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<a NAME="E1:89_1_1"/><span class="lab"><font color="Green" title="E41">A1</font></span>: 
 <a href="funct_7.html#K5" title="FUNCT_7:func.5">firstdom</a> <font color="Maroon" title="c3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#D9" target="_self" onmouseover="rs('funct_7/D9')" onmouseout="rh()">Def8</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="89_1_1_1"><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"> ( <font color="Maroon" title="c3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p1">(<span class="default"><a href="funct_7.html#K3" title="FUNCT_7:func.3">compose</a> (<font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c2">X</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )</span></a><div class="add"><span class="kw">assume </span><a NAME="E1:89_1_1_1"/>
<font color="Maroon" title="c3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</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">  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p1">(<span class="default"><a href="funct_7.html#K3" title="FUNCT_7:func.3">compose</a> (<font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c2">X</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><br/><a NAME="E2:89_1_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  <a href="funct_7.html#K3" title="FUNCT_7:func.3">compose</a> (<font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c2">X</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <font color="Maroon" title="c2">X</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:89_1_1"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="ref" href="funct_7.html#D6" target="_self" onmouseover="rs('funct_7/D6')" onmouseout="rh()">Def5</a>, <a class="ref" href="funct_7.html#T39" target="_self" onmouseover="rs('funct_7/T39')" onmouseout="rh()">Th38</a></span>;<br/><span class="kw">hence </span><a NAME="E3:89_1_1_1"/>
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p1">(<span class="default"><a href="funct_7.html#K3" title="FUNCT_7:func.3">compose</a> (<font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c2">X</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</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"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E3:89_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A2</font></span>: 
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p1">(<span class="default"><a href="funct_7.html#K3" title="FUNCT_7:func.3">compose</a> (<font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c2">X</font>)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="funct_7.html#K6" title="FUNCT_7:func.6">lastrng</a> <font color="Maroon" title="c3">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#T59" target="_self" onmouseover="rs('funct_7/T59')" onmouseout="rh()">Th58</a></span>;<br/>
<a NAME="E4:89_1_1"/>
 <a href="funct_7.html#K6" title="FUNCT_7:func.6">lastrng</a> <font color="Maroon" title="c3">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">Y</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#D9" target="_self" onmouseover="rs('funct_7/D9')" onmouseout="rh()">Def8</a></span>;<br/>
<a NAME="E5:89_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A3</font></span>: 
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p1">(<span class="default"><a href="funct_7.html#K3" title="FUNCT_7:func.3">compose</a> (<font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c2">X</font>)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">Y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:89_1_1"><span class="lab"><font color="Green" title="E42">A2</font></span></a></span>;<br/>
<a NAME="E6:89_1_1"/>
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><a href="funct_7.html#K3" title="FUNCT_7:func.3">compose</a> (<font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c2">X</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:89_1_1"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="ref" href="funct_7.html#T62" target="_self" onmouseover="rs('funct_7/T62')" onmouseout="rh()">Th61</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:89_1_1"/>
 <a href="funct_7.html#K3" title="FUNCT_7:func.3">compose</a> (<font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c2">X</font>) is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:89_1_1"><span class="lab"><font color="Green" title="E43">A3</font></span></a>, <a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T4" onmouseover="rs('relset_1/T4')" onmouseout="rh()">RELSET_1:4</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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