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<span class="kw">let </span><font color="Maroon" title="c1">B1</font>, <font color="Maroon" title="c2">B2</font>, <font color="Maroon" title="c3">A1</font>, <font color="Maroon" title="c4">A2</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c3">A1</font>,<font color="Maroon" title="c1">B1</font><br/> for <font color="Olive" title="b2">g</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c4">A2</font>,<font color="Maroon" title="c2">B2</font><br/> for <font color="Olive" title="b3">Y1</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c3">A1</font><br/> for <font color="Olive" title="b4">Y2</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c4">A2</font> holds  <span class="p1"><a href="cat_2.html#K9" title="CAT_2:func.9">[:</a><span class="default"><font color="Olive" title="b1">f</font>,<font color="Olive" title="b2">g</font></span><a href="cat_2.html#K9" title="CAT_2:func.9">:]</a></span> <a href="nattra_1.html#K1" title="NATTRA_1:func.1">|</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Olive" title="b3">Y1</font>,<font color="Olive" title="b4">Y2</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1"><a href="cat_2.html#K9" title="CAT_2:func.9">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">f</font> <a href="nattra_1.html#K1" title="NATTRA_1:func.1">|</a> <font color="Olive" title="b3">Y1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b2">g</font> <a href="nattra_1.html#K1" title="NATTRA_1:func.1">|</a> <font color="Olive" title="b4">Y2</font></span>)</span></span><a href="cat_2.html#K9" title="CAT_2:func.9">:]</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c5">f</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c3">A1</font>,<font color="Maroon" title="c1">B1</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">g</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c4">A2</font>,<font color="Maroon" title="c2">B2</font><br/> for <font color="Olive" title="b2">Y1</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c3">A1</font><br/> for <font color="Olive" title="b3">Y2</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c4">A2</font> holds  <span class="p1"><a href="cat_2.html#K9" title="CAT_2:func.9">[:</a><span class="default"><font color="Maroon" title="c5">f</font>,<font color="Olive" title="b1">g</font></span><a href="cat_2.html#K9" title="CAT_2:func.9">:]</a></span> <a href="nattra_1.html#K1" title="NATTRA_1:func.1">|</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Olive" title="b2">Y1</font>,<font color="Olive" title="b3">Y2</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1"><a href="cat_2.html#K9" title="CAT_2:func.9">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="nattra_1.html#K1" title="NATTRA_1:func.1">|</a> <font color="Olive" title="b2">Y1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b1">g</font> <a href="nattra_1.html#K1" title="NATTRA_1:func.1">|</a> <font color="Olive" title="b3">Y2</font></span>)</span></span><a href="cat_2.html#K9" title="CAT_2:func.9">:]</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c6">g</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c4">A2</font>,<font color="Maroon" title="c2">B2</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">Y1</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c3">A1</font><br/> for <font color="Olive" title="b2">Y2</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c4">A2</font> holds  <span class="p1"><a href="cat_2.html#K9" title="CAT_2:func.9">[:</a><span class="default"><font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">g</font></span><a href="cat_2.html#K9" title="CAT_2:func.9">:]</a></span> <a href="nattra_1.html#K1" title="NATTRA_1:func.1">|</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Olive" title="b1">Y1</font>,<font color="Olive" title="b2">Y2</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1"><a href="cat_2.html#K9" title="CAT_2:func.9">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="nattra_1.html#K1" title="NATTRA_1:func.1">|</a> <font color="Olive" title="b1">Y1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="nattra_1.html#K1" title="NATTRA_1:func.1">|</a> <font color="Olive" title="b2">Y2</font></span>)</span></span><a href="cat_2.html#K9" title="CAT_2:func.9">:]</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c7">Y1</font> be  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c3">A1</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">Y2</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c4">A2</font> holds  <span class="p1"><a href="cat_2.html#K9" title="CAT_2:func.9">[:</a><span class="default"><font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">g</font></span><a href="cat_2.html#K9" title="CAT_2:func.9">:]</a></span> <a href="nattra_1.html#K1" title="NATTRA_1:func.1">|</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c7">Y1</font>,<font color="Olive" title="b1">Y2</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1"><a href="cat_2.html#K9" title="CAT_2:func.9">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="nattra_1.html#K1" title="NATTRA_1:func.1">|</a> <font color="Maroon" title="c7">Y1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="nattra_1.html#K1" title="NATTRA_1:func.1">|</a> <font color="Olive" title="b1">Y2</font></span>)</span></span><a href="cat_2.html#K9" title="CAT_2:func.9">:]</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c8">Y2</font> be  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c4">A2</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="cat_2.html#K9" title="CAT_2:func.9">[:</a><span class="default"><font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">g</font></span><a href="cat_2.html#K9" title="CAT_2:func.9">:]</a></span> <a href="nattra_1.html#K1" title="NATTRA_1:func.1">|</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c7">Y1</font>,<font color="Maroon" title="c8">Y2</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1"><a href="cat_2.html#K9" title="CAT_2:func.9">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="nattra_1.html#K1" title="NATTRA_1:func.1">|</a> <font color="Maroon" title="c7">Y1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="nattra_1.html#K1" title="NATTRA_1:func.1">|</a> <font color="Maroon" title="c8">Y2</font></span>)</span></span><a href="cat_2.html#K9" title="CAT_2:func.9">:]</a></span></span><br/>















<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_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="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c7">Y1</font>,<font color="Maroon" title="c8">Y2</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span>; <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="cat_2.html#K9" title="CAT_2:func.9">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="nattra_1.html#K1" title="NATTRA_1:func.1">|</a> <font color="Maroon" title="c7">Y1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="nattra_1.html#K1" title="NATTRA_1:func.1">|</a> <font color="Maroon" title="c8">Y2</font></span>)</span></span><a href="cat_2.html#K9" title="CAT_2:func.9">:]</a></span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="cat_2.html#K9" title="CAT_2:func.9">[:</a><span class="default"><font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">g</font></span><a href="cat_2.html#K9" title="CAT_2:func.9">:]</a></span> <a href="nattra_1.html#K1" title="NATTRA_1:func.1">|</a> <span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c7">Y1</font>,<font color="Maroon" title="c8">Y2</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">a</font></span><br/><span class="kw">consider </span><font color="Maroon" title="c10">a1</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c7">Y1</font>, <font color="Maroon" title="c11">a2</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c8">Y2</font><span class="kw"> such that </span><br/><a NAME="E2:2_1"/><span class="lab"><font color="Green" title="E2">A1</font></span>: 
<font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">a1</font>,<font color="Maroon" title="c11">a2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="domain_1.html#T9" title="DOMAIN_1:th.9">DOMAIN_1:9</a></span>;<br/><a NAME="E3:2_1"/><span class="lab"><font color="Green" title="E3">A2</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="nattra_1.html#K1" title="NATTRA_1:func.1">|</a> <font color="Maroon" title="c8">Y2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">a2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></span>;<br/><a NAME="E4:2_1"/><span class="lab"><font color="Green" title="E4">A3</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="nattra_1.html#K1" title="NATTRA_1:func.1">|</a> <font color="Maroon" title="c7">Y1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">a1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></span>;<br/><span class="kw">thus </span><span class="p1"><a href="cat_2.html#K9" title="CAT_2:func.9">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="nattra_1.html#K1" title="NATTRA_1:func.1">|</a> <font color="Maroon" title="c7">Y1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="nattra_1.html#K1" title="NATTRA_1:func.1">|</a> <font color="Maroon" title="c8">Y2</font></span>)</span></span><a href="cat_2.html#K9" title="CAT_2:func.9">:]</a></span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">a</font> = 
<span class="p1"><a href="cat_2.html#K9" title="CAT_2:func.9">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="nattra_1.html#K1" title="NATTRA_1:func.1">|</a> <font color="Maroon" title="c7">Y1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="nattra_1.html#K1" title="NATTRA_1:func.1">|</a> <font color="Maroon" title="c8">Y2</font></span>)</span></span><a href="cat_2.html#K9" title="CAT_2:func.9">:]</a></span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c10">a1</font>,<font color="Maroon" title="c11">a2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1"><span class="lab"><font color="Green" title="E2">A1</font></span></a></span>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">a1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">a2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1"><span class="lab"><font color="Green" title="E4">A3</font></span></a>, <a class="txt" href="#E3:2_1"><span class="lab"><font color="Green" title="E3">A2</font></span></a>, <a class="ref" href="funct_3.html#T96" title="FUNCT_3:th.96">FUNCT_3:96</a></span>
<br/>.= 
<span class="p1"><a href="cat_2.html#K9" title="CAT_2:func.9">[:</a><span class="default"><font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">g</font></span><a href="cat_2.html#K9" title="CAT_2:func.9">:]</a></span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c10">a1</font>,<font color="Maroon" title="c11">a2</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_3.html#T96" title="FUNCT_3:th.96">FUNCT_3:96</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="cat_2.html#K9" title="CAT_2:func.9">[:</a><span class="default"><font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">g</font></span><a href="cat_2.html#K9" title="CAT_2:func.9">:]</a></span> <a href="nattra_1.html#K1" title="NATTRA_1:func.1">|</a> <span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c7">Y1</font>,<font color="Maroon" title="c8">Y2</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">a</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1"><span class="lab"><font color="Green" title="E2">A1</font></span></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</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="E2:2"/>
<span class="p1"><a href="cat_2.html#K9" title="CAT_2:func.9">[:</a><span class="default"><font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">g</font></span><a href="cat_2.html#K9" title="CAT_2:func.9">:]</a></span> <a href="nattra_1.html#K1" title="NATTRA_1:func.1">|</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c7">Y1</font>,<font color="Maroon" title="c8">Y2</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1"><a href="cat_2.html#K9" title="CAT_2:func.9">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="nattra_1.html#K1" title="NATTRA_1:func.1">|</a> <font color="Maroon" title="c7">Y1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="nattra_1.html#K1" title="NATTRA_1:func.1">|</a> <font color="Maroon" title="c8">Y2</font></span>)</span></span><a href="cat_2.html#K9" title="CAT_2:func.9">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</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/>


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