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<b>consider </b><font color="Maroon" title="c2">U1</font> being   <a href="unialg_1.html#NM2" title="UNIALG_1:NM.2">Universal_Algebra</a>;<br/>
<b>set </b><font color="Maroon" title="c3">f</font> = <font color="Maroon" title="c1">J</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c2">U1</font>;<br/>
<a NAME="E2:28_1_1"/><i><font color="Green" title="E17">A1</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">J</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c2">U1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">J</font>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c4">f</font> = <font color="Maroon" title="c1">J</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c2">U1</font> as    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">J</font> <b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c4">f</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> is <a href="pralg_1.html#V3" title="PRALG_1:attr.3">equal-signature</a> &amp; <font color="Maroon" title="c4">f</font> is <a href="pralg_1.html#V1" title="PRALG_1:attr.1">Univ_Alg-yielding</a> )</span><br/>

<a NAME="E4:28_1_1"/>
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">f</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">f</font> holds <br/>for <font color="Olive" title="b3">U1</font>, <font color="Olive" title="b4">U2</font> being  <a href="unialg_1.html#NM2" title="UNIALG_1:NM.2">Universal_Algebra</a>  st <font color="Olive" title="b3">U1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b4">U2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">y</font> holds <br/> <a href="unialg_1.html#K4" title="UNIALG_1:func.4">signature</a> <font color="Olive" title="b3">U1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="unialg_1.html#K4" title="UNIALG_1:func.4">signature</a> <font color="Olive" title="b4">U2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c6">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">f</font> &amp; <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">f</font> implies for <font color="Olive" title="b1">U1</font>, <font color="Olive" title="b2">U2</font> being  <a href="unialg_1.html#NM2" title="UNIALG_1:NM.2">Universal_Algebra</a>  st <font color="Olive" title="b1">U1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> &amp; <font color="Olive" title="b2">U2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">y</font> holds <br/> <a href="unialg_1.html#K4" title="UNIALG_1:func.4">signature</a> <font color="Olive" title="b1">U1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="unialg_1.html#K4" title="UNIALG_1:func.4">signature</a> <font color="Olive" title="b2">U2</font> )</span><br/>



<b>assume </b><a NAME="E1:28_1_1_1"/>
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">f</font> &amp; <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">f</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">U1</font>, <font color="Olive" title="b2">U2</font> being  <a href="unialg_1.html#NM2" title="UNIALG_1:NM.2">Universal_Algebra</a>  st <font color="Olive" title="b1">U1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> &amp; <font color="Olive" title="b2">U2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">y</font> holds <br/> <a href="unialg_1.html#K4" title="UNIALG_1:func.4">signature</a> <font color="Olive" title="b1">U1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="unialg_1.html#K4" title="UNIALG_1:func.4">signature</a> <font color="Olive" title="b2">U2</font></span><br/>

<a NAME="E2:28_1_1_1"/><b>then </b><i><font color="Green" title="E18">A2</font></i>: 
( <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">U1</font> &amp; <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">U1</font> )
 
<b>by </b><i><a class="txt" href="#E2:28_1_1"><i><font color="Green" title="E17">A1</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/>
<b>let </b><font color="Maroon" title="c7">U2</font>, <font color="Maroon" title="c8">U3</font> be   <a href="unialg_1.html#NM2" title="UNIALG_1:NM.2">Universal_Algebra</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c7">U2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> &amp; <font color="Maroon" title="c8">U3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">y</font> implies  <a href="unialg_1.html#K4" title="UNIALG_1:func.4">signature</a> <font color="Maroon" title="c7">U2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="unialg_1.html#K4" title="UNIALG_1:func.4">signature</a> <font color="Maroon" title="c8">U3</font> )</span><br/>



<b>assume </b><a NAME="E3:28_1_1_1"/>
( <font color="Maroon" title="c7">U2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> &amp; <font color="Maroon" title="c8">U3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">y</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="unialg_1.html#K4" title="UNIALG_1:func.4">signature</a> <font color="Maroon" title="c7">U2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="unialg_1.html#K4" title="UNIALG_1:func.4">signature</a> <font color="Maroon" title="c8">U3</font></span><br/>

<b>hence </b><a NAME="E4:28_1_1_1"/>
 <a href="unialg_1.html#K4" title="UNIALG_1:func.4">signature</a> <font color="Maroon" title="c7">U2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="unialg_1.html#K4" title="UNIALG_1:func.4">signature</a> <font color="Maroon" title="c8">U3</font>
 <b>by </b><i><a class="txt" href="#E2:28_1_1_1"><i><font color="Green" title="E18">A2</font></i></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>hence </b><a NAME="E5:28_1_1"/>
<font color="Maroon" title="c4">f</font> is <a href="pralg_1.html#V3" title="PRALG_1:attr.3">equal-signature</a>
 <b>by </b><i><a class="ref" href="pralg_1.html#D12" target="_self" title="PRALG_1:def.12">Def12</a></i>; <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> is <a href="pralg_1.html#V1" title="PRALG_1:attr.1">Univ_Alg-yielding</a></span><br/>

<a NAME="E6:28_1_1"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">f</font> holds <br/><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> is   <a href="unialg_1.html#NM2" title="UNIALG_1:NM.2">Universal_Algebra</a>
 
<b>by </b><i><a class="txt" href="#E2:28_1_1"><i><font color="Green" title="E17">A1</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/>
<b>hence </b><a NAME="E7:28_1_1"/>
<font color="Maroon" title="c4">f</font> is <a href="pralg_1.html#V1" title="PRALG_1:attr.1">Univ_Alg-yielding</a>
 <b>by </b><i><a class="ref" href="pralg_1.html#D10" target="_self" title="PRALG_1:def.10">Def10</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>
