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<b>let </b><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> ;<br/><b>let </b><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>;<br/><b>let </b><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>;<br/><b>let </b><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>;<br/><b>let </b><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>;<br/>











<b>thus </b><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> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:41_1"/>
( <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> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="binop_1.html#D7" title="BINOP_1:def.7">BINOP_1:def 7</a><br/>

<b>hence </b><a NAME="E2:41_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> )
 <b>by </b><i><a class="ref" href="filter_1.html#T25" target="_self" title="FILTER_1:th.25">Th25</a>, <a class="ref" href="filter_1.html#T26" target="_self" title="FILTER_1:th.26">Th26</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="binop_1.html#D7" title="BINOP_1:def.7">BINOP_1:def 7</a><br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:41"/>
( <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> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="binop_1.html#D7" title="BINOP_1:def.7">BINOP_1:def 7</a><br/>

<b>hence </b><a NAME="E3:41"/>
( <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> )
 <b>by </b><i><a class="ref" href="filter_1.html#T25" target="_self" title="FILTER_1:th.25">Th25</a>, <a class="ref" href="filter_1.html#T26" target="_self" title="FILTER_1:th.26">Th26</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="binop_1.html#D7" title="BINOP_1:def.7">BINOP_1:def 7</a><br/>


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