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<b>let </b><font color="Maroon" title="c1">Y</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <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">z</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">Y</font><br/> for <font color="Olive" title="b2">PA</font>, <font color="Olive" title="b3">PB</font> being   <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of <font color="Maroon" title="c1">Y</font>  st <font color="Olive" title="b2">PA</font> <a href="partit1.html#NR1" title="PARTIT1:NR.1">'&lt;'</a> <font color="Olive" title="b3">PB</font> holds <br/> <a href="bvfunc_1.html#K20" title="BVFUNC_1:func.20">EqClass</a> <font color="Olive" title="b1">z</font>,<font color="Olive" title="b2">PA</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="bvfunc_1.html#K20" title="BVFUNC_1:func.20">EqClass</a> <font color="Olive" title="b1">z</font>,<font color="Olive" title="b3">PB</font></span><br/><b>let </b><font color="Maroon" title="c2">z</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">Y</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">PA</font>, <font color="Olive" title="b2">PB</font> being   <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of <font color="Maroon" title="c1">Y</font>  st <font color="Olive" title="b1">PA</font> <a href="partit1.html#NR1" title="PARTIT1:NR.1">'&lt;'</a> <font color="Olive" title="b2">PB</font> holds <br/> <a href="bvfunc_1.html#K20" title="BVFUNC_1:func.20">EqClass</a> <font color="Maroon" title="c2">z</font>,<font color="Olive" title="b1">PA</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="bvfunc_1.html#K20" title="BVFUNC_1:func.20">EqClass</a> <font color="Maroon" title="c2">z</font>,<font color="Olive" title="b2">PB</font></span><br/><b>let </b><font color="Maroon" title="c3">PA</font>, <font color="Maroon" title="c4">PB</font> be    <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of <font color="Maroon" title="c1">Y</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">PA</font> <a href="partit1.html#NR1" title="PARTIT1:NR.1">'&lt;'</a> <font color="Maroon" title="c4">PB</font> implies  <a href="bvfunc_1.html#K20" title="BVFUNC_1:func.20">EqClass</a> <font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">PA</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="bvfunc_1.html#K20" title="BVFUNC_1:func.20">EqClass</a> <font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c4">PB</font> )</span><br/>





<b>assume </b><a NAME="E1:1"/><i><font color="Green" title="E2">A1</font></i>: 
<font color="Maroon" title="c3">PA</font> <a href="partit1.html#NR1" title="PARTIT1:NR.1">'&lt;'</a> <font color="Maroon" title="c4">PB</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="bvfunc_1.html#K20" title="BVFUNC_1:func.20">EqClass</a> <font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">PA</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="bvfunc_1.html#K20" title="BVFUNC_1:func.20">EqClass</a> <font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c4">PB</font></span><br/>

<a NAME="E2:1"/><i><font color="Green" title="E3">A2</font></i>: 
( <font color="Maroon" title="c2">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bvfunc_1.html#K20" title="BVFUNC_1:func.20">EqClass</a> <font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">PA</font> &amp;  <a href="bvfunc_1.html#K20" title="BVFUNC_1:func.20">EqClass</a> <font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">PA</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">PA</font> )
 
<b>by </b><i><a class="ref" href="eqrel_1.html#D8" title="EQREL_1:def.8">EQREL_1:def 8</a></i>;<br/>
<a NAME="E3:1"/>
 ex <font color="Olive" title="b1">b</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">PB</font> &amp;  <a href="bvfunc_1.html#K20" title="BVFUNC_1:func.20">EqClass</a> <font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">PA</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">b</font> )
 
<b>by </b><i><a class="txt" href="bvfunc11.html#E1:1"><i><font color="Green" title="E2">A1</font></i></a>, <a class="ref" href="setfam_1.html#D2" title="SETFAM_1:def.2">SETFAM_1:def 2</a></i>;<br/>
<b>hence </b><a NAME="E4:1"/>
 <a href="bvfunc_1.html#K20" title="BVFUNC_1:func.20">EqClass</a> <font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">PA</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="bvfunc_1.html#K20" title="BVFUNC_1:func.20">EqClass</a> <font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c4">PB</font>
 <b>by </b><i><a class="txt" href="bvfunc11.html#E2:1"><i><font color="Green" title="E3">A2</font></i></a>, <a class="ref" href="eqrel_1.html#D8" title="EQREL_1:def.8">EQREL_1:def 8</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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