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<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">K</font>, <font color="Olive" title="b2">L</font>, <font color="Olive" title="b3">M</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  <font color="Olive" title="b1">K</font> <a href="substlat.html#K4" title="SUBSTLAT:func.4">^</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">L</font> <a href="substlat.html#K4" title="SUBSTLAT:func.4">^</a> <font color="Olive" title="b3">M</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">K</font> <a href="substlat.html#K4" title="SUBSTLAT:func.4">^</a> <font color="Olive" title="b2">L</font></span>)</span> <a href="substlat.html#K4" title="SUBSTLAT:func.4">^</a> <font color="Olive" title="b3">M</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">K</font>, <font color="Maroon" title="c4">L</font>, <font color="Maroon" title="c5">M</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">K</font> <a href="substlat.html#K4" title="SUBSTLAT:func.4">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">L</font> <a href="substlat.html#K4" title="SUBSTLAT:func.4">^</a> <font color="Maroon" title="c5">M</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">K</font> <a href="substlat.html#K4" title="SUBSTLAT:func.4">^</a> <font color="Maroon" title="c4">L</font></span>)</span> <a href="substlat.html#K4" title="SUBSTLAT:func.4">^</a> <font color="Maroon" title="c5">M</font></span><br/>





<a NAME="E1:31"/><span class="lab"><font color="Green" title="E23">A1</font></span>: 
( <font color="Maroon" title="c4">L</font> <a href="substlat.html#K4" title="SUBSTLAT:func.4">^</a> <font color="Maroon" title="c5">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">M</font> <a href="substlat.html#K4" title="SUBSTLAT:func.4">^</a> <font color="Maroon" title="c4">L</font> &amp; <font color="Maroon" title="c3">K</font> <a href="substlat.html#K4" title="SUBSTLAT:func.4">^</a> <font color="Maroon" title="c4">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">L</font> <a href="substlat.html#K4" title="SUBSTLAT:func.4">^</a> <font color="Maroon" title="c3">K</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="substlat.html#T3" target="_self" title="SUBSTLAT:th.3">Th3</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E24">A2</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c6">K</font>, <font color="Maroon" title="c7">L</font>, <font color="Maroon" title="c8">M</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">K</font> <a href="substlat.html#K4" title="SUBSTLAT:func.4">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">L</font> <a href="substlat.html#K4" title="SUBSTLAT:func.4">^</a> <font color="Maroon" title="c8">M</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">K</font> <a href="substlat.html#K4" title="SUBSTLAT:func.4">^</a> <font color="Maroon" title="c7">L</font></span>)</span> <a href="substlat.html#K4" title="SUBSTLAT:func.4">^</a> <font color="Maroon" title="c8">M</font></span><br/><a NAME="E1:31_1"/><span class="lab"><font color="Green" title="E25">A3</font></span>: 
( <font color="Maroon" title="c6">K</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <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> &amp; <font color="Maroon" title="c7">L</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <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 class="kw">by</span> <span class="lab"><a class="ref" href="finsub_1.html#D5" title="FINSUB_1:def.5">FINSUB_1:def 5</a></span>;<br/><a NAME="E2:31_1"/><span class="lab"><font color="Green" title="E26">A4</font></span>: 
<font color="Maroon" title="c8">M</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <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 class="kw">by</span> <span class="lab"><a class="ref" href="finsub_1.html#D5" title="FINSUB_1:def.5">FINSUB_1:def 5</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c9">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">K</font> <a href="substlat.html#K4" title="SUBSTLAT:func.4">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">L</font> <a href="substlat.html#K4" title="SUBSTLAT:func.4">^</a> <font color="Maroon" title="c8">M</font></span>)</span> implies <font color="Maroon" title="c9">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">K</font> <a href="substlat.html#K4" title="SUBSTLAT:func.4">^</a> <font color="Maroon" title="c7">L</font></span>)</span> <a href="substlat.html#K4" title="SUBSTLAT:func.4">^</a> <font color="Maroon" title="c8">M</font> )</span><br/><a NAME="E1:31_1_1"/><span class="lab"><font color="Green" title="E27">A5</font></span>: 
<font color="Maroon" title="c6">K</font> <a href="substlat.html#K4" title="SUBSTLAT:func.4">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">L</font> <a href="substlat.html#K4" title="SUBSTLAT:func.4">^</a> <font color="Maroon" title="c8">M</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <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 class="kw">by</span> <span class="lab"><a class="ref" href="finsub_1.html#D5" title="FINSUB_1:def.5">FINSUB_1:def 5</a></span>;<br/><span class="kw">assume </span><a NAME="E2:31_1_1"/><span class="lab"><font color="Green" title="E28">A6</font></span>: 
<font color="Maroon" title="c9">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">K</font> <a href="substlat.html#K4" title="SUBSTLAT:func.4">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">L</font> <a href="substlat.html#K4" title="SUBSTLAT:func.4">^</a> <font color="Maroon" title="c8">M</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">K</font> <a href="substlat.html#K4" title="SUBSTLAT:func.4">^</a> <font color="Maroon" title="c7">L</font></span>)</span> <a href="substlat.html#K4" title="SUBSTLAT:func.4">^</a> <font color="Maroon" title="c8">M</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">b</font>, <font color="Maroon" title="c11">c</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E4:31_1_1"/><span class="lab"><font color="Green" title="E29">A7</font></span>: 
<font color="Maroon" title="c10">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">K</font>
 <span class="kw">and </span><br/><a NAME="E5:31_1_1"/><span class="lab"><font color="Green" title="E30">A8</font></span>: 
<font color="Maroon" title="c11">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">L</font> <a href="substlat.html#K4" title="SUBSTLAT:func.4">^</a> <font color="Maroon" title="c8">M</font>
 <span class="kw">and </span><br/><a NAME="E6:31_1_1"/><span class="lab"><font color="Green" title="E31">A9</font></span>: 
<font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c11">c</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="substlat.html#T15" target="_self" title="SUBSTLAT:th.15">Th15</a></span>;<br/><a NAME="E7:31_1_1"/><span class="lab"><font color="Green" title="E32">A10</font></span>: 
<font color="Maroon" title="c11">c</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">b</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c11">c</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c12">b1</font>, <font color="Maroon" title="c13">c1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E9:31_1_1"/><span class="lab"><font color="Green" title="E33">A11</font></span>: 
<font color="Maroon" title="c12">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">L</font>
 <span class="kw">and </span><br/><a NAME="E10:31_1_1"/><span class="lab"><font color="Green" title="E34">A12</font></span>: 
<font color="Maroon" title="c13">c1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">M</font>
 <span class="kw">and </span><br/><a NAME="E11:31_1_1"/><span class="lab"><font color="Green" title="E35">A13</font></span>: 
<font color="Maroon" title="c11">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">b1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c13">c1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:31_1_1"><span class="lab"><font color="Green" title="E30">A8</font></span></a>, <a class="ref" href="substlat.html#T15" target="_self" title="SUBSTLAT:th.15">Th15</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c14">b2</font> = <font color="Maroon" title="c10">b</font>, <font color="Maroon" title="c15">b12</font> = <font color="Maroon" title="c12">b1</font> as   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">C</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31_1"><span class="lab"><font color="Green" title="E25">A3</font></span></a>, <a class="txt" href="#E4:31_1_1"><span class="lab"><font color="Green" title="E29">A7</font></span></a>, <a class="txt" href="#E9:31_1_1"><span class="lab"><font color="Green" title="E33">A11</font></span></a>, <a class="ref" href="partfun1.html#T120" title="PARTFUN1:th.120">PARTFUN1:120</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c16">b9</font> = <font color="Maroon" title="c10">b</font>, <font color="Maroon" title="c17">b19</font> = <font color="Maroon" title="c12">b1</font>, <font color="Maroon" title="c18">c19</font> = <font color="Maroon" title="c13">c1</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <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 class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31_1"><span class="lab"><font color="Green" title="E25">A3</font></span></a>, <a class="txt" href="#E2:31_1"><span class="lab"><font color="Green" title="E26">A4</font></span></a>, <a class="txt" href="#E4:31_1_1"><span class="lab"><font color="Green" title="E29">A7</font></span></a>, <a class="txt" href="#E9:31_1_1"><span class="lab"><font color="Green" title="E33">A11</font></span></a>, <a class="txt" href="#E10:31_1_1"><span class="lab"><font color="Green" title="E34">A12</font></span></a></span>;<br/><a NAME="E14:31_1_1"/>
<font color="Maroon" title="c12">b1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:31_1_1"><span class="lab"><font color="Green" title="E35">A13</font></span></a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></span>;<br/><a NAME="E15:31_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A14</font></span>: 
( <font color="Maroon" title="c10">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">b</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c11">c</font> &amp; <font color="Maroon" title="c12">b1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">b</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c11">c</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:31_1_1"><span class="lab"><font color="Green" title="E32">A10</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></span>;<br/><a NAME="E16:31_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A15</font></span>: 
<font color="Maroon" title="c16">b9</font> <a href="partfun1.html#R1" title="PARTFUN1:pred.1">tolerates</a> <font color="Maroon" title="c17">b19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:31_1_1"><span class="lab"><font color="Green" title="E28">A6</font></span></a>, <a class="txt" href="#E6:31_1_1"><span class="lab"><font color="Green" title="E31">A9</font></span></a>, <a class="txt" href="#E1:31_1_1"><span class="lab"><font color="Green" title="E27">A5</font></span></a>, <a class="ref" href="partfun1.html#T139" title="PARTFUN1:th.139">PARTFUN1:139</a></span>;<br/><a NAME="E17:31_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c16">b9</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c17">b19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">b9</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c17">b19</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T31" title="FUNCT_4:th.31">FUNCT_4:31</a></span>;<br/><a NAME="E18:31_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">b2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c15">b12</font> is   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">C</font>
 ;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c19">c2</font> = <font color="Maroon" title="c16">b9</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c17">b19</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <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 class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#T119" title="PARTFUN1:th.119">PARTFUN1:119</a></span>;<br/><a NAME="E20:31_1_1"/><span class="lab"><font color="Green" title="E38">A16</font></span>: 
( <font color="Maroon" title="c10">b</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">b1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c13">c1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">b</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c12">b1</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c13">c1</font> &amp; <font color="Maroon" title="c19">c2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">K</font> <a href="substlat.html#K4" title="SUBSTLAT:func.4">^</a> <font color="Maroon" title="c7">L</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:31_1_1"><span class="lab"><font color="Green" title="E29">A7</font></span></a>, <a class="txt" href="#E9:31_1_1"><span class="lab"><font color="Green" title="E33">A11</font></span></a>, <a class="txt" href="#E16:31_1_1"><span class="lab"><font color="Green" title="E37">A15</font></span></a>, <a class="ref" href="xboole_1.html#T4" title="XBOOLE_1:th.4">XBOOLE_1:4</a></span>;<br/><a NAME="E21:31_1_1"/>
<font color="Maroon" title="c13">c1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:31_1_1"><span class="lab"><font color="Green" title="E35">A13</font></span></a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></span>;<br/><a NAME="E22:31_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A17</font></span>: 
<font color="Maroon" title="c13">c1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">b</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c11">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:31_1_1"><span class="lab"><font color="Green" title="E32">A10</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><a NAME="E23:31_1_1"/>
<font color="Maroon" title="c19">c2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">b</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c11">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:31_1_1"><span class="lab"><font color="Green" title="E36">A14</font></span></a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></span>;<br/><a NAME="E24:31_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c19">c2</font> <a href="partfun1.html#R1" title="PARTFUN1:pred.1">tolerates</a> <font color="Maroon" title="c18">c19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:31_1_1"><span class="lab"><font color="Green" title="E28">A6</font></span></a>, <a class="txt" href="#E6:31_1_1"><span class="lab"><font color="Green" title="E31">A9</font></span></a>, <a class="txt" href="#E1:31_1_1"><span class="lab"><font color="Green" title="E27">A5</font></span></a>, <a class="txt" href="#E22:31_1_1"><span class="lab"><font color="Green" title="E39">A17</font></span></a>, <a class="ref" href="partfun1.html#T139" title="PARTFUN1:th.139">PARTFUN1:139</a></span>;<br/><span class="kw">hence </span><a NAME="E25:31_1_1"/>
<font color="Maroon" title="c9">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">K</font> <a href="substlat.html#K4" title="SUBSTLAT:func.4">^</a> <font color="Maroon" title="c7">L</font></span>)</span> <a href="substlat.html#K4" title="SUBSTLAT:func.4">^</a> <font color="Maroon" title="c8">M</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:31_1_1"><span class="lab"><font color="Green" title="E31">A9</font></span></a>, <a class="txt" href="#E10:31_1_1"><span class="lab"><font color="Green" title="E34">A12</font></span></a>, <a class="txt" href="#E11:31_1_1"><span class="lab"><font color="Green" title="E35">A13</font></span></a>, <a class="txt" href="#E20:31_1_1"><span class="lab"><font color="Green" title="E38">A16</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E4:31_1"/>
<font color="Maroon" title="c6">K</font> <a href="substlat.html#K4" title="SUBSTLAT:func.4">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">L</font> <a href="substlat.html#K4" title="SUBSTLAT:func.4">^</a> <font color="Maroon" title="c8">M</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">K</font> <a href="substlat.html#K4" title="SUBSTLAT:func.4">^</a> <font color="Maroon" title="c7">L</font></span>)</span> <a href="substlat.html#K4" title="SUBSTLAT:func.4">^</a> <font color="Maroon" title="c8">M</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="substlat.html#E11"><span class="lab"><font color="Green" title="E8">Lm2</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E3:31"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A18</font></span>: 
<font color="Maroon" title="c3">K</font> <a href="substlat.html#K4" title="SUBSTLAT:func.4">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">L</font> <a href="substlat.html#K4" title="SUBSTLAT:func.4">^</a> <font color="Maroon" title="c5">M</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">K</font> <a href="substlat.html#K4" title="SUBSTLAT:func.4">^</a> <font color="Maroon" title="c4">L</font></span>)</span> <a href="substlat.html#K4" title="SUBSTLAT:func.4">^</a> <font color="Maroon" title="c5">M</font>
 
;<br/>
<a NAME="E4:31"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c3">K</font> <a href="substlat.html#K4" title="SUBSTLAT:func.4">^</a> <font color="Maroon" title="c4">L</font></span>)</span> <a href="substlat.html#K4" title="SUBSTLAT:func.4">^</a> <font color="Maroon" title="c5">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">M</font> <a href="substlat.html#K4" title="SUBSTLAT:func.4">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">K</font> <a href="substlat.html#K4" title="SUBSTLAT:func.4">^</a> <font color="Maroon" title="c4">L</font></span>)</span> &amp; <font color="Maroon" title="c3">K</font> <a href="substlat.html#K4" title="SUBSTLAT:func.4">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">L</font> <a href="substlat.html#K4" title="SUBSTLAT:func.4">^</a> <font color="Maroon" title="c5">M</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">L</font> <a href="substlat.html#K4" title="SUBSTLAT:func.4">^</a> <font color="Maroon" title="c5">M</font></span>)</span> <a href="substlat.html#K4" title="SUBSTLAT:func.4">^</a> <font color="Maroon" title="c3">K</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="substlat.html#T3" target="_self" title="SUBSTLAT:th.3">Th3</a></span>;<br/>
<a NAME="E5:31"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">K</font> <a href="substlat.html#K4" title="SUBSTLAT:func.4">^</a> <font color="Maroon" title="c4">L</font></span>)</span> <a href="substlat.html#K4" title="SUBSTLAT:func.4">^</a> <font color="Maroon" title="c5">M</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">K</font> <a href="substlat.html#K4" title="SUBSTLAT:func.4">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">L</font> <a href="substlat.html#K4" title="SUBSTLAT:func.4">^</a> <font color="Maroon" title="c5">M</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E23">A1</font></span></a>, <a class="txt" href="#E2:31"><span class="lab"><font color="Green" title="E24">A2</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:31"/>
<font color="Maroon" title="c3">K</font> <a href="substlat.html#K4" title="SUBSTLAT:func.4">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">L</font> <a href="substlat.html#K4" title="SUBSTLAT:func.4">^</a> <font color="Maroon" title="c5">M</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">K</font> <a href="substlat.html#K4" title="SUBSTLAT:func.4">^</a> <font color="Maroon" title="c4">L</font></span>)</span> <a href="substlat.html#K4" title="SUBSTLAT:func.4">^</a> <font color="Maroon" title="c5">M</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:31"><span class="lab"><font color="Green" title="E25">A18</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
