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<b>let </b><font color="Maroon" title="c4">O1</font>, <font color="Maroon" title="c5">O2</font> be  <a href="pralg_1.html#V4" title="PRALG_1:attr.4">equal-arity</a>  <a href="pralg_1.html#M1" title="PRALG_1:mode.1">ManySortedOperation</a> of  <a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</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">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">J</font><br/> for <font color="Olive" title="b2">o</font> being  <a href="unialg_2.html#NM5" title="UNIALG_2:NM.5">operation</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="pralg_1.html#K11" title="PRALG_1:func.11">.</a> <font color="Olive" title="b1">j</font></span>)</span>  st the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="pralg_1.html#K11" title="PRALG_1:func.11">.</a> <font color="Olive" title="b1">j</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">o</font> holds <br/><font color="Maroon" title="c4">O1</font> <a href="pralg_1.html#K14" title="PRALG_1:func.14">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">o</font> ) &amp; ( for <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">J</font><br/> for <font color="Olive" title="b2">o</font> being  <a href="unialg_2.html#NM5" title="UNIALG_2:NM.5">operation</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="pralg_1.html#K11" title="PRALG_1:func.11">.</a> <font color="Olive" title="b1">j</font></span>)</span>  st the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="pralg_1.html#K11" title="PRALG_1:func.11">.</a> <font color="Olive" title="b1">j</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">o</font> holds <br/><font color="Maroon" title="c5">O2</font> <a href="pralg_1.html#K14" title="PRALG_1:func.14">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">o</font> ) implies <font color="Maroon" title="c4">O1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">O2</font> )</span><br/>

<b>assume </b><a NAME="E1:48_1_2"/><i><font color="Green" title="E27">A10</font></i>: 
( ( for <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">J</font><br/> for <font color="Olive" title="b2">o</font> being  <a href="unialg_2.html#NM5" title="UNIALG_2:NM.5">operation</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="pralg_1.html#K11" title="PRALG_1:func.11">.</a> <font color="Olive" title="b1">j</font></span>)</span>  st the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="pralg_1.html#K11" title="PRALG_1:func.11">.</a> <font color="Olive" title="b1">j</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">o</font> holds <br/><font color="Maroon" title="c4">O1</font> <a href="pralg_1.html#K14" title="PRALG_1:func.14">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">o</font> ) &amp; ( for <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">J</font><br/> for <font color="Olive" title="b2">o</font> being  <a href="unialg_2.html#NM5" title="UNIALG_2:NM.5">operation</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="pralg_1.html#K11" title="PRALG_1:func.11">.</a> <font color="Olive" title="b1">j</font></span>)</span>  st the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="pralg_1.html#K11" title="PRALG_1:func.11">.</a> <font color="Olive" title="b1">j</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">o</font> holds <br/><font color="Maroon" title="c5">O2</font> <a href="pralg_1.html#K14" title="PRALG_1:func.14">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">o</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">O1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">O2</font></span><br/>

<a NAME="E2:48_1_2"/>
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> <font color="Maroon" title="c1">J</font> holds <br/><font color="Maroon" title="c4">O1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">O2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">x</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="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">J</font> implies <font color="Maroon" title="c4">O1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">O2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> )</span><br/>

<b>assume </b><a NAME="E1:48_1_2_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">J</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">O1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">O2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font></span><br/>

<b>then reconsider </b><font color="Maroon" title="c7">x1</font> = <font color="Maroon" title="c6">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">J</font> ;<br/>
<a NAME="E3:48_1_2_1"/>
<font color="Maroon" title="c3">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="unialg_1.html#K4" title="UNIALG_1:func.4">signature</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="pralg_1.html#K11" title="PRALG_1:func.11">.</a> <font color="Maroon" title="c7">x1</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E1:48"><i><font color="Green" title="E26">A1</font></i></a>, <a class="ref" href="pralg_1.html#D14" target="_self" title="PRALG_1:def.14">Def14</a></i>;<br/>
<a NAME="E4:48_1_2_1"/><b>then </b>
<font color="Maroon" title="c3">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="unialg_1.html#K4" title="UNIALG_1:func.4">signature</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="pralg_1.html#K11" title="PRALG_1:func.11">.</a> <font color="Maroon" title="c7">x1</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E5:48_1_2_1"/><b>then </b>
<font color="Maroon" title="c3">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="pralg_1.html#K11" title="PRALG_1:func.11">.</a> <font color="Maroon" title="c7">x1</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="unialg_1.html#D11" title="UNIALG_1:def.11">UNIALG_1:def 11</a></i>;<br/>
<a NAME="E6:48_1_2_1"/><b>then </b>
<font color="Maroon" title="c3">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="pralg_1.html#K11" title="PRALG_1:func.11">.</a> <font color="Maroon" title="c7">x1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c8">o</font> = the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="pralg_1.html#K11" title="PRALG_1:func.11">.</a> <font color="Maroon" title="c7">x1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">n</font> as   <a href="unialg_2.html#NM5" title="UNIALG_2:NM.5">operation</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="pralg_1.html#K11" title="PRALG_1:func.11">.</a> <font color="Maroon" title="c7">x1</font></span>)</span> <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E8:48_1_2_1"/>
( <font color="Maroon" title="c4">O1</font> <a href="pralg_1.html#K14" title="PRALG_1:func.14">.</a> <font color="Maroon" title="c7">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">o</font> &amp; <font color="Maroon" title="c5">O2</font> <a href="pralg_1.html#K14" title="PRALG_1:func.14">.</a> <font color="Maroon" title="c7">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">o</font> )
 
<b>by </b><i><a class="txt" href="#E1:48_1_2"><i><font color="Green" title="E27">A10</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:48_1_2_1"/>
<font color="Maroon" title="c4">O1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">O2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font>
 ; <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="E3:48_1_2"/>
<font color="Maroon" title="c4">O1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">O2</font>
 <b>by </b><i><a class="ref" href="pboole.html#T3" title="PBOOLE:th.3">PBOOLE:3</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>
