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<span class="kw">let </span><font color="Maroon" title="c1">D</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">S</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="c1">D</font><br/>  for <font color="Olive" title="b2">f</font> being   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c1">D</font><br/>  for <font color="Olive" title="b3">g</font> being   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Olive" title="b1">S</font><br/>  for <font color="Olive" title="b4">d</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</font><br/>  for <font color="Olive" title="b5">d9</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Olive" title="b1">S</font>  st <font color="Olive" title="b3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Olive" title="b1">S</font> &amp; <font color="Olive" title="b5">d9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">d</font> holds <br/>( ( <font color="Olive" title="b4">d</font> <a href="binop_1.html#R1" title="BINOP_1:pred.1">is_a_left_unity_wrt</a> <font color="Olive" title="b2">f</font> implies <font color="Olive" title="b5">d9</font> <a href="binop_1.html#R1" title="BINOP_1:pred.1">is_a_left_unity_wrt</a> <font color="Olive" title="b3">g</font> ) &amp; ( <font color="Olive" title="b4">d</font> <a href="binop_1.html#R2" title="BINOP_1:pred.2">is_a_right_unity_wrt</a> <font color="Olive" title="b2">f</font> implies <font color="Olive" title="b5">d9</font> <a href="binop_1.html#R2" title="BINOP_1:pred.2">is_a_right_unity_wrt</a> <font color="Olive" title="b3">g</font> ) &amp; ( <font color="Olive" title="b4">d</font> <a href="binop_1.html#R3" title="BINOP_1:pred.3">is_a_unity_wrt</a> <font color="Olive" title="b2">f</font> implies <font color="Olive" title="b5">d9</font> <a href="binop_1.html#R3" title="BINOP_1:pred.3">is_a_unity_wrt</a> <font color="Olive" title="b3">g</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">S</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="c1">D</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">f</font> being   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c1">D</font><br/>  for <font color="Olive" title="b2">g</font> being   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c2">S</font><br/>  for <font color="Olive" title="b3">d</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</font><br/>  for <font color="Olive" title="b4">d9</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font>  st <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c2">S</font> &amp; <font color="Olive" title="b4">d9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">d</font> holds <br/>( ( <font color="Olive" title="b3">d</font> <a href="binop_1.html#R1" title="BINOP_1:pred.1">is_a_left_unity_wrt</a> <font color="Olive" title="b1">f</font> implies <font color="Olive" title="b4">d9</font> <a href="binop_1.html#R1" title="BINOP_1:pred.1">is_a_left_unity_wrt</a> <font color="Olive" title="b2">g</font> ) &amp; ( <font color="Olive" title="b3">d</font> <a href="binop_1.html#R2" title="BINOP_1:pred.2">is_a_right_unity_wrt</a> <font color="Olive" title="b1">f</font> implies <font color="Olive" title="b4">d9</font> <a href="binop_1.html#R2" title="BINOP_1:pred.2">is_a_right_unity_wrt</a> <font color="Olive" title="b2">g</font> ) &amp; ( <font color="Olive" title="b3">d</font> <a href="binop_1.html#R3" title="BINOP_1:pred.3">is_a_unity_wrt</a> <font color="Olive" title="b1">f</font> implies <font color="Olive" title="b4">d9</font> <a href="binop_1.html#R3" title="BINOP_1:pred.3">is_a_unity_wrt</a> <font color="Olive" title="b2">g</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">f</font> be   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c1">D</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">g</font> being   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c2">S</font><br/>  for <font color="Olive" title="b2">d</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</font><br/>  for <font color="Olive" title="b3">d9</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font>  st <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c2">S</font> &amp; <font color="Olive" title="b3">d9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">d</font> holds <br/>( ( <font color="Olive" title="b2">d</font> <a href="binop_1.html#R1" title="BINOP_1:pred.1">is_a_left_unity_wrt</a> <font color="Maroon" title="c3">f</font> implies <font color="Olive" title="b3">d9</font> <a href="binop_1.html#R1" title="BINOP_1:pred.1">is_a_left_unity_wrt</a> <font color="Olive" title="b1">g</font> ) &amp; ( <font color="Olive" title="b2">d</font> <a href="binop_1.html#R2" title="BINOP_1:pred.2">is_a_right_unity_wrt</a> <font color="Maroon" title="c3">f</font> implies <font color="Olive" title="b3">d9</font> <a href="binop_1.html#R2" title="BINOP_1:pred.2">is_a_right_unity_wrt</a> <font color="Olive" title="b1">g</font> ) &amp; ( <font color="Olive" title="b2">d</font> <a href="binop_1.html#R3" title="BINOP_1:pred.3">is_a_unity_wrt</a> <font color="Maroon" title="c3">f</font> implies <font color="Olive" title="b3">d9</font> <a href="binop_1.html#R3" title="BINOP_1:pred.3">is_a_unity_wrt</a> <font color="Olive" title="b1">g</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">g</font> be   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c2">S</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">d</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</font><br/>  for <font color="Olive" title="b2">d9</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font>  st <font color="Maroon" title="c4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c2">S</font> &amp; <font color="Olive" title="b2">d9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">d</font> holds <br/>( ( <font color="Olive" title="b1">d</font> <a href="binop_1.html#R1" title="BINOP_1:pred.1">is_a_left_unity_wrt</a> <font color="Maroon" title="c3">f</font> implies <font color="Olive" title="b2">d9</font> <a href="binop_1.html#R1" title="BINOP_1:pred.1">is_a_left_unity_wrt</a> <font color="Maroon" title="c4">g</font> ) &amp; ( <font color="Olive" title="b1">d</font> <a href="binop_1.html#R2" title="BINOP_1:pred.2">is_a_right_unity_wrt</a> <font color="Maroon" title="c3">f</font> implies <font color="Olive" title="b2">d9</font> <a href="binop_1.html#R2" title="BINOP_1:pred.2">is_a_right_unity_wrt</a> <font color="Maroon" title="c4">g</font> ) &amp; ( <font color="Olive" title="b1">d</font> <a href="binop_1.html#R3" title="BINOP_1:pred.3">is_a_unity_wrt</a> <font color="Maroon" title="c3">f</font> implies <font color="Olive" title="b2">d9</font> <a href="binop_1.html#R3" title="BINOP_1:pred.3">is_a_unity_wrt</a> <font color="Maroon" title="c4">g</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c5">d</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">d9</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font>  st <font color="Maroon" title="c4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c2">S</font> &amp; <font color="Olive" title="b1">d9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">d</font> holds <br/>( ( <font color="Maroon" title="c5">d</font> <a href="binop_1.html#R1" title="BINOP_1:pred.1">is_a_left_unity_wrt</a> <font color="Maroon" title="c3">f</font> implies <font color="Olive" title="b1">d9</font> <a href="binop_1.html#R1" title="BINOP_1:pred.1">is_a_left_unity_wrt</a> <font color="Maroon" title="c4">g</font> ) &amp; ( <font color="Maroon" title="c5">d</font> <a href="binop_1.html#R2" title="BINOP_1:pred.2">is_a_right_unity_wrt</a> <font color="Maroon" title="c3">f</font> implies <font color="Olive" title="b1">d9</font> <a href="binop_1.html#R2" title="BINOP_1:pred.2">is_a_right_unity_wrt</a> <font color="Maroon" title="c4">g</font> ) &amp; ( <font color="Maroon" title="c5">d</font> <a href="binop_1.html#R3" title="BINOP_1:pred.3">is_a_unity_wrt</a> <font color="Maroon" title="c3">f</font> implies <font color="Olive" title="b1">d9</font> <a href="binop_1.html#R3" title="BINOP_1:pred.3">is_a_unity_wrt</a> <font color="Maroon" title="c4">g</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c6">d9</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c2">S</font> &amp; <font color="Maroon" title="c6">d9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">d</font> implies ( ( <font color="Maroon" title="c5">d</font> <a href="binop_1.html#R1" title="BINOP_1:pred.1">is_a_left_unity_wrt</a> <font color="Maroon" title="c3">f</font> implies <font color="Maroon" title="c6">d9</font> <a href="binop_1.html#R1" title="BINOP_1:pred.1">is_a_left_unity_wrt</a> <font color="Maroon" title="c4">g</font> ) &amp; ( <font color="Maroon" title="c5">d</font> <a href="binop_1.html#R2" title="BINOP_1:pred.2">is_a_right_unity_wrt</a> <font color="Maroon" title="c3">f</font> implies <font color="Maroon" title="c6">d9</font> <a href="binop_1.html#R2" title="BINOP_1:pred.2">is_a_right_unity_wrt</a> <font color="Maroon" title="c4">g</font> ) &amp; ( <font color="Maroon" title="c5">d</font> <a href="binop_1.html#R3" title="BINOP_1:pred.3">is_a_unity_wrt</a> <font color="Maroon" title="c3">f</font> implies <font color="Maroon" title="c6">d9</font> <a href="binop_1.html#R3" title="BINOP_1:pred.3">is_a_unity_wrt</a> <font color="Maroon" title="c4">g</font> ) ) )</span><br/>











<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:2"/><span class="lab"><font color="Green" title="E2">A1</font></span>: 
<font color="Maroon" title="c4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c2">S</font>
 <span class="kw">and </span><br/><a NAME="E2:2"/><span class="lab"><font color="Green" title="E3">A2</font></span>: 
<font color="Maroon" title="c6">d9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">d</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c5">d</font> <a href="binop_1.html#R1" title="BINOP_1:pred.1">is_a_left_unity_wrt</a> <font color="Maroon" title="c3">f</font> implies <font color="Maroon" title="c6">d9</font> <a href="binop_1.html#R1" title="BINOP_1:pred.1">is_a_left_unity_wrt</a> <font color="Maroon" title="c4">g</font> ) &amp; ( <font color="Maroon" title="c5">d</font> <a href="binop_1.html#R2" title="BINOP_1:pred.2">is_a_right_unity_wrt</a> <font color="Maroon" title="c3">f</font> implies <font color="Maroon" title="c6">d9</font> <a href="binop_1.html#R2" title="BINOP_1:pred.2">is_a_right_unity_wrt</a> <font color="Maroon" title="c4">g</font> ) &amp; ( <font color="Maroon" title="c5">d</font> <a href="binop_1.html#R3" title="BINOP_1:pred.3">is_a_unity_wrt</a> <font color="Maroon" title="c3">f</font> implies <font color="Maroon" title="c6">d9</font> <a href="binop_1.html#R3" title="BINOP_1:pred.3">is_a_unity_wrt</a> <font color="Maroon" title="c4">g</font> ) )</span><br/>

<a NAME="E3:2"/><span class="lab"><font color="Green" title="E4">A3</font></span>: 
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c2">S</font></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">thus </span><a NAME="E4:2"/>
( <font color="Maroon" title="c5">d</font> <a href="binop_1.html#R1" title="BINOP_1:pred.1">is_a_left_unity_wrt</a> <font color="Maroon" title="c3">f</font> implies <font color="Maroon" title="c6">d9</font> <a href="binop_1.html#R1" title="BINOP_1:pred.1">is_a_left_unity_wrt</a> <font color="Maroon" title="c4">g</font> )
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c5">d</font> <a href="binop_1.html#R2" title="BINOP_1:pred.2">is_a_right_unity_wrt</a> <font color="Maroon" title="c3">f</font> implies <font color="Maroon" title="c6">d9</font> <a href="binop_1.html#R2" title="BINOP_1:pred.2">is_a_right_unity_wrt</a> <font color="Maroon" title="c4">g</font> ) &amp; ( <font color="Maroon" title="c5">d</font> <a href="binop_1.html#R3" title="BINOP_1:pred.3">is_a_unity_wrt</a> <font color="Maroon" title="c3">f</font> implies <font color="Maroon" title="c6">d9</font> <a href="binop_1.html#R3" title="BINOP_1:pred.3">is_a_unity_wrt</a> <font color="Maroon" title="c4">g</font> ) )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:2_1"/><span class="lab"><font color="Green" title="E5">A4</font></span>: 
 for <font color="Olive" title="b1">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</font> holds  <font color="Maroon" title="c3">f</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c5">d</font>,<font color="Olive" title="b1">a</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font>
 ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="binop_1.html#D16" onmouseover="rs('binop_1/D16')" onmouseout="rh()">BINOP_1:def 16</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">d9</font> <a href="binop_1.html#R1" title="BINOP_1:pred.1">is_a_left_unity_wrt</a> <font color="Maroon" title="c4">g</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c7">a</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="binop_1.html#D16" onmouseover="rs('binop_1/D16')" onmouseout="rh()">BINOP_1:def 16</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">g</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c6">d9</font>,<font color="Maroon" title="c7">a</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a</font></span><br/>

<span class="kw">thus </span><font color="Maroon" title="c4">g</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c6">d9</font>,<font color="Maroon" title="c7">a</font>) = 
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c6">d9</font>,<font color="Maroon" title="c7">a</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2"><span class="lab"><font color="Green" title="E2">A1</font></span></a>, <a class="txt" href="#E3:2"><span class="lab"><font color="Green" title="E4">A3</font></span></a>, <a class="ref" href="funct_1.html#T47" onmouseover="rs('funct_1/T47')" onmouseout="rh()">FUNCT_1:47</a></span>
<br/>.= 
<font color="Maroon" title="c3">f</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c7">a</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2"><span class="lab"><font color="Green" title="E3">A2</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c7">a</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1"><span class="lab"><font color="Green" title="E5">A4</font></span></a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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">thus </span><a NAME="E5:2"/>
( <font color="Maroon" title="c5">d</font> <a href="binop_1.html#R2" title="BINOP_1:pred.2">is_a_right_unity_wrt</a> <font color="Maroon" title="c3">f</font> implies <font color="Maroon" title="c6">d9</font> <a href="binop_1.html#R2" title="BINOP_1:pred.2">is_a_right_unity_wrt</a> <font color="Maroon" title="c4">g</font> )
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">d</font> <a href="binop_1.html#R3" title="BINOP_1:pred.3">is_a_unity_wrt</a> <font color="Maroon" title="c3">f</font> implies <font color="Maroon" title="c6">d9</font> <a href="binop_1.html#R3" title="BINOP_1:pred.3">is_a_unity_wrt</a> <font color="Maroon" title="c4">g</font> )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:2_2"/><span class="lab"><font color="Green" title="E5">A5</font></span>: 
 for <font color="Olive" title="b1">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</font> holds  <font color="Maroon" title="c3">f</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Olive" title="b1">a</font>,<font color="Maroon" title="c5">d</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font>
 ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="binop_1.html#D17" onmouseover="rs('binop_1/D17')" onmouseout="rh()">BINOP_1:def 17</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">d9</font> <a href="binop_1.html#R2" title="BINOP_1:pred.2">is_a_right_unity_wrt</a> <font color="Maroon" title="c4">g</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c7">a</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="binop_1.html#D17" onmouseover="rs('binop_1/D17')" onmouseout="rh()">BINOP_1:def 17</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">g</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c6">d9</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a</font></span><br/>

<span class="kw">thus </span><font color="Maroon" title="c4">g</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c6">d9</font>) = 
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c6">d9</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2"><span class="lab"><font color="Green" title="E2">A1</font></span></a>, <a class="txt" href="#E3:2"><span class="lab"><font color="Green" title="E4">A3</font></span></a>, <a class="ref" href="funct_1.html#T47" onmouseover="rs('funct_1/T47')" onmouseout="rh()">FUNCT_1:47</a></span>
<br/>.= 
<font color="Maroon" title="c3">f</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c5">d</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2"><span class="lab"><font color="Green" title="E3">A2</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c7">a</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_2"><span class="lab"><font color="Green" title="E5">A5</font></span></a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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><a NAME="E6:2"/><span class="lab"><font color="Green" title="E5">A6</font></span>: 
<font color="Maroon" title="c5">d</font> <a href="binop_1.html#R3" title="BINOP_1:pred.3">is_a_unity_wrt</a> <font color="Maroon" title="c3">f</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">d9</font> <a href="binop_1.html#R3" title="BINOP_1:pred.3">is_a_unity_wrt</a> <font color="Maroon" title="c4">g</font></span><br/>

<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_3"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">s</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font> holds <br/> ( <font color="Maroon" title="c4">g</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c6">d9</font>,<font color="Olive" title="b1">s</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">s</font> &amp; <font color="Maroon" title="c4">g</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Olive" title="b1">s</font>,<font color="Maroon" title="c6">d9</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">s</font> )</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">s</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">g</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c6">d9</font>,<font color="Maroon" title="c7">s</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">s</font> &amp; <font color="Maroon" title="c4">g</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c7">s</font>,<font color="Maroon" title="c6">d9</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">s</font> )</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c8">s9</font> = <font color="Maroon" title="c7">s</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</font> ;<br/><a NAME="E2:2_3"/><span class="lab"><font color="Green" title="E6">A7</font></span>: 
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c2">S</font></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/><a NAME="E3:2_3"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c6">d9</font>,<font color="Maroon" title="c7">s</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c2">S</font></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span>
 ;<br/><span class="kw">hence </span><font color="Maroon" title="c4">g</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c6">d9</font>,<font color="Maroon" title="c7">s</font>) = 
<font color="Maroon" title="c3">f</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c8">s9</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2"><span class="lab"><font color="Green" title="E2">A1</font></span></a>, <a class="txt" href="#E2:2"><span class="lab"><font color="Green" title="E3">A2</font></span></a>, <a class="txt" href="#E2:2_3"><span class="lab"><font color="Green" title="E6">A7</font></span></a>, <a class="ref" href="funct_1.html#T47" onmouseover="rs('funct_1/T47')" onmouseout="rh()">FUNCT_1:47</a></span>
<br/>.= 
<font color="Maroon" title="c7">s</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:2"><span class="lab"><font color="Green" title="E5">A6</font></span></a>, <a class="ref" href="binop_1.html#T3" onmouseover="rs('binop_1/T3')" onmouseout="rh()">BINOP_1:3</a></span>
;<br/> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">g</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c7">s</font>,<font color="Maroon" title="c6">d9</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">s</font></span><br/><a NAME="E5:2_3"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">s</font>,<font color="Maroon" title="c6">d9</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c2">S</font></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span>
 ;<br/><span class="kw">hence </span><font color="Maroon" title="c4">g</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c7">s</font>,<font color="Maroon" title="c6">d9</font>) = 
<font color="Maroon" title="c3">f</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c8">s9</font>,<font color="Maroon" title="c5">d</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2"><span class="lab"><font color="Green" title="E2">A1</font></span></a>, <a class="txt" href="#E2:2"><span class="lab"><font color="Green" title="E3">A2</font></span></a>, <a class="txt" href="#E2:2_3"><span class="lab"><font color="Green" title="E6">A7</font></span></a>, <a class="ref" href="funct_1.html#T47" onmouseover="rs('funct_1/T47')" onmouseout="rh()">FUNCT_1:47</a></span>
<br/>.= 
<font color="Maroon" title="c7">s</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:2"><span class="lab"><font color="Green" title="E5">A6</font></span></a>, <a class="ref" href="binop_1.html#T3" onmouseover="rs('binop_1/T3')" onmouseout="rh()">BINOP_1:3</a></span>
;<br/> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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">hence </span><a NAME="E8:2"/>
<font color="Maroon" title="c6">d9</font> <a href="binop_1.html#R3" title="BINOP_1:pred.3">is_a_unity_wrt</a> <font color="Maroon" title="c4">g</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="binop_1.html#T3" onmouseover="rs('binop_1/T3')" onmouseout="rh()">BINOP_1:3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
