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<b>let </b><font color="Maroon" title="c1">V</font>, <font color="Maroon" title="c2">C</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>, <font color="Olive" title="b2">B</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <span class="p1">(<span class="default"><a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">C</font></span>)</span> holds   <a href="substlat.html#K3" title="SUBSTLAT:func.3">mi</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">A</font> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <font color="Olive" title="b2">B</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="substlat.html#K3" title="SUBSTLAT:func.3">mi</a> <font color="Olive" title="b1">A</font></span>)</span> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <font color="Olive" title="b2">B</font></span><br/><b>let </b><font color="Maroon" title="c3">A</font>, <font color="Maroon" title="c4">B</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <span class="p1">(<span class="default"><a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">C</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="substlat.html#K3" title="SUBSTLAT:func.3">mi</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="substlat.html#K3" title="SUBSTLAT:func.3">mi</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <font color="Maroon" title="c4">B</font></span><br/>







<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">a</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">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="substlat.html#K3" title="SUBSTLAT:func.3">mi</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <font color="Maroon" title="c4">B</font></span>)</span> implies <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="substlat.html#K3" title="SUBSTLAT:func.3">mi</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <font color="Maroon" title="c4">B</font> )</span><br/><b>assume </b><a NAME="E1:22_1"/><i><font color="Green" title="E14">A1</font></i>: 
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="substlat.html#K3" title="SUBSTLAT:func.3">mi</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <font color="Maroon" title="c4">B</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="substlat.html#K3" title="SUBSTLAT:func.3">mi</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <font color="Maroon" title="c4">B</font></span><br/><a NAME="E2:22_1"/><b>then </b><i><font color="Green" title="E15">A2</font></i>: 
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <font color="Maroon" title="c4">B</font>
 <b>by </b><i><a class="ref" href="substlat.html#T6" target="_self" title="SUBSTLAT:th.6">Th6</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c6">a'</font> = <font color="Maroon" title="c5">a</font> as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <b>by </b><i><a class="txt" href="substlat.html#E1:22_1"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="substlat.html#E2"><i><font color="Green" title="E1">Lm1</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:22_1_1_1"/>
( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> or <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> )
 </span><b>by </b><i><a class="txt" href="substlat.html#E2:22_1"><i><font color="Green" title="E15">A2</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_1_1_1"><b>suppose </b></a><a NAME="E1:22_1_1_1_1"/><i><font color="Green" title="E16">A3</font></i>: 
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="substlat.html#K3" title="SUBSTLAT:func.3">mi</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <font color="Maroon" title="c4">B</font></span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">b</font> be  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</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"> ( <font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Maroon" title="c7">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">a</font> implies <font color="Maroon" title="c7">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> )</span><br/><b>assume </b><a NAME="E1:22_1_1_1_1_1"/>
<font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c7">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">a</font> implies <font color="Maroon" title="c7">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> )</span><br/><a NAME="E2:22_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <font color="Maroon" title="c4">B</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><b>hence </b><a NAME="E3:22_1_1_1_1_1"/>
( <font color="Maroon" title="c7">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">a</font> implies <font color="Maroon" title="c7">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> )
 <b>by </b><i><a class="txt" href="substlat.html#E1:22_1"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="substlat.html#T6" target="_self" title="SUBSTLAT:th.6">Th6</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><a NAME="E3:22_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c6">a'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="substlat.html#K3" title="SUBSTLAT:func.3">mi</a> <font color="Maroon" title="c3">A</font>
 <b>by </b><i><a class="txt" href="substlat.html#E1:22_1_1_1_1"><i><font color="Green" title="E16">A3</font></i></a>, <a class="ref" href="substlat.html#T7" target="_self" title="SUBSTLAT:th.7">Th7</a></i>;<br/><b>hence </b><a NAME="E4:22_1_1_1_1"/>
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="substlat.html#K3" title="SUBSTLAT:func.3">mi</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <font color="Maroon" title="c4">B</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_1_1_2"><b>suppose </b></a><a NAME="E1:22_1_1_1_2"/>
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="substlat.html#K3" title="SUBSTLAT:func.3">mi</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <font color="Maroon" title="c4">B</font></span><br/><div class="add"><b>hence </b><a NAME="E2:22_1_1_1_2"/>
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="substlat.html#K3" title="SUBSTLAT:func.3">mi</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <font color="Maroon" title="c4">B</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</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></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:22_1"/>
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="substlat.html#K3" title="SUBSTLAT:func.3">mi</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <font color="Maroon" title="c4">B</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="E2:22"/>
 <a href="substlat.html#K3" title="SUBSTLAT:func.3">mi</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="substlat.html#K3" title="SUBSTLAT:func.3">mi</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <font color="Maroon" title="c4">B</font>
 <b>by </b><i><a class="txt" href="substlat.html#E11"><i><font color="Green" title="E8">Lm2</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/>


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