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<span class="kw">let </span><font color="Maroon" title="c1">X</font>, <font color="Maroon" title="c2">Y1</font>, <font color="Maroon" title="c3">Y2</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">f1</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y1</font><br/> for <font color="Olive" title="b2">f2</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c3">Y2</font> holds  <span class="p1"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Olive" title="b1">f1</font>,<font color="Olive" title="b2">f2</font></span><a href="funct_3.html#K13" title="FUNCT_3:func.13">:&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Olive" title="b1">f1</font>,<font color="Olive" title="b2">f2</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="funct_3.html#K12" title="FUNCT_3:func.12">delta</a> <font color="Maroon" title="c1">X</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c4">f1</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y1</font>; <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">f2</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c3">Y2</font> holds  <span class="p1"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c4">f1</font>,<font color="Olive" title="b1">f2</font></span><a href="funct_3.html#K13" title="FUNCT_3:func.13">:&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c4">f1</font>,<font color="Olive" title="b1">f2</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="funct_3.html#K12" title="FUNCT_3:func.12">delta</a> <font color="Maroon" title="c1">X</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c5">f2</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c3">Y2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c4">f1</font>,<font color="Maroon" title="c5">f2</font></span><a href="funct_3.html#K13" title="FUNCT_3:func.13">:&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c4">f1</font>,<font color="Maroon" title="c5">f2</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="funct_3.html#K12" title="FUNCT_3:func.12">delta</a> <font color="Maroon" title="c1">X</font></span>)</span></span><br/>









<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="115_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:115_1"/>
( <font color="Maroon" title="c2">Y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c3">Y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or ( <font color="Maroon" title="c2">Y1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c3">Y2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="115_1_1"><span class="kw">suppose </span></a><a NAME="E1:115_1_1"/>
( <font color="Maroon" title="c2">Y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c3">Y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c4">f1</font>,<font color="Maroon" title="c5">f2</font></span><a href="funct_3.html#K13" title="FUNCT_3:func.13">:&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c4">f1</font>,<font color="Maroon" title="c5">f2</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="funct_3.html#K12" title="FUNCT_3:func.12">delta</a> <font color="Maroon" title="c1">X</font></span>)</span></span><br/><div class="add"><a NAME="E2:115_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A1</font></span>: 
(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 ;<br/><span class="lab"><font color="Green" title="E50">A2</font></span>:  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c4">f1</font>,<font color="Maroon" title="c5">f2</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span> = 
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">f1</font></span>)</span>,<span class="p2">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">f2</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_3.html#D9" target="_self" title="FUNCT_3:def.9">Def9</a></span>
<br/>.= 
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:115_1_1"><span class="lab"><font color="Green" title="E49">A1</font></span></a>, <a class="ref" href="zfmisc_1.html#T113" title="ZFMISC_1:th.113">ZFMISC_1:113</a></span>
;<br/> <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c4">f1</font>,<font color="Maroon" title="c5">f2</font></span><a href="funct_3.html#K13" title="FUNCT_3:func.13">:&gt;</a></span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">f1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">f2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_3.html#D8" target="_self" title="FUNCT_3:def.8">Def8</a></span>
<br/>.= 
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:115_1_1"><span class="lab"><font color="Green" title="E49">A1</font></span></a></span>
;<br/><span class="kw">hence </span><span class="p1"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c4">f1</font>,<font color="Maroon" title="c5">f2</font></span><a href="funct_3.html#K13" title="FUNCT_3:func.13">:&gt;</a></span> = 
<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="funct_3.html#K12" title="FUNCT_3:func.12">delta</a> <font color="Maroon" title="c1">X</font></span>)</span>

<br/>.= 
<span class="p1"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c4">f1</font>,<font color="Maroon" title="c5">f2</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="funct_3.html#K12" title="FUNCT_3:func.12">delta</a> <font color="Maroon" title="c1">X</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:115_1_1"><span class="lab"><font color="Green" title="E50">A2</font></span></a>, <a class="ref" href="relat_1.html#T64" title="RELAT_1:th.64">RELAT_1:64</a></span>
;<br/> <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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="115_1_2"><span class="kw">suppose </span></a><a NAME="E1:115_1_2"/>
( <font color="Maroon" title="c2">Y1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c3">Y2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c4">f1</font>,<font color="Maroon" title="c5">f2</font></span><a href="funct_3.html#K13" title="FUNCT_3:func.13">:&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c4">f1</font>,<font color="Maroon" title="c5">f2</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="funct_3.html#K12" title="FUNCT_3:func.12">delta</a> <font color="Maroon" title="c1">X</font></span>)</span></span><br/><div class="add"><a NAME="E2:115_1_2"/><span class="kw">then </span>
(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E3:115_1_2"/>
<span class="p1"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c4">f1</font>,<font color="Maroon" title="c5">f2</font></span><a href="funct_3.html#K13" title="FUNCT_3:func.13">:&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c4">f1</font>,<font color="Maroon" title="c5">f2</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="funct_3.html#K12" title="FUNCT_3:func.12">delta</a> <font color="Maroon" title="c1">X</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_3.html#T89" target="_self" title="FUNCT_3:th.89">Th89</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></div><span class="kw">end;</span></div>

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