
<b>by </b><i><a class="txt" href="msuhom_1.html#E5:25"><i><font color="Green" title="E24">A4</font></i></a>, <a class="ref" href="msuhom_1.html#T9" target="_self" title="MSUHOM_1:th.9">Th9</a></i>;<br/>
<a NAME="E13:25"/><i><font color="Green" title="E27">A7</font></i>: 
 <a href="msualg_1.html#K8" title="MSUALG_1:func.8">MSSorts</a> <font color="Maroon" title="c2">U2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</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="ref" href="msualg_1.html#D14" title="MSUALG_1:def.14">MSUALG_1:def 14</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">g</font> = <span class="p1">(<span class="default"><a href="pboole.html#K8" title="PBOOLE:func.8">*--&gt;</a> 0</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K6" title="MSUALG_1:func.6">signature</a> <font color="Maroon" title="c1">U1</font></span>)</span> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K6" title="MSUALG_1:func.6">signature</a> <font color="Maroon" title="c1">U1</font></span>)</span>,<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  <b>by </b><i><a class="ref" href="msualg_1.html#T7" title="MSUALG_1:th.7">MSUALG_1:7</a></i>;<br/>
<a NAME="E15:25"/><i><font color="Green" title="E28">A8</font></i>: 
( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="msualg_1.html#K7" title="MSUALG_1:func.7">MSSign</a> <font color="Maroon" title="c1">U1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <span class="p1">(<span class="default"><a href="msualg_1.html#K7" title="MSUALG_1:func.7">MSSign</a> <font color="Maroon" title="c1">U1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K6" title="MSUALG_1:func.6">signature</a> <font color="Maroon" title="c1">U1</font></span>)</span> &amp; the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">Arity</a> of <span class="p1">(<span class="default"><a href="msualg_1.html#K7" title="MSUALG_1:func.7">MSSign</a> <font color="Maroon" title="c1">U1</font></span>)</span> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c9">g</font> &amp; the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">ResultSort</a> of <span class="p1">(<span class="default"><a href="msualg_1.html#K7" title="MSUALG_1:func.7">MSSign</a> <font color="Maroon" title="c1">U1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K6" title="MSUALG_1:func.6">signature</a> <font color="Maroon" title="c1">U1</font></span>)</span></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> 0 )
 
<b>by </b><i><a class="ref" href="msualg_1.html#D13" title="MSUALG_1:def.13">MSUALG_1:def 13</a></i>;<br/>
<a NAME="E16:25"/><i><font color="Green" title="E29">A9</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="msualg_1.html#K7" title="MSUALG_1:func.7">MSSign</a> <font color="Maroon" title="c1">U1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="msualg_1.html#D13" title="MSUALG_1:def.13">MSUALG_1:def 13</a></i>;<br/>
<a NAME="E17:25"/><i><font color="Green" title="E30">A10</font></i>: 
<span class="p1">(<span class="default"><a href="msualg_1.html#K8" title="MSUALG_1:func.8">MSSorts</a> <font color="Maroon" title="c2">U2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 0 <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="msuhom_1.html#E2:25"><i><font color="Green" title="E22">A2</font></i></a>, <a class="txt" href="msuhom_1.html#E13:25"><i><font color="Green" title="E27">A7</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/>
<span class="p1">(<span class="default"><a href="msuhom_1.html#K3" title="MSUHOM_1:func.3">MSAlg</a> <font color="Maroon" title="c3">h</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 0 = 
<span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <font color="Maroon" title="c3">h</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 0
<b>by </b><i><a class="txt" href="msuhom_1.html#E5:25"><i><font color="Green" title="E24">A4</font></i></a>, <a class="ref" href="msuhom_1.html#D3" target="_self" title="MSUHOM_1:def.3">Def3</a></i>
<br/>.= 
<font color="Maroon" title="c3">h</font>
<b>by </b><i><a class="txt" href="msuhom_1.html#E2:25"><i><font color="Green" title="E22">A2</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
;<br/>
<a NAME="E19:25"/><b>then </b>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c8">h'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <font color="Maroon" title="c2">U2</font></span>)</span> <a href="msuhom_1.html#K2" title="MSUHOM_1:func.2">Over</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K7" title="MSUALG_1:func.7">MSSign</a> <font color="Maroon" title="c1">U1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 0
 
<b>by </b><i><a class="txt" href="msuhom_1.html#E10:25"><i><font color="Green" title="E25">A5</font></i></a>, <a class="txt" href="msuhom_1.html#E12:25"><i><font color="Green" title="E26">A6</font></i></a>, <a class="txt" href="msuhom_1.html#E15:25"><i><font color="Green" title="E28">A8</font></i></a>, <a class="txt" href="msuhom_1.html#E17:25"><i><font color="Green" title="E30">A10</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E20:25"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msuhom_1.html#K3" title="MSUHOM_1:func.3">MSAlg</a> <font color="Maroon" title="c3">h</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <font color="Maroon" title="c2">U2</font></span>)</span> <a href="msuhom_1.html#K2" title="MSUHOM_1:func.2">Over</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K7" title="MSUALG_1:func.7">MSSign</a> <font color="Maroon" title="c1">U1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font>
 <b>by </b><i><a class="txt" href="msuhom_1.html#E16:25"><i><font color="Green" title="E29">A9</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>


</div>
