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<span class="kw">let </span><font color="Maroon" title="c1">D1</font>, <font color="Maroon" title="c2">D2</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="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">a1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D1</font><br/>  for <font color="Olive" title="b2">a2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">D2</font><br/>  for <font color="Olive" title="b3">f1</font> being   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c1">D1</font><br/>  for <font color="Olive" title="b4">f2</font> being   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c2">D2</font> holds <br/> ( ( <font color="Olive" title="b1">a1</font> <a href="binop_1.html#R3" title="BINOP_1:pred.3">is_a_unity_wrt</a> <font color="Olive" title="b3">f1</font> &amp; <font color="Olive" title="b2">a2</font> <a href="binop_1.html#R3" title="BINOP_1:pred.3">is_a_unity_wrt</a> <font color="Olive" title="b4">f2</font> ) iff <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">a1</font>,<font color="Olive" title="b2">a2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="binop_1.html#R3" title="BINOP_1:pred.3">is_a_unity_wrt</a> <span class="p1"><a href="filter_1.html#K6" title="FILTER_1:func.6">|:</a><span class="default"><font color="Olive" title="b3">f1</font>,<font color="Olive" title="b4">f2</font></span><a href="filter_1.html#K6" title="FILTER_1:func.6">:|</a></span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">a1</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">a2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">D2</font><br/>  for <font color="Olive" title="b2">f1</font> being   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c1">D1</font><br/>  for <font color="Olive" title="b3">f2</font> being   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c2">D2</font> holds <br/> ( ( <font color="Maroon" title="c3">a1</font> <a href="binop_1.html#R3" title="BINOP_1:pred.3">is_a_unity_wrt</a> <font color="Olive" title="b2">f1</font> &amp; <font color="Olive" title="b1">a2</font> <a href="binop_1.html#R3" title="BINOP_1:pred.3">is_a_unity_wrt</a> <font color="Olive" title="b3">f2</font> ) iff <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Olive" title="b1">a2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="binop_1.html#R3" title="BINOP_1:pred.3">is_a_unity_wrt</a> <span class="p1"><a href="filter_1.html#K6" title="FILTER_1:func.6">|:</a><span class="default"><font color="Olive" title="b2">f1</font>,<font color="Olive" title="b3">f2</font></span><a href="filter_1.html#K6" title="FILTER_1:func.6">:|</a></span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">a2</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">D2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">f1</font> being   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c1">D1</font><br/>  for <font color="Olive" title="b2">f2</font> being   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c2">D2</font> holds <br/> ( ( <font color="Maroon" title="c3">a1</font> <a href="binop_1.html#R3" title="BINOP_1:pred.3">is_a_unity_wrt</a> <font color="Olive" title="b1">f1</font> &amp; <font color="Maroon" title="c4">a2</font> <a href="binop_1.html#R3" title="BINOP_1:pred.3">is_a_unity_wrt</a> <font color="Olive" title="b2">f2</font> ) iff <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c4">a2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="binop_1.html#R3" title="BINOP_1:pred.3">is_a_unity_wrt</a> <span class="p1"><a href="filter_1.html#K6" title="FILTER_1:func.6">|:</a><span class="default"><font color="Olive" title="b1">f1</font>,<font color="Olive" title="b2">f2</font></span><a href="filter_1.html#K6" title="FILTER_1:func.6">:|</a></span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c5">f1</font> be   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c1">D1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">f2</font> being   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c2">D2</font> holds <br/> ( ( <font color="Maroon" title="c3">a1</font> <a href="binop_1.html#R3" title="BINOP_1:pred.3">is_a_unity_wrt</a> <font color="Maroon" title="c5">f1</font> &amp; <font color="Maroon" title="c4">a2</font> <a href="binop_1.html#R3" title="BINOP_1:pred.3">is_a_unity_wrt</a> <font color="Olive" title="b1">f2</font> ) iff <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c4">a2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="binop_1.html#R3" title="BINOP_1:pred.3">is_a_unity_wrt</a> <span class="p1"><a href="filter_1.html#K6" title="FILTER_1:func.6">|:</a><span class="default"><font color="Maroon" title="c5">f1</font>,<font color="Olive" title="b1">f2</font></span><a href="filter_1.html#K6" title="FILTER_1:func.6">:|</a></span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c6">f2</font> be   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c2">D2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c3">a1</font> <a href="binop_1.html#R3" title="BINOP_1:pred.3">is_a_unity_wrt</a> <font color="Maroon" title="c5">f1</font> &amp; <font color="Maroon" title="c4">a2</font> <a href="binop_1.html#R3" title="BINOP_1:pred.3">is_a_unity_wrt</a> <font color="Maroon" title="c6">f2</font> ) iff <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c4">a2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="binop_1.html#R3" title="BINOP_1:pred.3">is_a_unity_wrt</a> <span class="p1"><a href="filter_1.html#K6" title="FILTER_1:func.6">|:</a><span class="default"><font color="Maroon" title="c5">f1</font>,<font color="Maroon" title="c6">f2</font></span><a href="filter_1.html#K6" title="FILTER_1:func.6">:|</a></span> )</span><br/>











<span class="kw">thus </span><a NAME="E1:41"/>
( <font color="Maroon" title="c3">a1</font> <a href="binop_1.html#R3" title="BINOP_1:pred.3">is_a_unity_wrt</a> <font color="Maroon" title="c5">f1</font> &amp; <font color="Maroon" title="c4">a2</font> <a href="binop_1.html#R3" title="BINOP_1:pred.3">is_a_unity_wrt</a> <font color="Maroon" title="c6">f2</font> implies <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c4">a2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="binop_1.html#R3" title="BINOP_1:pred.3">is_a_unity_wrt</a> <span class="p1"><a href="filter_1.html#K6" title="FILTER_1:func.6">|:</a><span class="default"><font color="Maroon" title="c5">f1</font>,<font color="Maroon" title="c6">f2</font></span><a href="filter_1.html#K6" title="FILTER_1:func.6">:|</a></span> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c4">a2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="binop_1.html#R3" title="BINOP_1:pred.3">is_a_unity_wrt</a> <span class="p1"><a href="filter_1.html#K6" title="FILTER_1:func.6">|:</a><span class="default"><font color="Maroon" title="c5">f1</font>,<font color="Maroon" title="c6">f2</font></span><a href="filter_1.html#K6" title="FILTER_1:func.6">:|</a></span> implies ( <font color="Maroon" title="c3">a1</font> <a href="binop_1.html#R3" title="BINOP_1:pred.3">is_a_unity_wrt</a> <font color="Maroon" title="c5">f1</font> &amp; <font color="Maroon" title="c4">a2</font> <a href="binop_1.html#R3" title="BINOP_1:pred.3">is_a_unity_wrt</a> <font color="Maroon" title="c6">f2</font> ) )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:41_1"/><span class="lab"><font color="Green" title="E27">A1</font></span>: 
<font color="Maroon" title="c3">a1</font> <a href="binop_1.html#R1" title="BINOP_1:pred.1">is_a_left_unity_wrt</a> <font color="Maroon" title="c5">f1</font>
 <span class="kw">and </span><br/><a NAME="E2:41_1"/><span class="lab"><font color="Green" title="E28">A2</font></span>: 
<font color="Maroon" title="c3">a1</font> <a href="binop_1.html#R2" title="BINOP_1:pred.2">is_a_right_unity_wrt</a> <font color="Maroon" title="c5">f1</font>
 <span class="kw">and </span><br/><a NAME="E3:41_1"/><span class="lab"><font color="Green" title="E29">A3</font></span>: 
<font color="Maroon" title="c4">a2</font> <a href="binop_1.html#R1" title="BINOP_1:pred.1">is_a_left_unity_wrt</a> <font color="Maroon" title="c6">f2</font>
 <span class="kw">and </span><br/><a NAME="E4:41_1"/><span class="lab"><font color="Green" title="E30">A4</font></span>: 
<font color="Maroon" title="c4">a2</font> <a href="binop_1.html#R2" title="BINOP_1:pred.2">is_a_right_unity_wrt</a> <font color="Maroon" title="c6">f2</font>
 ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="binop_1.html#D7" title="BINOP_1:def.7">BINOP_1:def 7</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c4">a2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="binop_1.html#R3" title="BINOP_1:pred.3">is_a_unity_wrt</a> <span class="p1"><a href="filter_1.html#K6" title="FILTER_1:func.6">|:</a><span class="default"><font color="Maroon" title="c5">f1</font>,<font color="Maroon" title="c6">f2</font></span><a href="filter_1.html#K6" title="FILTER_1:func.6">:|</a></span></span><br/>

<span class="kw">thus </span><a NAME="E5:41_1"/>
( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c4">a2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="binop_1.html#R1" title="BINOP_1:pred.1">is_a_left_unity_wrt</a> <span class="p1"><a href="filter_1.html#K6" title="FILTER_1:func.6">|:</a><span class="default"><font color="Maroon" title="c5">f1</font>,<font color="Maroon" title="c6">f2</font></span><a href="filter_1.html#K6" title="FILTER_1:func.6">:|</a></span> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c4">a2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="binop_1.html#R2" title="BINOP_1:pred.2">is_a_right_unity_wrt</a> <span class="p1"><a href="filter_1.html#K6" title="FILTER_1:func.6">|:</a><span class="default"><font color="Maroon" title="c5">f1</font>,<font color="Maroon" title="c6">f2</font></span><a href="filter_1.html#K6" title="FILTER_1:func.6">:|</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:41_1"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="txt" href="#E2:41_1"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="txt" href="#E3:41_1"><span class="lab"><font color="Green" title="E29">A3</font></span></a>, <a class="txt" href="#E4:41_1"><span class="lab"><font color="Green" title="E30">A4</font></span></a>, <a class="ref" href="filter_1.html#T24" target="_self" title="FILTER_1:th.24">Th25</a>, <a class="ref" href="filter_1.html#T25" target="_self" title="FILTER_1:th.25">Th26</a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="binop_1.html#D7" title="BINOP_1:def.7">BINOP_1:def 7</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:41"/><span class="lab"><font color="Green" title="E27">A5</font></span>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c4">a2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="binop_1.html#R1" title="BINOP_1:pred.1">is_a_left_unity_wrt</a> <span class="p1"><a href="filter_1.html#K6" title="FILTER_1:func.6">|:</a><span class="default"><font color="Maroon" title="c5">f1</font>,<font color="Maroon" title="c6">f2</font></span><a href="filter_1.html#K6" title="FILTER_1:func.6">:|</a></span>
 <span class="kw">and </span><br/><a NAME="E3:41"/><span class="lab"><font color="Green" title="E28">A6</font></span>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c4">a2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="binop_1.html#R2" title="BINOP_1:pred.2">is_a_right_unity_wrt</a> <span class="p1"><a href="filter_1.html#K6" title="FILTER_1:func.6">|:</a><span class="default"><font color="Maroon" title="c5">f1</font>,<font color="Maroon" title="c6">f2</font></span><a href="filter_1.html#K6" title="FILTER_1:func.6">:|</a></span>
 ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="binop_1.html#D7" title="BINOP_1:def.7">BINOP_1:def 7</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">a1</font> <a href="binop_1.html#R3" title="BINOP_1:pred.3">is_a_unity_wrt</a> <font color="Maroon" title="c5">f1</font> &amp; <font color="Maroon" title="c4">a2</font> <a href="binop_1.html#R3" title="BINOP_1:pred.3">is_a_unity_wrt</a> <font color="Maroon" title="c6">f2</font> )</span><br/>

<span class="kw">thus </span><a NAME="E4:41"/>
( <font color="Maroon" title="c3">a1</font> <a href="binop_1.html#R1" title="BINOP_1:pred.1">is_a_left_unity_wrt</a> <font color="Maroon" title="c5">f1</font> &amp; <font color="Maroon" title="c3">a1</font> <a href="binop_1.html#R2" title="BINOP_1:pred.2">is_a_right_unity_wrt</a> <font color="Maroon" title="c5">f1</font> &amp; <font color="Maroon" title="c4">a2</font> <a href="binop_1.html#R1" title="BINOP_1:pred.1">is_a_left_unity_wrt</a> <font color="Maroon" title="c6">f2</font> &amp; <font color="Maroon" title="c4">a2</font> <a href="binop_1.html#R2" title="BINOP_1:pred.2">is_a_right_unity_wrt</a> <font color="Maroon" title="c6">f2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:41"><span class="lab"><font color="Green" title="E27">A5</font></span></a>, <a class="txt" href="#E3:41"><span class="lab"><font color="Green" title="E28">A6</font></span></a>, <a class="ref" href="filter_1.html#T24" target="_self" title="FILTER_1:th.24">Th25</a>, <a class="ref" href="filter_1.html#T25" target="_self" title="FILTER_1:th.25">Th26</a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="binop_1.html#D7" title="BINOP_1:def.7">BINOP_1:def 7</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
