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<b>let </b><font color="Maroon" title="c1">I</font>, <font color="Maroon" title="c2">J</font>, <font color="Maroon" title="c3">K</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"> for <font color="Olive" title="b1">A</font> being   <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font><br/> for <font color="Olive" title="b2">B</font> being   <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c2">J</font><br/> for <font color="Olive" title="b3">C</font> being   <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c3">K</font>  st <font color="Olive" title="b1">A</font> <a href="altcat_2.html#R2" title="ALTCAT_2:pred.2">cc=</a> <font color="Olive" title="b2">B</font> &amp; <font color="Olive" title="b2">B</font> <a href="altcat_2.html#R2" title="ALTCAT_2:pred.2">cc=</a> <font color="Olive" title="b3">C</font> holds <br/><font color="Olive" title="b1">A</font> <a href="altcat_2.html#R2" title="ALTCAT_2:pred.2">cc=</a> <font color="Olive" title="b3">C</font></span><br/><b>let </b><font color="Maroon" title="c4">A</font> be    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</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">B</font> being   <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c2">J</font><br/> for <font color="Olive" title="b2">C</font> being   <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c3">K</font>  st <font color="Maroon" title="c4">A</font> <a href="altcat_2.html#R2" title="ALTCAT_2:pred.2">cc=</a> <font color="Olive" title="b1">B</font> &amp; <font color="Olive" title="b1">B</font> <a href="altcat_2.html#R2" title="ALTCAT_2:pred.2">cc=</a> <font color="Olive" title="b2">C</font> holds <br/><font color="Maroon" title="c4">A</font> <a href="altcat_2.html#R2" title="ALTCAT_2:pred.2">cc=</a> <font color="Olive" title="b2">C</font></span><br/><b>let </b><font color="Maroon" title="c5">B</font> be    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c2">J</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">C</font> being   <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c3">K</font>  st <font color="Maroon" title="c4">A</font> <a href="altcat_2.html#R2" title="ALTCAT_2:pred.2">cc=</a> <font color="Maroon" title="c5">B</font> &amp; <font color="Maroon" title="c5">B</font> <a href="altcat_2.html#R2" title="ALTCAT_2:pred.2">cc=</a> <font color="Olive" title="b1">C</font> holds <br/><font color="Maroon" title="c4">A</font> <a href="altcat_2.html#R2" title="ALTCAT_2:pred.2">cc=</a> <font color="Olive" title="b1">C</font></span><br/><b>let </b><font color="Maroon" title="c6">C</font> be    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c3">K</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">A</font> <a href="altcat_2.html#R2" title="ALTCAT_2:pred.2">cc=</a> <font color="Maroon" title="c5">B</font> &amp; <font color="Maroon" title="c5">B</font> <a href="altcat_2.html#R2" title="ALTCAT_2:pred.2">cc=</a> <font color="Maroon" title="c6">C</font> implies <font color="Maroon" title="c4">A</font> <a href="altcat_2.html#R2" title="ALTCAT_2:pred.2">cc=</a> <font color="Maroon" title="c6">C</font> )</span><br/>







<b>assume </b><b>that </b><br/><a NAME="E1:15"/><i><font color="Green" title="E5">A1</font></i>: 
<font color="Maroon" title="c4">A</font> <a href="altcat_2.html#R2" title="ALTCAT_2:pred.2">cc=</a> <font color="Maroon" title="c5">B</font>
 <b>and </b><br/><a NAME="E2:15"/><i><font color="Green" title="E6">A2</font></i>: 
<font color="Maroon" title="c5">B</font> <a href="altcat_2.html#R2" title="ALTCAT_2:pred.2">cc=</a> <font color="Maroon" title="c6">C</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">A</font> <a href="altcat_2.html#R2" title="ALTCAT_2:pred.2">cc=</a> <font color="Maroon" title="c6">C</font></span><br/>

<a NAME="E3:15"/><i><font color="Green" title="E7">A3</font></i>: 
<font color="Maroon" title="c1">I</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">J</font>
 
<b>by </b><i><a class="txt" href="#E1:15"><i><font color="Green" title="E5">A1</font></i></a>, <a class="ref" href="altcat_2.html#D2" target="_self" title="ALTCAT_2:def.2">Def2</a></i>;<br/>
<a NAME="E4:15"/>
<font color="Maroon" title="c2">J</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">K</font>
 
<b>by </b><i><a class="txt" href="#E2:15"><i><font color="Green" title="E6">A2</font></i></a>, <a class="ref" href="altcat_2.html#D2" target="_self" title="ALTCAT_2:def.2">Def2</a></i>;<br/>
<b>hence </b><a NAME="E5:15"/>
<font color="Maroon" title="c1">I</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">K</font>
 <b>by </b><i><a class="txt" href="#E3:15"><i><font color="Green" title="E7">A3</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_2.html#D2" target="_self" title="ALTCAT_2:def.2">ALTCAT_2:def 2</a> <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">i</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> holds <br/><font color="Maroon" title="c4">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span><br/>

<b>let </b><font color="Maroon" title="c7">i</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="c7">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> implies <font color="Maroon" title="c4">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> )</span><br/>

<b>assume </b><a NAME="E6:15"/>
<font color="Maroon" title="c7">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</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">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font></span><br/>

<a NAME="E7:15"/><b>then </b>
( <font color="Maroon" title="c4">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> &amp; <font color="Maroon" title="c5">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">C</font> <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="txt" href="#E1:15"><i><font color="Green" title="E5">A1</font></i></a>, <a class="txt" href="#E2:15"><i><font color="Green" title="E6">A2</font></i></a>, <a class="txt" href="#E3:15"><i><font color="Green" title="E7">A3</font></i></a>, <a class="ref" href="altcat_2.html#D2" target="_self" title="ALTCAT_2:def.2">Def2</a></i>;<br/>
<b>hence </b><a NAME="E8:15"/>
<font color="Maroon" title="c4">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">C</font> <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="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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