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<span class="kw">let </span><font color="Maroon" title="c1">X</font> be   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="struct_0.html#L1" title="STRUCT_0:struct.1">1-sorted</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">I</font> being   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> <br/>  for <font color="Olive" title="b2">J</font> being   <a href="waybel_3.html#V4" title="WAYBEL_3:attr.4">non-Empty</a>   <a href="waybel18.html#V1" title="WAYBEL18:attr.1">TopStruct-yielding</a>  <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of <font color="Olive" title="b1">I</font><br/>  for <font color="Olive" title="b3">f</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><font color="Olive" title="b1">I</font> <a href="waybel26.html#NK2" title="WAYBEL26:NK.2">-TOP_prod</a> <font color="Olive" title="b2">J</font></span>)</span><br/>  for <font color="Olive" title="b4">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b1">I</font> holds  <span class="p1">(<span class="default"><a href="funct_6.html#K9" title="FUNCT_6:func.9">commute</a> <font color="Olive" title="b3">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="waybel18.html#K6" title="WAYBEL18:func.6">proj</a> (<font color="Olive" title="b2">J</font>,<font color="Olive" title="b4">i</font>)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b3">f</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">I</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">J</font> being   <a href="waybel_3.html#V4" title="WAYBEL_3:attr.4">non-Empty</a>   <a href="waybel18.html#V1" title="WAYBEL18:attr.1">TopStruct-yielding</a>  <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of <font color="Maroon" title="c2">I</font><br/>  for <font color="Olive" title="b2">f</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="waybel26.html#NK2" title="WAYBEL26:NK.2">-TOP_prod</a> <font color="Olive" title="b1">J</font></span>)</span><br/>  for <font color="Olive" title="b3">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">I</font> holds  <span class="p1">(<span class="default"><a href="funct_6.html#K9" title="FUNCT_6:func.9">commute</a> <font color="Olive" title="b2">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="waybel18.html#K6" title="WAYBEL18:func.6">proj</a> (<font color="Olive" title="b1">J</font>,<font color="Olive" title="b3">i</font>)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b2">f</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">J</font> be   <a href="waybel_3.html#V4" title="WAYBEL_3:attr.4">non-Empty</a>   <a href="waybel18.html#V1" title="WAYBEL18:attr.1">TopStruct-yielding</a>  <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of <font color="Maroon" title="c2">I</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="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="waybel26.html#NK2" title="WAYBEL26:NK.2">-TOP_prod</a> <font color="Maroon" title="c3">J</font></span>)</span><br/>  for <font color="Olive" title="b2">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">I</font> holds  <span class="p1">(<span class="default"><a href="funct_6.html#K9" title="FUNCT_6:func.9">commute</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="waybel18.html#K6" title="WAYBEL18:func.6">proj</a> (<font color="Maroon" title="c3">J</font>,<font color="Olive" title="b2">i</font>)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b1">f</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">f</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="waybel26.html#NK2" title="WAYBEL26:NK.2">-TOP_prod</a> <font color="Maroon" title="c3">J</font></span>)</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">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">I</font> holds  <span class="p1">(<span class="default"><a href="funct_6.html#K9" title="FUNCT_6:func.9">commute</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="waybel18.html#K6" title="WAYBEL18:func.6">proj</a> (<font color="Maroon" title="c3">J</font>,<font color="Olive" title="b1">i</font>)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</font></span><br/>







<a NAME="E1:39"/><span class="lab"><font color="Green" title="E32">A1</font></span>: 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="waybel26.html#NK2" title="WAYBEL26:NK.2">-TOP_prod</a> <font color="Maroon" title="c3">J</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K14" title="PRALG_1:func.14">Carrier</a> <font color="Maroon" title="c3">J</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel18.html#D3" onmouseover="rs('waybel18/D3')" onmouseout="rh()">WAYBEL18:def 3</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c5">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">I</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"> <span class="p1">(<span class="default"><a href="funct_6.html#K9" title="FUNCT_6:func.9">commute</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="waybel18.html#K6" title="WAYBEL18:func.6">proj</a> (<font color="Maroon" title="c3">J</font>,<font color="Maroon" title="c5">i</font>)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</font></span><br/>

<a NAME="E2:39"/><span class="lab"><font color="Green" title="E33">A2</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K14" title="PRALG_1:func.14">Carrier</a> <font color="Maroon" title="c3">J</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">I</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D2" onmouseover="rs('partfun1/D2')" onmouseout="rh()">PARTFUN1:def 2</a></span>;<br/>
<a NAME="E3:39"/><span class="lab"><font color="Green" title="E34">A3</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c4">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> (<font color="Maroon" title="c2">I</font>,<span class="p1">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K14" title="PRALG_1:func.14">Carrier</a> <font color="Maroon" title="c3">J</font></span>)</span></span>)</span>)
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="39_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">g</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</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"> (  not <font color="Maroon" title="c6">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c4">f</font> or <font color="Maroon" title="c6">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> (<font color="Maroon" title="c2">I</font>,<span class="p1">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K14" title="PRALG_1:func.14">Carrier</a> <font color="Maroon" title="c3">J</font></span>)</span></span>)</span>) )</span><br/>

<span class="kw">assume </span><a NAME="E1:39_1"/>
<font color="Maroon" title="c6">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c4">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">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> (<font color="Maroon" title="c2">I</font>,<span class="p1">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K14" title="PRALG_1:func.14">Carrier</a> <font color="Maroon" title="c3">J</font></span>)</span></span>)</span>)</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">h</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E3:39_1"/><span class="lab"><font color="Green" title="E35">A4</font></span>: 
<font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">h</font>
 <span class="kw">and </span><br/><a NAME="E4:39_1"/><span class="lab"><font color="Green" title="E36">A5</font></span>: 
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c7">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">I</font>
 <span class="kw">and </span><br/><a NAME="E5:39_1"/><span class="lab"><font color="Green" title="E37">A6</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> holds <br/><font color="Maroon" title="c7">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K14" title="PRALG_1:func.14">Carrier</a> <font color="Maroon" title="c3">J</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="txt" href="#E2:39"><span class="lab"><font color="Green" title="E33">A2</font></span></a>, <a class="ref" href="card_3.html#D5" onmouseover="rs('card_3/D5')" onmouseout="rh()">CARD_3:def 5</a></span>;<br/>
<a NAME="E6:39_1"/>
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c7">h</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K14" title="PRALG_1:func.14">Carrier</a> <font color="Maroon" title="c3">J</font></span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="39_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</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"> (  not <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c7">h</font> or <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K14" title="PRALG_1:func.14">Carrier</a> <font color="Maroon" title="c3">J</font></span>)</span> )</span><br/>

<a NAME="E1:39_1_1"/><span class="lab"><font color="Green" title="E38">A7</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K14" title="PRALG_1:func.14">Carrier</a> <font color="Maroon" title="c3">J</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">I</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D2" onmouseover="rs('partfun1/D2')" onmouseout="rh()">PARTFUN1:def 2</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:39_1_1"/>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c7">h</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="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K14" title="PRALG_1:func.14">Carrier</a> <font color="Maroon" title="c3">J</font></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E4:39_1_1"/><span class="lab"><font color="Green" title="E39">A8</font></span>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c7">h</font>
 <span class="kw">and </span><br/><a NAME="E5:39_1_1"/><span class="lab"><font color="Green" title="E40">A9</font></span>: 
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/>
<a NAME="E6:39_1_1"/>
<font color="Maroon" title="c7">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K14" title="PRALG_1:func.14">Carrier</a> <font color="Maroon" title="c3">J</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:39_1"><span class="lab"><font color="Green" title="E36">A5</font></span></a>, <a class="txt" href="#E5:39_1"><span class="lab"><font color="Green" title="E37">A6</font></span></a>, <a class="txt" href="#E4:39_1_1"><span class="lab"><font color="Green" title="E39">A8</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:39_1_1"/>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K14" title="PRALG_1:func.14">Carrier</a> <font color="Maroon" title="c3">J</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:39_1"><span class="lab"><font color="Green" title="E36">A5</font></span></a>, <a class="txt" href="#E4:39_1_1"><span class="lab"><font color="Green" title="E39">A8</font></span></a>, <a class="txt" href="#E5:39_1_1"><span class="lab"><font color="Green" title="E40">A9</font></span></a>, <a class="txt" href="#E1:39_1_1"><span class="lab"><font color="Green" title="E38">A7</font></span></a>, <a class="ref" href="card_5.html#T2" onmouseover="rs('card_5/T2')" onmouseout="rh()">CARD_5:2</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">hence </span><a NAME="E7:39_1"/>
<font color="Maroon" title="c6">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> (<font color="Maroon" title="c2">I</font>,<span class="p1">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K14" title="PRALG_1:func.14">Carrier</a> <font color="Maroon" title="c3">J</font></span>)</span></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:39_1"><span class="lab"><font color="Green" title="E35">A4</font></span></a>, <a class="txt" href="#E4:39_1"><span class="lab"><font color="Green" title="E36">A5</font></span></a>, <a class="ref" href="funct_2.html#D2" onmouseover="rs('funct_2/D2')" onmouseout="rh()">FUNCT_2:def 2</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>
<a NAME="E4:39"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>
 
<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="E5:39"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A10</font></span>: 
<font color="Maroon" title="c4">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> (<font color="Maroon" title="c2">I</font>,<span class="p2">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p3">(<span class="default"><a href="pralg_1.html#K14" title="PRALG_1:func.14">Carrier</a> <font color="Maroon" title="c3">J</font></span>)</span></span>)</span>)</span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:39"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="ref" href="funct_2.html#D2" onmouseover="rs('funct_2/D2')" onmouseout="rh()">FUNCT_2:def 2</a></span>;<br/>
<a NAME="E6:39"/><span class="kw">then </span>
 <a href="funct_6.html#K9" title="FUNCT_6:func.9">commute</a> <font color="Maroon" title="c4">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> (<font color="Maroon" title="c2">I</font>,<span class="p1">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,<span class="p2">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p3">(<span class="default"><a href="pralg_1.html#K14" title="PRALG_1:func.14">Carrier</a> <font color="Maroon" title="c3">J</font></span>)</span></span>)</span>)</span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_6.html#T55" onmouseover="rs('funct_6/T55')" onmouseout="rh()">FUNCT_6:55</a></span>;<br/>
<a NAME="E7:39"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A11</font></span>: 
 ex <font color="Olive" title="b1">g</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>(  <a href="funct_6.html#K9" title="FUNCT_6:func.9">commute</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> &amp;  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">I</font> &amp;  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Olive" title="b1">g</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K14" title="PRALG_1:func.14">Carrier</a> <font color="Maroon" title="c3">J</font></span>)</span></span>)</span>) )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D2" onmouseover="rs('funct_2/D2')" onmouseout="rh()">FUNCT_2:def 2</a></span>;<br/>
<a NAME="E8:39"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="funct_6.html#K9" title="FUNCT_6:func.9">commute</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p1">(<span class="default"><a href="funct_6.html#K9" title="FUNCT_6:func.9">commute</a> <font color="Maroon" title="c4">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">g</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E10:39"/><span class="lab"><font color="Green" title="E37">A12</font></span>: 
<span class="p1">(<span class="default"><a href="funct_6.html#K9" title="FUNCT_6:func.9">commute</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g</font>
 <span class="kw">and </span><br/><a NAME="E11:39"/><span class="lab"><font color="Green" title="E38">A13</font></span>: 
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>
 <span class="kw">and </span><br/><a NAME="E12:39"/>
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c6">g</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K14" title="PRALG_1:func.14">Carrier</a> <font color="Maroon" title="c3">J</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:39"><span class="lab"><font color="Green" title="E36">A11</font></span></a>, <a class="ref" href="funct_2.html#D2" onmouseover="rs('funct_2/D2')" onmouseout="rh()">FUNCT_2:def 2</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E39">A14</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="39_2"><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">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font> holds <br/><font color="Maroon" title="c6">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel18.html#K6" title="WAYBEL18:func.6">proj</a> (<font color="Maroon" title="c3">J</font>,<font color="Maroon" title="c5">i</font>)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font> implies <font color="Maroon" title="c6">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel18.html#K6" title="WAYBEL18:func.6">proj</a> (<font color="Maroon" title="c3">J</font>,<font color="Maroon" title="c5">i</font>)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> )</span><br/><a NAME="E1:39_2"/><span class="lab"><font color="Green" title="E40">A15</font></span>: 
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><a href="card_3.html#K12" title="CARD_3:func.12">proj</a> (<span class="p2">(<span class="default"><a href="pralg_1.html#K14" title="PRALG_1:func.14">Carrier</a> <font color="Maroon" title="c3">J</font></span>)</span>,<font color="Maroon" title="c5">i</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K14" title="PRALG_1:func.14">Carrier</a> <font color="Maroon" title="c3">J</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_3.html#D16" onmouseover="rs('card_3/D16')" onmouseout="rh()">CARD_3:def 16</a></span>;<br/><span class="kw">assume </span><a NAME="E2:39_2"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</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">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel18.html#K6" title="WAYBEL18:func.6">proj</a> (<font color="Maroon" title="c3">J</font>,<font color="Maroon" title="c5">i</font>)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font></span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">a</font> = <font color="Maroon" title="c7">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font> ;<br/><span class="kw">consider </span><font color="Maroon" title="c9">h</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E5:39_2"/><span class="lab"><font color="Green" title="E41">A16</font></span>: 
<font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">h</font>
 <span class="kw">and </span><br/><a NAME="E6:39_2"/>
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c9">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">I</font>
 <span class="kw">and </span><br/><a NAME="E7:39_2"/>
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> holds <br/><font color="Maroon" title="c9">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K14" title="PRALG_1:func.14">Carrier</a> <font color="Maroon" title="c3">J</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="txt" href="#E2:39"><span class="lab"><font color="Green" title="E33">A2</font></span></a>, <a class="ref" href="card_3.html#D5" onmouseover="rs('card_3/D5')" onmouseout="rh()">CARD_3:def 5</a></span>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel18.html#K6" title="WAYBEL18:func.6">proj</a> (<font color="Maroon" title="c3">J</font>,<font color="Maroon" title="c5">i</font>)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">a</font> = 
<span class="p1">(<span class="default"><a href="waybel18.html#K6" title="WAYBEL18:func.6">proj</a> (<font color="Maroon" title="c3">J</font>,<font color="Maroon" title="c5">i</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">a</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T15" onmouseover="rs('funct_2/T15')" onmouseout="rh()">FUNCT_2:15</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="card_3.html#K12" title="CARD_3:func.12">proj</a> (<span class="p2">(<span class="default"><a href="pralg_1.html#K14" title="PRALG_1:func.14">Carrier</a> <font color="Maroon" title="c3">J</font></span>)</span>,<font color="Maroon" title="c5">i</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">a</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel18.html#D4" onmouseover="rs('waybel18/D4')" onmouseout="rh()">WAYBEL18:def 4</a></span>
<br/>.= 
<font color="Maroon" title="c9">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="txt" href="#E5:39_2"><span class="lab"><font color="Green" title="E41">A16</font></span></a>, <a class="txt" href="#E1:39_2"><span class="lab"><font color="Green" title="E40">A15</font></span></a>, <a class="ref" href="card_3.html#D16" onmouseover="rs('card_3/D16')" onmouseout="rh()">CARD_3:def 16</a></span>
;<br/><span class="kw">hence </span><a NAME="E9:39_2"/>
<font color="Maroon" title="c6">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel18.html#K6" title="WAYBEL18:func.6">proj</a> (<font color="Maroon" title="c3">J</font>,<font color="Maroon" title="c5">i</font>)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:39"><span class="lab"><font color="Green" title="E35">A10</font></span></a>, <a class="txt" href="#E10:39"><span class="lab"><font color="Green" title="E37">A12</font></span></a>, <a class="txt" href="#E5:39_2"><span class="lab"><font color="Green" title="E41">A16</font></span></a>, <a class="ref" href="funct_6.html#T56" onmouseover="rs('funct_6/T56')" onmouseout="rh()">FUNCT_6:56</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>
<a NAME="E14:39"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel18.html#K6" title="WAYBEL18:func.6">proj</a> (<font color="Maroon" title="c3">J</font>,<font color="Maroon" title="c5">i</font>)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>
 
<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">hence </span><a NAME="E15:39"/>
<span class="p1">(<span class="default"><a href="funct_6.html#K9" title="FUNCT_6:func.9">commute</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="waybel18.html#K6" title="WAYBEL18:func.6">proj</a> (<font color="Maroon" title="c3">J</font>,<font color="Maroon" title="c5">i</font>)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:39"><span class="lab"><font color="Green" title="E37">A12</font></span></a>, <a class="txt" href="#E11:39"><span class="lab"><font color="Green" title="E38">A13</font></span></a>, <a class="txt" href="#E13:39"><span class="lab"><font color="Green" title="E39">A14</font></span></a>, <a class="ref" href="funct_1.html#T2" onmouseover="rs('funct_1/T2')" onmouseout="rh()">FUNCT_1:2</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/>


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