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<span class="kw">let </span><font color="Maroon" title="c1">X</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">f</font>, <font color="Olive" title="b2">g</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a> holds <br/> (  <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="mesfunc1.html#K3" title="MESFUNC1:func.3">+</a> <font color="Olive" title="b2">g</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="mesfunc1.html#K3" title="MESFUNC1:func.3">+</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <font color="Olive" title="b2">g</font></span>)</span> &amp;  <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">-</a> <font color="Olive" title="b2">g</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="mesfunc1.html#K3" title="MESFUNC1:func.3">+</a> <font color="Olive" title="b2">g</font> &amp;  <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">-</a> <font color="Olive" title="b2">g</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b2">g</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">-</a> <font color="Olive" title="b1">f</font> &amp;  <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="mesfunc1.html#K3" title="MESFUNC1:func.3">+</a> <font color="Olive" title="b2">g</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b1">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">-</a> <font color="Olive" title="b2">g</font> &amp;  <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="mesfunc1.html#K3" title="MESFUNC1:func.3">+</a> <font color="Olive" title="b2">g</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b1">f</font> <a href="mesfunc1.html#K3" title="MESFUNC1:func.3">+</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <font color="Olive" title="b2">g</font></span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">f</font>, <font color="Maroon" title="c3">g</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</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="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="mesfunc1.html#K3" title="MESFUNC1:func.3">+</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mesfunc1.html#K3" title="MESFUNC1:func.3">+</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <font color="Maroon" title="c3">g</font></span>)</span> &amp;  <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">-</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mesfunc1.html#K3" title="MESFUNC1:func.3">+</a> <font color="Maroon" title="c3">g</font> &amp;  <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">-</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c3">g</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">-</a> <font color="Maroon" title="c2">f</font> &amp;  <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mesfunc1.html#K3" title="MESFUNC1:func.3">+</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c2">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">-</a> <font color="Maroon" title="c3">g</font> &amp;  <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mesfunc1.html#K3" title="MESFUNC1:func.3">+</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c2">f</font> <a href="mesfunc1.html#K3" title="MESFUNC1:func.3">+</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <font color="Maroon" title="c3">g</font></span>)</span> )</span><br/>



<a NAME="E1:88"/><span class="lab"><font color="Green" title="E60">A1</font></span>: 
( <font color="Maroon" title="c2">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c2">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c3">g</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c3">g</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="measur11.html#E77"><span class="lab"><font color="Green" title="E58">LEM10</font></span></a></span>;<br/>
<a NAME="E2:88"/><span class="lab"><font color="Green" title="E61">A2</font></span>: 
(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <font color="Maroon" title="c2">f</font></span>)</span> &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <font color="Maroon" title="c3">g</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc1.html#D7" onmouseover="rs('mesfunc1/D7')" onmouseout="rh()">MESFUNC1:def 7</a></span>;<br/>
<a NAME="E3:88"/><span class="lab"><font color="Green" title="E62">A3</font></span>: 
(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="mesfunc1.html#K3" title="MESFUNC1:func.3">+</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="mesfunc1.html#K3" title="MESFUNC1:func.3">+</a> <font color="Maroon" title="c3">g</font></span>)</span> &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">f</font> &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">g</font> &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">-</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">-</a> <font color="Maroon" title="c3">g</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc1.html#D7" onmouseover="rs('mesfunc1/D7')" onmouseout="rh()">MESFUNC1:def 7</a></span>;<br/>
<a NAME="E4:88"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A4</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="mesfunc1.html#K3" title="MESFUNC1:func.3">+</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc1.html#D3" onmouseover="rs('mesfunc1/D3')" onmouseout="rh()">MESFUNC1:def 3</a></span>;<br/>
<a NAME="E5:88"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A5</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="mesfunc1.html#K3" title="MESFUNC1:func.3">+</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mesfunc1.html#K3" title="MESFUNC1:func.3">+</a> <span class="p2">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:88"><span class="lab"><font color="Green" title="E60">A1</font></span></a>, <a class="txt" href="#E2:88"><span class="lab"><font color="Green" title="E61">A2</font></span></a>, <a class="ref" href="mesfunc1.html#D3" onmouseover="rs('mesfunc1/D3')" onmouseout="rh()">MESFUNC1:def 3</a></span>;<br/>
<a NAME="E6:88"/><span class="lab"><font color="Green" title="E65">A6</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">-</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:88"><span class="lab"><font color="Green" title="E62">A3</font></span></a>, <a class="ref" href="mesfunc1.html#D4" onmouseover="rs('mesfunc1/D4')" onmouseout="rh()">MESFUNC1:def 4</a></span>;<br/>
<a NAME="E7:88"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A7</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">-</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mesfunc1.html#K3" title="MESFUNC1:func.3">+</a> <font color="Maroon" title="c3">g</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:88"><span class="lab"><font color="Green" title="E60">A1</font></span></a>, <a class="txt" href="#E2:88"><span class="lab"><font color="Green" title="E61">A2</font></span></a>, <a class="ref" href="mesfunc1.html#D3" onmouseover="rs('mesfunc1/D3')" onmouseout="rh()">MESFUNC1:def 3</a></span>;<br/>
<a NAME="E8:88"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">C1</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mesfunc1.html#K3" title="MESFUNC1:func.3">+</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">-</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc1.html#D7" onmouseover="rs('mesfunc1/D7')" onmouseout="rh()">MESFUNC1:def 7</a></span>;<br/>
<a NAME="E9:88"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A10</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mesfunc1.html#K3" title="MESFUNC1:func.3">+</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">-</a> <font color="Maroon" title="c3">g</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc1.html#D7" onmouseover="rs('mesfunc1/D7')" onmouseout="rh()">MESFUNC1:def 7</a></span>;<br/>
<a NAME="E10:88"/><span class="kw">then </span>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mesfunc1.html#K3" title="MESFUNC1:func.3">+</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc1.html#D4" onmouseover="rs('mesfunc1/D4')" onmouseout="rh()">MESFUNC1:def 4</a></span>;<br/>
<a NAME="E11:88"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A12</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mesfunc1.html#K3" title="MESFUNC1:func.3">+</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="mesfunc1.html#K3" title="MESFUNC1:func.3">+</a> <span class="p2">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:88"><span class="lab"><font color="Green" title="E60">A1</font></span></a>, <a class="txt" href="#E2:88"><span class="lab"><font color="Green" title="E61">A2</font></span></a>, <a class="ref" href="mesfunc1.html#D3" onmouseover="rs('mesfunc1/D3')" onmouseout="rh()">MESFUNC1:def 3</a></span>;<br/>
<a NAME="E12:88"/><span class="lab"><font color="Green" title="E70">A8</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">-</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">-</a> <font color="Maroon" title="c2">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:88"><span class="lab"><font color="Green" title="E65">A6</font></span></a>, <a class="ref" href="mesfunc1.html#D4" onmouseover="rs('mesfunc1/D4')" onmouseout="rh()">MESFUNC1:def 4</a></span>;<br/>
<a NAME="E13:88"/><span class="lab"><font color="Green" title="E71">A9</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mesfunc1.html#K3" title="MESFUNC1:func.3">+</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mesfunc1.html#K3" title="MESFUNC1:func.3">+</a> <font color="Maroon" title="c3">g</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc1.html#D7" onmouseover="rs('mesfunc1/D7')" onmouseout="rh()">MESFUNC1:def 7</a></span>;<br/>
<a NAME="E14:88"/><span class="lab"><font color="Green" title="E72">B3</font></span>: 
(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="mesfunc1.html#K3" title="MESFUNC1:func.3">+</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">f</font> &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="mesfunc1.html#K3" title="MESFUNC1:func.3">+</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">g</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:88"><span class="lab"><font color="Green" title="E63">A4</font></span></a>, <a class="ref" href="xboole_1.html#T18" onmouseover="rs('xboole_1/T18')" onmouseout="rh()">XBOOLE_1:18</a>, <a class="ref" href="xboole_1.html#T36" onmouseover="rs('xboole_1/T36')" onmouseout="rh()">XBOOLE_1:36</a></span>;<br/>
<a NAME="E15:88"/><span class="lab"><font color="Green" title="E73">B4</font></span>: 
(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">-</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">f</font> &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">-</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">g</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:88"><span class="lab"><font color="Green" title="E65">A6</font></span></a>, <a class="ref" href="xboole_1.html#T18" onmouseover="rs('xboole_1/T18')" onmouseout="rh()">XBOOLE_1:18</a>, <a class="ref" href="xboole_1.html#T36" onmouseover="rs('xboole_1/T36')" onmouseout="rh()">XBOOLE_1:36</a></span>;<br/>
<a NAME="E16:88"/><span class="lab"><font color="Green" title="E74">B5</font></span>: 
(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mesfunc1.html#K3" title="MESFUNC1:func.3">+</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <font color="Maroon" title="c2">f</font></span>)</span> &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mesfunc1.html#K3" title="MESFUNC1:func.3">+</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">g</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:88"><span class="lab"><font color="Green" title="E67">C1</font></span></a>, <a class="txt" href="#E3:88"><span class="lab"><font color="Green" title="E62">A3</font></span></a>, <a class="txt" href="#E6:88"><span class="lab"><font color="Green" title="E65">A6</font></span></a>, <a class="ref" href="xboole_1.html#T18" onmouseover="rs('xboole_1/T18')" onmouseout="rh()">XBOOLE_1:18</a>, <a class="ref" href="xboole_1.html#T36" onmouseover="rs('xboole_1/T36')" onmouseout="rh()">XBOOLE_1:36</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="88_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">  for <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="mesfunc1.html#K3" title="MESFUNC1:func.3">+</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="mesfunc1.html#K3" title="MESFUNC1:func.3">+</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</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="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mesfunc1.html#K3" title="MESFUNC1:func.3">+</a> <span class="p2">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b1">x</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</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="c4">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="mesfunc1.html#K3" title="MESFUNC1:func.3">+</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> implies <span class="p1">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="mesfunc1.html#K3" title="MESFUNC1:func.3">+</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c4">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="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mesfunc1.html#K3" title="MESFUNC1:func.3">+</a> <span class="p2">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c4">x</font> )</span><br/><span class="kw">assume </span><a NAME="E1:88_1"/><span class="lab"><font color="Green" title="E75">B2</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="mesfunc1.html#K3" title="MESFUNC1:func.3">+</a> <font color="Maroon" title="c3">g</font></span>)</span></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"> <span class="p1">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="mesfunc1.html#K3" title="MESFUNC1:func.3">+</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c4">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="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mesfunc1.html#K3" title="MESFUNC1:func.3">+</a> <span class="p2">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c4">x</font></span><br/><span class="kw">then </span><span class="p1">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="mesfunc1.html#K3" title="MESFUNC1:func.3">+</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c4">x</font> = 
 <a href="xxreal_3.html#K7" title="XXREAL_3:func.7">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="mesfunc1.html#K3" title="MESFUNC1:func.3">+</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c4">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc1.html#D7" onmouseover="rs('mesfunc1/D7')" onmouseout="rh()">MESFUNC1:def 7</a></span>
<br/>.= 
 <a href="xxreal_3.html#K7" title="XXREAL_3:func.7">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="xxreal_3.html#K9" title="XXREAL_3:func.9">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c4">x</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:88"><span class="lab"><font color="Green" title="E62">A3</font></span></a>, <a class="txt" href="#E1:88_1"><span class="lab"><font color="Green" title="E75">B2</font></span></a>, <a class="ref" href="mesfunc1.html#D3" onmouseover="rs('mesfunc1/D3')" onmouseout="rh()">MESFUNC1:def 3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="xxreal_3.html#K7" title="XXREAL_3:func.7">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c4">x</font></span>)</span></span>)</span> <a href="xxreal_3.html#K9" title="XXREAL_3:func.9">+</a> <span class="p1">(<span class="default"><a href="xxreal_3.html#K7" title="XXREAL_3:func.7">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c4">x</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_3.html#T9" onmouseover="rs('xxreal_3/T9')" onmouseout="rh()">XXREAL_3:9</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="xxreal_3.html#K9" title="XXREAL_3:func.9">+</a> <span class="p1">(<span class="default"><a href="xxreal_3.html#K7" title="XXREAL_3:func.7">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c4">x</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:88"><span class="lab"><font color="Green" title="E61">A2</font></span></a>, <a class="txt" href="#E1:88_1"><span class="lab"><font color="Green" title="E75">B2</font></span></a>, <a class="txt" href="#E14:88"><span class="lab"><font color="Green" title="E72">B3</font></span></a>, <a class="ref" href="mesfunc1.html#D7" onmouseover="rs('mesfunc1/D7')" onmouseout="rh()">MESFUNC1:def 7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="xxreal_3.html#K9" title="XXREAL_3:func.9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c4">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:88"><span class="lab"><font color="Green" title="E61">A2</font></span></a>, <a class="txt" href="#E1:88_1"><span class="lab"><font color="Green" title="E75">B2</font></span></a>, <a class="txt" href="#E14:88"><span class="lab"><font color="Green" title="E72">B3</font></span></a>, <a class="ref" href="mesfunc1.html#D7" onmouseover="rs('mesfunc1/D7')" onmouseout="rh()">MESFUNC1:def 7</a></span>
;<br/><span class="kw">hence </span><a NAME="E3:88_1"/>
<span class="p1">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="mesfunc1.html#K3" title="MESFUNC1:func.3">+</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c4">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="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mesfunc1.html#K3" title="MESFUNC1:func.3">+</a> <span class="p2">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c4">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:88_1"><span class="lab"><font color="Green" title="E75">B2</font></span></a>, <a class="txt" href="#E5:88"><span class="lab"><font color="Green" title="E64">A5</font></span></a>, <a class="ref" href="mesfunc1.html#D3" onmouseover="rs('mesfunc1/D3')" onmouseout="rh()">MESFUNC1:def 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><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E18:88"/>
 <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="mesfunc1.html#K3" title="MESFUNC1:func.3">+</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mesfunc1.html#K3" title="MESFUNC1:func.3">+</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <font color="Maroon" title="c3">g</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:88"><span class="lab"><font color="Green" title="E64">A5</font></span></a>, <a class="ref" href="partfun1.html#T5" onmouseover="rs('partfun1/T5')" onmouseout="rh()">PARTFUN1:5</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"> (  <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">-</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mesfunc1.html#K3" title="MESFUNC1:func.3">+</a> <font color="Maroon" title="c3">g</font> &amp;  <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">-</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c3">g</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">-</a> <font color="Maroon" title="c2">f</font> &amp;  <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mesfunc1.html#K3" title="MESFUNC1:func.3">+</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c2">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">-</a> <font color="Maroon" title="c3">g</font> &amp;  <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mesfunc1.html#K3" title="MESFUNC1:func.3">+</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c2">f</font> <a href="mesfunc1.html#K3" title="MESFUNC1:func.3">+</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <font color="Maroon" title="c3">g</font></span>)</span> )</span><br/>

<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="88_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="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">-</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">-</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</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="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mesfunc1.html#K3" title="MESFUNC1:func.3">+</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b1">x</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</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="c4">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">-</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> implies <span class="p1">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">-</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c4">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="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mesfunc1.html#K3" title="MESFUNC1:func.3">+</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c4">x</font> )</span><br/><span class="kw">assume </span><a NAME="E1:88_2"/><span class="lab"><font color="Green" title="E75">B2</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">-</a> <font color="Maroon" title="c3">g</font></span>)</span></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"> <span class="p1">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">-</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c4">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="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mesfunc1.html#K3" title="MESFUNC1:func.3">+</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c4">x</font></span><br/><span class="kw">then </span><span class="p1">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">-</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c4">x</font> = 
 <a href="xxreal_3.html#K7" title="XXREAL_3:func.7">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">-</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c4">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc1.html#D7" onmouseover="rs('mesfunc1/D7')" onmouseout="rh()">MESFUNC1:def 7</a></span>
<br/>.= 
 <a href="xxreal_3.html#K7" title="XXREAL_3:func.7">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="xxreal_3.html#K10" title="XXREAL_3:func.10">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c4">x</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:88"><span class="lab"><font color="Green" title="E62">A3</font></span></a>, <a class="txt" href="#E1:88_2"><span class="lab"><font color="Green" title="E75">B2</font></span></a>, <a class="ref" href="mesfunc1.html#D4" onmouseover="rs('mesfunc1/D4')" onmouseout="rh()">MESFUNC1:def 4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="xxreal_3.html#K7" title="XXREAL_3:func.7">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c4">x</font></span>)</span></span>)</span> <a href="xxreal_3.html#K9" title="XXREAL_3:func.9">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c4">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_3.html#T26" onmouseover="rs('xxreal_3/T26')" onmouseout="rh()">XXREAL_3:26</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="xxreal_3.html#K9" title="XXREAL_3:func.9">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c4">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:88"><span class="lab"><font color="Green" title="E61">A2</font></span></a>, <a class="txt" href="#E15:88"><span class="lab"><font color="Green" title="E73">B4</font></span></a>, <a class="txt" href="#E1:88_2"><span class="lab"><font color="Green" title="E75">B2</font></span></a>, <a class="ref" href="mesfunc1.html#D7" onmouseover="rs('mesfunc1/D7')" onmouseout="rh()">MESFUNC1:def 7</a></span>
;<br/><span class="kw">hence </span><a NAME="E3:88_2"/>
<span class="p1">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">-</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c4">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="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mesfunc1.html#K3" title="MESFUNC1:func.3">+</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c4">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:88_2"><span class="lab"><font color="Green" title="E75">B2</font></span></a>, <a class="txt" href="#E7:88"><span class="lab"><font color="Green" title="E66">A7</font></span></a>, <a class="ref" href="mesfunc1.html#D3" onmouseover="rs('mesfunc1/D3')" onmouseout="rh()">MESFUNC1:def 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><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E20:88"/>
 <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">-</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mesfunc1.html#K3" title="MESFUNC1:func.3">+</a> <font color="Maroon" title="c3">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:88"><span class="lab"><font color="Green" title="E66">A7</font></span></a>, <a class="ref" href="partfun1.html#T5" onmouseover="rs('partfun1/T5')" onmouseout="rh()">PARTFUN1:5</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"> (  <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">-</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c3">g</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">-</a> <font color="Maroon" title="c2">f</font> &amp;  <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mesfunc1.html#K3" title="MESFUNC1:func.3">+</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c2">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">-</a> <font color="Maroon" title="c3">g</font> &amp;  <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mesfunc1.html#K3" title="MESFUNC1:func.3">+</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c2">f</font> <a href="mesfunc1.html#K3" title="MESFUNC1:func.3">+</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <font color="Maroon" title="c3">g</font></span>)</span> )</span><br/>

<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="88_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">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">-</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">-</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">-</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b1">x</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</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="c4">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">-</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> implies <span class="p1">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">-</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">-</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c4">x</font> )</span><br/><span class="kw">assume </span><a NAME="E1:88_3"/><span class="lab"><font color="Green" title="E75">B2</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">-</a> <font color="Maroon" title="c3">g</font></span>)</span></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"> <span class="p1">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">-</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">-</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c4">x</font></span><br/><span class="kw">then </span><span class="p1">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">-</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c4">x</font> = 
 <a href="xxreal_3.html#K7" title="XXREAL_3:func.7">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">-</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c4">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc1.html#D7" onmouseover="rs('mesfunc1/D7')" onmouseout="rh()">MESFUNC1:def 7</a></span>
<br/>.= 
 <a href="xxreal_3.html#K7" title="XXREAL_3:func.7">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="xxreal_3.html#K10" title="XXREAL_3:func.10">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c4">x</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:88"><span class="lab"><font color="Green" title="E62">A3</font></span></a>, <a class="txt" href="#E1:88_3"><span class="lab"><font color="Green" title="E75">B2</font></span></a>, <a class="ref" href="mesfunc1.html#D4" onmouseover="rs('mesfunc1/D4')" onmouseout="rh()">MESFUNC1:def 4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="xxreal_3.html#K10" title="XXREAL_3:func.10">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c4">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_3.html#T26" onmouseover="rs('xxreal_3/T26')" onmouseout="rh()">XXREAL_3:26</a></span>
;<br/><span class="kw">hence </span><a NAME="E3:88_3"/>
<span class="p1">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">-</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">-</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c4">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:88_3"><span class="lab"><font color="Green" title="E75">B2</font></span></a>, <a class="txt" href="#E12:88"><span class="lab"><font color="Green" title="E70">A8</font></span></a>, <a class="ref" href="mesfunc1.html#D4" onmouseover="rs('mesfunc1/D4')" onmouseout="rh()">MESFUNC1:def 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/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E22:88"/>
 <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">-</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c3">g</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">-</a> <font color="Maroon" title="c2">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:88"><span class="lab"><font color="Green" title="E70">A8</font></span></a>, <a class="ref" href="partfun1.html#T5" onmouseover="rs('partfun1/T5')" onmouseout="rh()">PARTFUN1:5</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"> (  <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mesfunc1.html#K3" title="MESFUNC1:func.3">+</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c2">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">-</a> <font color="Maroon" title="c3">g</font> &amp;  <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mesfunc1.html#K3" title="MESFUNC1:func.3">+</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c2">f</font> <a href="mesfunc1.html#K3" title="MESFUNC1:func.3">+</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <font color="Maroon" title="c3">g</font></span>)</span> )</span><br/>

<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="88_4"><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="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mesfunc1.html#K3" title="MESFUNC1:func.3">+</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mesfunc1.html#K3" title="MESFUNC1:func.3">+</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">-</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b1">x</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</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="c4">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mesfunc1.html#K3" title="MESFUNC1:func.3">+</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> implies <span class="p1">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mesfunc1.html#K3" title="MESFUNC1:func.3">+</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">-</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c4">x</font> )</span><br/><span class="kw">assume </span><a NAME="E1:88_4"/><span class="lab"><font color="Green" title="E75">B2</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mesfunc1.html#K3" title="MESFUNC1:func.3">+</a> <font color="Maroon" title="c3">g</font></span>)</span></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"> <span class="p1">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mesfunc1.html#K3" title="MESFUNC1:func.3">+</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">-</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c4">x</font></span><br/><span class="kw">then </span><span class="p1">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mesfunc1.html#K3" title="MESFUNC1:func.3">+</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c4">x</font> = 
 <a href="xxreal_3.html#K7" title="XXREAL_3:func.7">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mesfunc1.html#K3" title="MESFUNC1:func.3">+</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c4">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc1.html#D7" onmouseover="rs('mesfunc1/D7')" onmouseout="rh()">MESFUNC1:def 7</a></span>
<br/>.= 
 <a href="xxreal_3.html#K7" title="XXREAL_3:func.7">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="xxreal_3.html#K9" title="XXREAL_3:func.9">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c4">x</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:88"><span class="lab"><font color="Green" title="E71">A9</font></span></a>, <a class="txt" href="#E1:88_4"><span class="lab"><font color="Green" title="E75">B2</font></span></a>, <a class="ref" href="mesfunc1.html#D3" onmouseover="rs('mesfunc1/D3')" onmouseout="rh()">MESFUNC1:def 3</a></span>
<br/>.= 
 <a href="xxreal_3.html#K7" title="XXREAL_3:func.7">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xxreal_3.html#K7" title="XXREAL_3:func.7">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c4">x</font></span>)</span></span>)</span> <a href="xxreal_3.html#K9" title="XXREAL_3:func.9">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c4">x</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:88"><span class="lab"><font color="Green" title="E74">B5</font></span></a>, <a class="txt" href="#E1:88_4"><span class="lab"><font color="Green" title="E75">B2</font></span></a>, <a class="ref" href="mesfunc1.html#D7" onmouseover="rs('mesfunc1/D7')" onmouseout="rh()">MESFUNC1:def 7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="xxreal_3.html#K10" title="XXREAL_3:func.10">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c4">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_3.html#T27" onmouseover="rs('xxreal_3/T27')" onmouseout="rh()">XXREAL_3:27</a></span>
;<br/><span class="kw">hence </span><a NAME="E3:88_4"/>
<span class="p1">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mesfunc1.html#K3" title="MESFUNC1:func.3">+</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">-</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c4">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:88_4"><span class="lab"><font color="Green" title="E75">B2</font></span></a>, <a class="txt" href="#E9:88"><span class="lab"><font color="Green" title="E68">A10</font></span></a>, <a class="ref" href="mesfunc1.html#D4" onmouseover="rs('mesfunc1/D4')" onmouseout="rh()">MESFUNC1:def 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/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E24:88"/>
 <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mesfunc1.html#K3" title="MESFUNC1:func.3">+</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c2">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">-</a> <font color="Maroon" title="c3">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:88"><span class="lab"><font color="Green" title="E68">A10</font></span></a>, <a class="ref" href="partfun1.html#T5" onmouseover="rs('partfun1/T5')" onmouseout="rh()">PARTFUN1:5</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">  <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mesfunc1.html#K3" title="MESFUNC1:func.3">+</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c2">f</font> <a href="mesfunc1.html#K3" title="MESFUNC1:func.3">+</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <font color="Maroon" title="c3">g</font></span>)</span></span><br/>

<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="88_5"><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="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mesfunc1.html#K3" title="MESFUNC1:func.3">+</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mesfunc1.html#K3" title="MESFUNC1:func.3">+</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="mesfunc1.html#K3" title="MESFUNC1:func.3">+</a> <span class="p2">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b1">x</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</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="c4">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mesfunc1.html#K3" title="MESFUNC1:func.3">+</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> implies <span class="p1">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mesfunc1.html#K3" title="MESFUNC1:func.3">+</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="mesfunc1.html#K3" title="MESFUNC1:func.3">+</a> <span class="p2">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c4">x</font> )</span><br/><span class="kw">assume </span><a NAME="E1:88_5"/><span class="lab"><font color="Green" title="E75">B2</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mesfunc1.html#K3" title="MESFUNC1:func.3">+</a> <font color="Maroon" title="c3">g</font></span>)</span></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"> <span class="p1">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mesfunc1.html#K3" title="MESFUNC1:func.3">+</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="mesfunc1.html#K3" title="MESFUNC1:func.3">+</a> <span class="p2">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c4">x</font></span><br/><span class="kw">then </span><span class="p1">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mesfunc1.html#K3" title="MESFUNC1:func.3">+</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c4">x</font> = 
 <a href="xxreal_3.html#K7" title="XXREAL_3:func.7">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mesfunc1.html#K3" title="MESFUNC1:func.3">+</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c4">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc1.html#D7" onmouseover="rs('mesfunc1/D7')" onmouseout="rh()">MESFUNC1:def 7</a></span>
<br/>.= 
 <a href="xxreal_3.html#K7" title="XXREAL_3:func.7">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="xxreal_3.html#K9" title="XXREAL_3:func.9">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c4">x</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:88"><span class="lab"><font color="Green" title="E71">A9</font></span></a>, <a class="txt" href="#E1:88_5"><span class="lab"><font color="Green" title="E75">B2</font></span></a>, <a class="ref" href="mesfunc1.html#D3" onmouseover="rs('mesfunc1/D3')" onmouseout="rh()">MESFUNC1:def 3</a></span>
<br/>.= 
 <a href="xxreal_3.html#K7" title="XXREAL_3:func.7">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xxreal_3.html#K7" title="XXREAL_3:func.7">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c4">x</font></span>)</span></span>)</span> <a href="xxreal_3.html#K9" title="XXREAL_3:func.9">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c4">x</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:88"><span class="lab"><font color="Green" title="E74">B5</font></span></a>, <a class="txt" href="#E1:88_5"><span class="lab"><font color="Green" title="E75">B2</font></span></a>, <a class="ref" href="mesfunc1.html#D7" onmouseover="rs('mesfunc1/D7')" onmouseout="rh()">MESFUNC1:def 7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="xxreal_3.html#K9" title="XXREAL_3:func.9">+</a> <span class="p1">(<span class="default"><a href="xxreal_3.html#K7" title="XXREAL_3:func.7">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c4">x</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_3.html#T27" onmouseover="rs('xxreal_3/T27')" onmouseout="rh()">XXREAL_3:27</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="xxreal_3.html#K9" title="XXREAL_3:func.9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c4">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:88_5"><span class="lab"><font color="Green" title="E75">B2</font></span></a>, <a class="txt" href="#E16:88"><span class="lab"><font color="Green" title="E74">B5</font></span></a>, <a class="txt" href="#E3:88"><span class="lab"><font color="Green" title="E62">A3</font></span></a>, <a class="ref" href="mesfunc1.html#D7" onmouseover="rs('mesfunc1/D7')" onmouseout="rh()">MESFUNC1:def 7</a></span>
;<br/><span class="kw">hence </span><a NAME="E3:88_5"/>
<span class="p1">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mesfunc1.html#K3" title="MESFUNC1:func.3">+</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="mesfunc1.html#K3" title="MESFUNC1:func.3">+</a> <span class="p2">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c4">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:88_5"><span class="lab"><font color="Green" title="E75">B2</font></span></a>, <a class="txt" href="#E11:88"><span class="lab"><font color="Green" title="E69">A12</font></span></a>, <a class="ref" href="mesfunc1.html#D3" onmouseover="rs('mesfunc1/D3')" onmouseout="rh()">MESFUNC1:def 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><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E26:88"/>
 <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mesfunc1.html#K3" title="MESFUNC1:func.3">+</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c2">f</font> <a href="mesfunc1.html#K3" title="MESFUNC1:func.3">+</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <font color="Maroon" title="c3">g</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:88"><span class="lab"><font color="Green" title="E69">A12</font></span></a>, <a class="ref" href="partfun1.html#T5" onmouseover="rs('partfun1/T5')" onmouseout="rh()">PARTFUN1:5</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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