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<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="17_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">  not  <a href="compos_1.html#K2" title="COMPOS_1:func.2">halt</a> <font color="Maroon" title="c1">S</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="valued_1.html#K63" title="VALUED_1:func.63">CutLastLoc</a> <font color="Maroon" title="c2">I</font></span>)</span></span></a><div class="add"><span class="kw">assume </span><a NAME="E1:17_1_1_1"/>
 <a href="compos_1.html#K2" title="COMPOS_1:func.2">halt</a> <font color="Maroon" title="c1">S</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="valued_1.html#K63" title="VALUED_1:func.63">CutLastLoc</a> <font color="Maroon" title="c2">I</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"> contradiction</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:17_1_1_1"/><span class="lab"><font color="Green" title="E8">A1</font></span>: 
<font color="Maroon" title="c3">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> <span class="p1">(<span class="default"><a href="valued_1.html#K63" title="VALUED_1:func.63">CutLastLoc</a> <font color="Maroon" title="c2">I</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:17_1_1_1"/><span class="lab"><font color="Green" title="E9">A2</font></span>: 
<span class="p1">(<span class="default"><a href="valued_1.html#K63" title="VALUED_1:func.63">CutLastLoc</a> <font color="Maroon" title="c2">I</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="compos_1.html#K2" title="COMPOS_1:func.2">halt</a> <font color="Maroon" title="c1">S</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="E5:17_1_1_1"/>
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><a href="valued_1.html#K63" title="VALUED_1:func.63">CutLastLoc</a> <font color="Maroon" title="c2">I</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">I</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T11" onmouseover="rs('relat_1/T11')" onmouseout="rh()">RELAT_1:11</a></span>;<br/><a NAME="E6:17_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E10">A3</font></span>: 
<font color="Maroon" title="c3">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="c2">I</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:17_1_1_1"><span class="lab"><font color="Green" title="E8">A1</font></span></a></span>;<br/><a NAME="E7:17_1_1_1"/><span class="lab"><font color="Green" title="E11">A4</font></span>: 
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">I</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D18" onmouseover="rs('relat_1/D18')" onmouseout="rh()">RELAT_1:def 18</a></span>;<br/><a NAME="E8:17_1_1_1"/>
<font color="Maroon" title="c2">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="compos_1.html#K2" title="COMPOS_1:func.2">halt</a> <font color="Maroon" title="c1">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:17_1_1_1"><span class="lab"><font color="Green" title="E9">A2</font></span></a>, <a class="txt" href="#E3:17_1_1_1"><span class="lab"><font color="Green" title="E8">A1</font></span></a>, <a class="ref" href="grfunc_1.html#T2" onmouseover="rs('grfunc_1/T2')" onmouseout="rh()">GRFUNC_1:2</a></span>;<br/><a NAME="E9:17_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="valued_1.html#K62" title="VALUED_1:func.62">LastLoc</a> <font color="Maroon" title="c2">I</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:17_1_1_1"><span class="lab"><font color="Green" title="E10">A3</font></span></a>, <a class="txt" href="#E7:17_1_1_1"><span class="lab"><font color="Green" title="E11">A4</font></span></a>, <a class="ref" href="compos_1.html#D15" onmouseover="rs('compos_1/D15')" onmouseout="rh()">COMPOS_1:def 15</a></span>;<br/><span class="kw">hence </span><a NAME="E10:17_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:17_1_1_1"><span class="lab"><font color="Green" title="E8">A1</font></span></a>, <a class="ref" href="compos_2.html#T17" target="_self" onmouseover="rs('compos_2/T17')" onmouseout="rh()">Th17</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="E2:17_1_1"/>
 <a href="valued_1.html#K63" title="VALUED_1:func.63">CutLastLoc</a> <font color="Maroon" title="c2">I</font> is  <a href="compos_1.html#V2" title="COMPOS_1:attr.2">halt-free</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/>


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