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<span class="kw">let </span><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">b</font> be   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</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">x1</font>, <font color="Olive" title="b2">y1</font>, <font color="Olive" title="b3">x2</font>, <font color="Olive" title="b4">y2</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a><br/>  for <font color="Olive" title="b5">Z</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a><br/>  for <font color="Olive" title="b6">f</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  st <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font> &amp; <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></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="Olive" title="b1">x1</font> &amp; <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></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="Olive" title="b2">y1</font> &amp; <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></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="Olive" title="b3">x2</font> &amp; <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></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="Olive" title="b4">y2</font> &amp; <font color="Olive" title="b1">x1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b3">x2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp; <font color="Olive" title="b2">y1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b4">y2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp; <font color="Olive" title="b5">Z</font> is  <a href="rcomp_1.html#V3" title="RCOMP_1:attr.3">open</a>  &amp; <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b5">Z</font> &amp; <font color="Olive" title="b1">x1</font> <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a> <font color="Olive" title="b5">Z</font> &amp; <font color="Olive" title="b2">y1</font> <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a> <font color="Olive" title="b5">Z</font> &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">x1</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <span class="p3">(<span class="default"><a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Olive" title="b2">y1</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></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="Olive" title="b6">f</font> &amp; <font color="Olive" title="b3">x2</font> <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a> <font color="Olive" title="b5">Z</font> &amp; <font color="Olive" title="b4">y2</font> <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a> <font color="Olive" title="b5">Z</font> &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b3">x2</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <span class="p3">(<span class="default"><a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Olive" title="b4">y2</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></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="Olive" title="b6">f</font> holds <br/> <a href="integr1c.html#K2" title="INTEGR1C:func.2">integral</a> (<font color="Olive" title="b6">f</font>,<font color="Olive" title="b1">x1</font>,<font color="Olive" title="b2">y1</font>,<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Olive" title="b5">Z</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integr1c.html#K2" title="INTEGR1C:func.2">integral</a> (<font color="Olive" title="b6">f</font>,<font color="Olive" title="b3">x2</font>,<font color="Olive" title="b4">y2</font>,<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Olive" title="b5">Z</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c3">x1</font>, <font color="Maroon" title="c4">y1</font>, <font color="Maroon" title="c5">x2</font>, <font color="Maroon" title="c6">y2</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</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">Z</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a><br/>  for <font color="Olive" title="b2">f</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  st <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font> &amp; <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></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">x1</font> &amp; <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></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="c4">y1</font> &amp; <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></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="c5">x2</font> &amp; <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></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="c6">y2</font> &amp; <font color="Maroon" title="c3">x1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c5">x2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp; <font color="Maroon" title="c4">y1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c6">y2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp; <font color="Olive" title="b1">Z</font> is  <a href="rcomp_1.html#V3" title="RCOMP_1:attr.3">open</a>  &amp; <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">Z</font> &amp; <font color="Maroon" title="c3">x1</font> <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a> <font color="Olive" title="b1">Z</font> &amp; <font color="Maroon" title="c4">y1</font> <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a> <font color="Olive" title="b1">Z</font> &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <span class="p3">(<span class="default"><a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c4">y1</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></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="Olive" title="b2">f</font> &amp; <font color="Maroon" title="c5">x2</font> <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a> <font color="Olive" title="b1">Z</font> &amp; <font color="Maroon" title="c6">y2</font> <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a> <font color="Olive" title="b1">Z</font> &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">x2</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <span class="p3">(<span class="default"><a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c6">y2</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></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="Olive" title="b2">f</font> holds <br/> <a href="integr1c.html#K2" title="INTEGR1C:func.2">integral</a> (<font color="Olive" title="b2">f</font>,<font color="Maroon" title="c3">x1</font>,<font color="Maroon" title="c4">y1</font>,<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Olive" title="b1">Z</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integr1c.html#K2" title="INTEGR1C:func.2">integral</a> (<font color="Olive" title="b2">f</font>,<font color="Maroon" title="c5">x2</font>,<font color="Maroon" title="c6">y2</font>,<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Olive" title="b1">Z</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c7">Z</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</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> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  st <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font> &amp; <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></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">x1</font> &amp; <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></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="c4">y1</font> &amp; <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></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="c5">x2</font> &amp; <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></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="c6">y2</font> &amp; <font color="Maroon" title="c3">x1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c5">x2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp; <font color="Maroon" title="c4">y1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c6">y2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp; <font color="Maroon" title="c7">Z</font> is  <a href="rcomp_1.html#V3" title="RCOMP_1:attr.3">open</a>  &amp; <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">Z</font> &amp; <font color="Maroon" title="c3">x1</font> <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a> <font color="Maroon" title="c7">Z</font> &amp; <font color="Maroon" title="c4">y1</font> <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a> <font color="Maroon" title="c7">Z</font> &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <span class="p3">(<span class="default"><a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c4">y1</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></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="Olive" title="b1">f</font> &amp; <font color="Maroon" title="c5">x2</font> <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a> <font color="Maroon" title="c7">Z</font> &amp; <font color="Maroon" title="c6">y2</font> <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a> <font color="Maroon" title="c7">Z</font> &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">x2</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <span class="p3">(<span class="default"><a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c6">y2</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></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="Olive" title="b1">f</font> holds <br/> <a href="integr1c.html#K2" title="INTEGR1C:func.2">integral</a> (<font color="Olive" title="b1">f</font>,<font color="Maroon" title="c3">x1</font>,<font color="Maroon" title="c4">y1</font>,<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c7">Z</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integr1c.html#K2" title="INTEGR1C:func.2">integral</a> (<font color="Olive" title="b1">f</font>,<font color="Maroon" title="c5">x2</font>,<font color="Maroon" title="c6">y2</font>,<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c7">Z</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c8">f</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font> &amp; <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></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">x1</font> &amp; <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></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="c4">y1</font> &amp; <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></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="c5">x2</font> &amp; <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></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="c6">y2</font> &amp; <font color="Maroon" title="c3">x1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c5">x2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp; <font color="Maroon" title="c4">y1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c6">y2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp; <font color="Maroon" title="c7">Z</font> is  <a href="rcomp_1.html#V3" title="RCOMP_1:attr.3">open</a>  &amp; <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">Z</font> &amp; <font color="Maroon" title="c3">x1</font> <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a> <font color="Maroon" title="c7">Z</font> &amp; <font color="Maroon" title="c4">y1</font> <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a> <font color="Maroon" title="c7">Z</font> &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <span class="p3">(<span class="default"><a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c4">y1</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></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="c8">f</font> &amp; <font color="Maroon" title="c5">x2</font> <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a> <font color="Maroon" title="c7">Z</font> &amp; <font color="Maroon" title="c6">y2</font> <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a> <font color="Maroon" title="c7">Z</font> &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">x2</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <span class="p3">(<span class="default"><a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c6">y2</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></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="c8">f</font> implies  <a href="integr1c.html#K2" title="INTEGR1C:func.2">integral</a> (<font color="Maroon" title="c8">f</font>,<font color="Maroon" title="c3">x1</font>,<font color="Maroon" title="c4">y1</font>,<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c7">Z</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integr1c.html#K2" title="INTEGR1C:func.2">integral</a> (<font color="Maroon" title="c8">f</font>,<font color="Maroon" title="c5">x2</font>,<font color="Maroon" title="c6">y2</font>,<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c7">Z</font>) )</span><br/>







<span class="kw">assume </span><a NAME="E1:7"/><span class="lab"><font color="Green" title="E6">A1</font></span>: 
( <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font> &amp; <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></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">x1</font> &amp; <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></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="c4">y1</font> &amp; <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></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="c5">x2</font> &amp; <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></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="c6">y2</font> &amp; <font color="Maroon" title="c3">x1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c5">x2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp; <font color="Maroon" title="c4">y1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c6">y2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp; <font color="Maroon" title="c7">Z</font> is  <a href="rcomp_1.html#V3" title="RCOMP_1:attr.3">open</a>  &amp; <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">Z</font> &amp; <font color="Maroon" title="c3">x1</font> <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a> <font color="Maroon" title="c7">Z</font> &amp; <font color="Maroon" title="c4">y1</font> <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a> <font color="Maroon" title="c7">Z</font> &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <span class="p3">(<span class="default"><a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c4">y1</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></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="c8">f</font> &amp; <font color="Maroon" title="c5">x2</font> <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a> <font color="Maroon" title="c7">Z</font> &amp; <font color="Maroon" title="c6">y2</font> <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a> <font color="Maroon" title="c7">Z</font> &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">x2</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <span class="p3">(<span class="default"><a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c6">y2</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></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="c8">f</font> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="integr1c.html#K2" title="INTEGR1C:func.2">integral</a> (<font color="Maroon" title="c8">f</font>,<font color="Maroon" title="c3">x1</font>,<font color="Maroon" title="c4">y1</font>,<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c7">Z</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integr1c.html#K2" title="INTEGR1C:func.2">integral</a> (<font color="Maroon" title="c8">f</font>,<font color="Maroon" title="c5">x2</font>,<font color="Maroon" title="c6">y2</font>,<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c7">Z</font>)</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c9">u01</font>, <font color="Maroon" title="c10">v01</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a><span class="kw"> such that </span><br/><a NAME="E3:7"/><span class="lab"><font color="Green" title="E7">A2</font></span>: 
( <font color="Maroon" title="c9">u01</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="comseq_3.html#K5" title="COMSEQ_3:func.5">Re</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="integr1c.html#K1" title="INTEGR1C:func.1">R2-to-C</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c3">x1</font>,<font color="Maroon" title="c4">y1</font></span><a href="funct_3.html#K13" title="FUNCT_3:func.13">:&gt;</a></span> &amp; <font color="Maroon" title="c10">v01</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="comseq_3.html#K6" title="COMSEQ_3:func.6">Im</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="integr1c.html#K1" title="INTEGR1C:func.1">R2-to-C</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c3">x1</font>,<font color="Maroon" title="c4">y1</font></span><a href="funct_3.html#K13" title="FUNCT_3:func.13">:&gt;</a></span> &amp;  <a href="integr1c.html#K2" title="INTEGR1C:func.2">integral</a> (<font color="Maroon" title="c8">f</font>,<font color="Maroon" title="c3">x1</font>,<font color="Maroon" title="c4">y1</font>,<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c7">Z</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra5.html#K4" title="INTEGRA5:func.4">integral</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">u01</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">v01</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>)</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra5.html#K4" title="INTEGRA5:func.4">integral</a> (<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">v01</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">u01</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="integr1c.html#D2" target="_self" onmouseover="rs('integr1c/D2')" onmouseout="rh()">Def2</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c11">u02</font>, <font color="Maroon" title="c12">v02</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a><span class="kw"> such that </span><br/><a NAME="E5:7"/><span class="lab"><font color="Green" title="E8">A3</font></span>: 
( <font color="Maroon" title="c11">u02</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="comseq_3.html#K5" title="COMSEQ_3:func.5">Re</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="integr1c.html#K1" title="INTEGR1C:func.1">R2-to-C</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c5">x2</font>,<font color="Maroon" title="c6">y2</font></span><a href="funct_3.html#K13" title="FUNCT_3:func.13">:&gt;</a></span> &amp; <font color="Maroon" title="c12">v02</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="comseq_3.html#K6" title="COMSEQ_3:func.6">Im</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="integr1c.html#K1" title="INTEGR1C:func.1">R2-to-C</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c5">x2</font>,<font color="Maroon" title="c6">y2</font></span><a href="funct_3.html#K13" title="FUNCT_3:func.13">:&gt;</a></span> &amp;  <a href="integr1c.html#K2" title="INTEGR1C:func.2">integral</a> (<font color="Maroon" title="c8">f</font>,<font color="Maroon" title="c5">x2</font>,<font color="Maroon" title="c6">y2</font>,<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c7">Z</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra5.html#K4" title="INTEGRA5:func.4">integral</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">u02</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">x2</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">v02</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">y2</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>)</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra5.html#K4" title="INTEGRA5:func.4">integral</a> (<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c12">v02</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">x2</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c11">u02</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p5">(<span class="default"><font color="Maroon" title="c6">y2</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="integr1c.html#D2" target="_self" onmouseover="rs('integr1c/D2')" onmouseout="rh()">Def2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c13">AB</font> = <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="measure5.html#V2" title="MEASURE5:attr.2">closed_interval</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="ref" href="measure5.html#T14" onmouseover="rs('measure5/T14')" onmouseout="rh()">MEASURE5:14</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E9">A4</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="7_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b</font> or <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">b</font> )</span></a><div class="add"><span class="kw">assume </span><a NAME="E1:7_1"/><span class="lab"><font color="Green" title="E10">A5</font></span>: 
(  not <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b</font> &amp;  not <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">b</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"> contradiction</span><br/><a NAME="E2:7_1"/><span class="kw">then </span>
(  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c1">a</font> or  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">b</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T55" onmouseover="rs('xreal_1/T55')" onmouseout="rh()">XREAL_1:55</a></span>;<br/><a NAME="E3:7_1"/><span class="kw">then </span>
( <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c1">a</font></span>)</span> or  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">b</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T29" onmouseover="rs('xreal_1/T29')" onmouseout="rh()">XREAL_1:29</a></span>;<br/><a NAME="E4:7_1"/><span class="kw">then </span>
( <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">b</font> or <font color="Maroon" title="c2">b</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">b</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T29" onmouseover="rs('xreal_1/T29')" onmouseout="rh()">XREAL_1:29</a></span>;<br/><span class="kw">hence </span><a NAME="E5:7_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7_1"><span class="lab"><font color="Green" title="E10">A5</font></span></a>, <a class="txt" href="#E1:7"><span class="lab"><font color="Green" title="E6">A1</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_2"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:7_2"/>
( <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">b</font> or <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:7"><span class="lab"><font color="Green" title="E9">A4</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_2_1"><span class="kw">suppose </span></a><a NAME="E1:7_2_1"/><span class="lab"><font color="Green" title="E10">A6</font></span>: 
<font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">b</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">  <a href="integr1c.html#K2" title="INTEGR1C:func.2">integral</a> (<font color="Maroon" title="c8">f</font>,<font color="Maroon" title="c3">x1</font>,<font color="Maroon" title="c4">y1</font>,<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c7">Z</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integr1c.html#K2" title="INTEGR1C:func.2">integral</a> (<font color="Maroon" title="c8">f</font>,<font color="Maroon" title="c5">x2</font>,<font color="Maroon" title="c6">y2</font>,<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c7">Z</font>)</span><br/><div class="add"><a NAME="E2:7_2_1"/><span class="lab"><font color="Green" title="E11">A7</font></span>: 
<font color="Maroon" title="c9">u01</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c11">u02</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="7_2_1_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:7_2_1_1"/><span class="lab"><font color="Green" title="E12">A8</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">u01</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</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> <span class="p1">(<span class="default"><font color="Maroon" title="c11">u02</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="7_2_1_1_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:7_2_1_1_1"/><span class="lab"><font color="Green" title="E13">A9</font></span>: 
 for <font color="Olive" title="b1">x0</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">u01</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span> holds <br/><font color="Olive" title="b1">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">u02</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="7_2_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c14">x0</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c14">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">u01</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span> implies <font color="Maroon" title="c14">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">u02</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:7_2_1_1_1_1"/><span class="lab"><font color="Green" title="E14">A10</font></span>: 
<font color="Maroon" title="c14">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">u01</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c14">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">u02</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span></span><br/>

<a NAME="E2:7_2_1_1_1_1"/>
<font color="Maroon" title="c14">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c9">u01</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c13">AB</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7_2_1_1_1_1"><span class="lab"><font color="Green" title="E14">A10</font></span></a>, <a class="ref" href="relat_1.html#T61" onmouseover="rs('relat_1/T61')" onmouseout="rh()">RELAT_1:61</a></span>;<br/>
<a NAME="E3:7_2_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E15">A11</font></span>: 
( <font color="Maroon" title="c14">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c9">u01</font> &amp; <font color="Maroon" title="c14">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">AB</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E4:7_2_1_1_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c14">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c3">x1</font>,<font color="Maroon" title="c4">y1</font></span><a href="funct_3.html#K13" title="FUNCT_3:func.13">:&gt;</a></span> &amp; <span class="p1"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c3">x1</font>,<font color="Maroon" title="c4">y1</font></span><a href="funct_3.html#K13" title="FUNCT_3:func.13">:&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x0</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"><span class="p2">(<span class="default"><a href="comseq_3.html#K5" title="COMSEQ_3:func.5">Re</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="integr1c.html#K1" title="INTEGR1C:func.1">R2-to-C</a></span>)</span> &amp; <font color="Maroon" title="c14">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">AB</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:7"><span class="lab"><font color="Green" title="E7">A2</font></span></a>, <a class="ref" href="funct_1.html#T11" onmouseover="rs('funct_1/T11')" onmouseout="rh()">FUNCT_1:11</a></span>;<br/>
<a NAME="E5:7_2_1_1_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c14">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">AB</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x0</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x0</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="comseq_3.html#K5" title="COMSEQ_3:func.5">Re</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="integr1c.html#K1" title="INTEGR1C:func.1">R2-to-C</a></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_3.html#D7" onmouseover="rs('funct_3/D7')" onmouseout="rh()">FUNCT_3:def 7</a></span>;<br/>
<a NAME="E6:7_2_1_1_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c14">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">AB</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x0</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x0</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="comseq_3.html#K5" title="COMSEQ_3:func.5">Re</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="integr1c.html#K1" title="INTEGR1C:func.1">R2-to-C</a></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>;<br/>
<a NAME="E7:7_2_1_1_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c14">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">AB</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">x2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x0</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">y2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x0</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="comseq_3.html#K5" title="COMSEQ_3:func.5">Re</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="integr1c.html#K1" title="INTEGR1C:func.1">R2-to-C</a></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>;<br/>
<a NAME="E8:7_2_1_1_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c14">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">AB</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">x2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x0</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">y2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x0</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="comseq_3.html#K5" title="COMSEQ_3:func.5">Re</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="integr1c.html#K1" title="INTEGR1C:func.1">R2-to-C</a></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>;<br/>
<a NAME="E9:7_2_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E16">A12</font></span>: 
( <font color="Maroon" title="c14">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">AB</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">x2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x0</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">y2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x0</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="comseq_3.html#K5" title="COMSEQ_3:func.5">Re</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="integr1c.html#K1" title="INTEGR1C:func.1">R2-to-C</a></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>;<br/>
<a NAME="E10:7_2_1_1_1_1"/>
<font color="Maroon" title="c14">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c6">y2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="txt" href="#E3:7_2_1_1_1_1"><span class="lab"><font color="Green" title="E15">A11</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E11:7_2_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c5">x2</font>,<font color="Maroon" title="c6">y2</font></span><a href="funct_3.html#K13" title="FUNCT_3:func.13">:&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_3.html#D7" onmouseover="rs('funct_3/D7')" onmouseout="rh()">FUNCT_3:def 7</a></span>;<br/>
<a NAME="E12:7_2_1_1_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c14">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">AB</font> &amp; <font color="Maroon" title="c14">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c5">x2</font>,<font color="Maroon" title="c6">y2</font></span><a href="funct_3.html#K13" title="FUNCT_3:func.13">:&gt;</a></span> &amp; <span class="p1"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c5">x2</font>,<font color="Maroon" title="c6">y2</font></span><a href="funct_3.html#K13" title="FUNCT_3:func.13">:&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x0</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"><span class="p2">(<span class="default"><a href="comseq_3.html#K5" title="COMSEQ_3:func.5">Re</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="integr1c.html#K1" title="INTEGR1C:func.1">R2-to-C</a></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:7_2_1_1_1_1"><span class="lab"><font color="Green" title="E16">A12</font></span></a>, <a class="ref" href="funct_3.html#D7" onmouseover="rs('funct_3/D7')" onmouseout="rh()">FUNCT_3:def 7</a></span>;<br/>
<a NAME="E13:7_2_1_1_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c14">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">AB</font> &amp; <font color="Maroon" title="c14">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c11">u02</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:7"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="ref" href="funct_1.html#T11" onmouseover="rs('funct_1/T11')" onmouseout="rh()">FUNCT_1:11</a></span>;<br/>
<a NAME="E14:7_2_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c11">u02</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c13">AB</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E15:7_2_1_1_1_1"/>
<font color="Maroon" title="c14">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">u02</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T61" onmouseover="rs('relat_1/T61')" onmouseout="rh()">RELAT_1:61</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:7_2_1_1_1"/>
 for <font color="Olive" title="b1">x0</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">u02</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span> holds <br/><font color="Olive" title="b1">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">u01</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="7_2_1_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c14">x0</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c14">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">u02</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span> implies <font color="Maroon" title="c14">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">u01</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:7_2_1_1_1_2"/><span class="lab"><font color="Green" title="E14">A13</font></span>: 
<font color="Maroon" title="c14">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">u02</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c14">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">u01</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span></span><br/>

<a NAME="E2:7_2_1_1_1_2"/>
<font color="Maroon" title="c14">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c11">u02</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c13">AB</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7_2_1_1_1_2"><span class="lab"><font color="Green" title="E14">A13</font></span></a>, <a class="ref" href="relat_1.html#T61" onmouseover="rs('relat_1/T61')" onmouseout="rh()">RELAT_1:61</a></span>;<br/>
<a NAME="E3:7_2_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E15">A14</font></span>: 
( <font color="Maroon" title="c14">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c11">u02</font> &amp; <font color="Maroon" title="c14">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">AB</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E4:7_2_1_1_1_2"/><span class="kw">then </span>
( <font color="Maroon" title="c14">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c5">x2</font>,<font color="Maroon" title="c6">y2</font></span><a href="funct_3.html#K13" title="FUNCT_3:func.13">:&gt;</a></span> &amp; <span class="p1"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c5">x2</font>,<font color="Maroon" title="c6">y2</font></span><a href="funct_3.html#K13" title="FUNCT_3:func.13">:&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x0</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"><span class="p2">(<span class="default"><a href="comseq_3.html#K5" title="COMSEQ_3:func.5">Re</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="integr1c.html#K1" title="INTEGR1C:func.1">R2-to-C</a></span>)</span> &amp; <font color="Maroon" title="c14">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">AB</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:7"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="ref" href="funct_1.html#T11" onmouseover="rs('funct_1/T11')" onmouseout="rh()">FUNCT_1:11</a></span>;<br/>
<a NAME="E5:7_2_1_1_1_2"/><span class="kw">then </span>
( <font color="Maroon" title="c14">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">AB</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">x2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x0</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">y2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x0</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="comseq_3.html#K5" title="COMSEQ_3:func.5">Re</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="integr1c.html#K1" title="INTEGR1C:func.1">R2-to-C</a></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_3.html#D7" onmouseover="rs('funct_3/D7')" onmouseout="rh()">FUNCT_3:def 7</a></span>;<br/>
<a NAME="E6:7_2_1_1_1_2"/><span class="kw">then </span>
( <font color="Maroon" title="c14">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">AB</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">x2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x0</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">y2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x0</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="comseq_3.html#K5" title="COMSEQ_3:func.5">Re</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="integr1c.html#K1" title="INTEGR1C:func.1">R2-to-C</a></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>;<br/>
<a NAME="E7:7_2_1_1_1_2"/><span class="kw">then </span>
( <font color="Maroon" title="c14">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">AB</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x0</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x0</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="comseq_3.html#K5" title="COMSEQ_3:func.5">Re</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="integr1c.html#K1" title="INTEGR1C:func.1">R2-to-C</a></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>;<br/>
<a NAME="E8:7_2_1_1_1_2"/><span class="kw">then </span>
( <font color="Maroon" title="c14">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">AB</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x0</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x0</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="comseq_3.html#K5" title="COMSEQ_3:func.5">Re</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="integr1c.html#K1" title="INTEGR1C:func.1">R2-to-C</a></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>;<br/>
<a NAME="E9:7_2_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E16">A15</font></span>: 
( <font color="Maroon" title="c14">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">AB</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x0</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x0</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="comseq_3.html#K5" title="COMSEQ_3:func.5">Re</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="integr1c.html#K1" title="INTEGR1C:func.1">R2-to-C</a></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>;<br/>
<a NAME="E10:7_2_1_1_1_2"/>
<font color="Maroon" title="c14">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">x1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">y1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="txt" href="#E3:7_2_1_1_1_2"><span class="lab"><font color="Green" title="E15">A14</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E11:7_2_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c14">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c3">x1</font>,<font color="Maroon" title="c4">y1</font></span><a href="funct_3.html#K13" title="FUNCT_3:func.13">:&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_3.html#D7" onmouseover="rs('funct_3/D7')" onmouseout="rh()">FUNCT_3:def 7</a></span>;<br/>
<a NAME="E12:7_2_1_1_1_2"/><span class="kw">then </span>
( <font color="Maroon" title="c14">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">AB</font> &amp; <font color="Maroon" title="c14">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c3">x1</font>,<font color="Maroon" title="c4">y1</font></span><a href="funct_3.html#K13" title="FUNCT_3:func.13">:&gt;</a></span> &amp; <span class="p1"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c3">x1</font>,<font color="Maroon" title="c4">y1</font></span><a href="funct_3.html#K13" title="FUNCT_3:func.13">:&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x0</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"><span class="p2">(<span class="default"><a href="comseq_3.html#K5" title="COMSEQ_3:func.5">Re</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="integr1c.html#K1" title="INTEGR1C:func.1">R2-to-C</a></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:7_2_1_1_1_2"><span class="lab"><font color="Green" title="E16">A15</font></span></a>, <a class="ref" href="funct_3.html#D7" onmouseover="rs('funct_3/D7')" onmouseout="rh()">FUNCT_3:def 7</a></span>;<br/>
<a NAME="E13:7_2_1_1_1_2"/><span class="kw">then </span>
( <font color="Maroon" title="c14">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">AB</font> &amp; <font color="Maroon" title="c14">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c9">u01</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:7"><span class="lab"><font color="Green" title="E7">A2</font></span></a>, <a class="ref" href="funct_1.html#T11" onmouseover="rs('funct_1/T11')" onmouseout="rh()">FUNCT_1:11</a></span>;<br/>
<a NAME="E14:7_2_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c14">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c9">u01</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c13">AB</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E15:7_2_1_1_1_2"/>
<font color="Maroon" title="c14">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">u01</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T61" onmouseover="rs('relat_1/T61')" onmouseout="rh()">RELAT_1:61</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="E3:7_2_1_1_1"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">u01</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</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> <span class="p1">(<span class="default"><font color="Maroon" title="c11">u02</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7_2_1_1_1"><span class="lab"><font color="Green" title="E13">A9</font></span></a>, <a class="ref" href="tarski.html#T2" onmouseover="rs('tarski/T2')" onmouseout="rh()">TARSKI:2</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:7_2_1_1"/>
 for <font color="Olive" title="b1">x0</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">u01</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c9">u01</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">u02</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x0</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="7_2_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c14">x0</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c14">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">u01</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c9">u01</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">u02</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x0</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:7_2_1_1_2"/><span class="lab"><font color="Green" title="E13">A16</font></span>: 
<font color="Maroon" title="c14">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">u01</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c9">u01</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">u02</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x0</font></span><br/>

<a NAME="E2:7_2_1_1_2"/>
<font color="Maroon" title="c14">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c9">u01</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c13">AB</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7_2_1_1_2"><span class="lab"><font color="Green" title="E13">A16</font></span></a>, <a class="ref" href="relat_1.html#T61" onmouseover="rs('relat_1/T61')" onmouseout="rh()">RELAT_1:61</a></span>;<br/>
<a NAME="E3:7_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E14">A17</font></span>: 
( <font color="Maroon" title="c14">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">AB</font> &amp; <font color="Maroon" title="c14">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c9">u01</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c15">x0</font> = <font color="Maroon" title="c14">x0</font> as   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> ;<br/>
<a NAME="E5:7_2_1_1_2"/><span class="lab"><font color="Green" title="E15">A18</font></span>: 
<font color="Maroon" title="c13">AB</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">x1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">y1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="ref" href="xboole_1.html#T19" onmouseover="rs('xboole_1/T19')" onmouseout="rh()">XBOOLE_1:19</a></span>;<br/>
<a NAME="E6:7_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c15">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">x1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">y1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:7_2_1_1_2"><span class="lab"><font color="Green" title="E14">A17</font></span></a></span>;<br/>
<a NAME="E7:7_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c15">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c3">x1</font>,<font color="Maroon" title="c4">y1</font></span><a href="funct_3.html#K13" title="FUNCT_3:func.13">:&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_3.html#D7" onmouseover="rs('funct_3/D7')" onmouseout="rh()">FUNCT_3:def 7</a></span>;<br/>
<a NAME="E8:7_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E16">A19</font></span>: 
<font color="Maroon" title="c9">u01</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</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="comseq_3.html#K5" title="COMSEQ_3:func.5">Re</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="integr1c.html#K1" title="INTEGR1C:func.1">R2-to-C</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c3">x1</font>,<font color="Maroon" title="c4">y1</font></span><a href="funct_3.html#K13" title="FUNCT_3:func.13">:&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:7"><span class="lab"><font color="Green" title="E7">A2</font></span></a>, <a class="ref" href="funct_1.html#T13" onmouseover="rs('funct_1/T13')" onmouseout="rh()">FUNCT_1:13</a></span>;<br/>
<a NAME="E9:7_2_1_1_2"/><span class="lab"><font color="Green" title="E17">A20</font></span>: 
<font color="Maroon" title="c13">AB</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c6">y2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="ref" href="xboole_1.html#T19" onmouseover="rs('xboole_1/T19')" onmouseout="rh()">XBOOLE_1:19</a></span>;<br/>
<a NAME="E10:7_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c15">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c6">y2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:7_2_1_1_2"><span class="lab"><font color="Green" title="E14">A17</font></span></a></span>;<br/>
<a NAME="E11:7_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c15">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c5">x2</font>,<font color="Maroon" title="c6">y2</font></span><a href="funct_3.html#K13" title="FUNCT_3:func.13">:&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_3.html#D7" onmouseover="rs('funct_3/D7')" onmouseout="rh()">FUNCT_3:def 7</a></span>;<br/>
<a NAME="E12:7_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c11">u02</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</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="comseq_3.html#K5" title="COMSEQ_3:func.5">Re</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="integr1c.html#K1" title="INTEGR1C:func.1">R2-to-C</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c5">x2</font>,<font color="Maroon" title="c6">y2</font></span><a href="funct_3.html#K13" title="FUNCT_3:func.13">:&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:7"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="ref" href="funct_1.html#T13" onmouseover="rs('funct_1/T13')" onmouseout="rh()">FUNCT_1:13</a></span>;<br/>
<a NAME="E13:7_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E18">A21</font></span>: 
<font color="Maroon" title="c11">u02</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</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="comseq_3.html#K5" title="COMSEQ_3:func.5">Re</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="integr1c.html#K1" title="INTEGR1C:func.1">R2-to-C</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">x2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">y2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:7_2_1_1_2"><span class="lab"><font color="Green" title="E14">A17</font></span></a>, <a class="txt" href="#E9:7_2_1_1_2"><span class="lab"><font color="Green" title="E17">A20</font></span></a>, <a class="ref" href="funct_3.html#T48" onmouseover="rs('funct_3/T48')" onmouseout="rh()">FUNCT_3:48</a></span>;<br/>
<span class="lab"><font color="Green" title="E19">A22</font></span>: <font color="Maroon" title="c3">x1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:7_2_1_1_2"><span class="lab"><font color="Green" title="E14">A17</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>
<br/>.= 
<font color="Maroon" title="c5">x2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:7_2_1_1_2"><span class="lab"><font color="Green" title="E14">A17</font></span></a>, <a class="txt" href="#E1:7"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>
;<br/>
<span class="lab"><font color="Green" title="E20">A23</font></span>: <font color="Maroon" title="c4">y1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:7_2_1_1_2"><span class="lab"><font color="Green" title="E14">A17</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>
<br/>.= 
<font color="Maroon" title="c6">y2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:7_2_1_1_2"><span class="lab"><font color="Green" title="E14">A17</font></span></a>, <a class="txt" href="#E1:7"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>
;<br/>
<a NAME="E16:7_2_1_1_2"/>
( <font color="Maroon" title="c9">u01</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">u01</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font> &amp; <font color="Maroon" title="c11">u02</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">u02</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:7_2_1_1_2"><span class="lab"><font color="Green" title="E14">A17</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>;<br/>
<span class="kw">hence </span><a NAME="E17:7_2_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">u01</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">u02</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:7_2_1_1_2"><span class="lab"><font color="Green" title="E14">A17</font></span></a>, <a class="txt" href="#E5:7_2_1_1_2"><span class="lab"><font color="Green" title="E15">A18</font></span></a>, <a class="txt" href="#E8:7_2_1_1_2"><span class="lab"><font color="Green" title="E16">A19</font></span></a>, <a class="txt" href="#E13:7_2_1_1_2"><span class="lab"><font color="Green" title="E18">A21</font></span></a>, <a class="txt" href="#E14:7_2_1_1_2"><span class="lab"><font color="Green" title="E19">A22</font></span></a>, <a class="txt" href="#E15:7_2_1_1_2"><span class="lab"><font color="Green" title="E20">A23</font></span></a>, <a class="ref" href="funct_3.html#T48" onmouseover="rs('funct_3/T48')" onmouseout="rh()">FUNCT_3:48</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="E3:7_2_1_1"/>
<font color="Maroon" title="c9">u01</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c11">u02</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7_2_1_1"><span class="lab"><font color="Green" title="E12">A8</font></span></a>, <a class="ref" href="funct_1.html#D11" onmouseover="rs('funct_1/D11')" onmouseout="rh()">FUNCT_1:def 11</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E3:7_2_1"/><span class="lab"><font color="Green" title="E12">A24</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">x2</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="7_2_1_2"><span class="kw">proof </span></a><div class="add">

<span class="lab"><font color="Green" title="E13">A25</font></span>:  <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"><font color="Maroon" title="c3">x1</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c13">AB</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T61" onmouseover="rs('relat_1/T61')" onmouseout="rh()">RELAT_1:61</a></span>
<br/>.= 
<font color="Maroon" title="c7">Z</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c13">AB</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="ref" href="fdiff_1.html#D7" onmouseover="rs('fdiff_1/D7')" onmouseout="rh()">FDIFF_1:def 7</a></span>
<br/>.= 
<font color="Maroon" title="c13">AB</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="ref" href="xboole_1.html#T28" onmouseover="rs('xboole_1/T28')" onmouseout="rh()">XBOOLE_1:28</a></span>
;<br/>
<span class="lab"><font color="Green" title="E14">A26</font></span>:  <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"><font color="Maroon" title="c5">x2</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">x2</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c13">AB</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T61" onmouseover="rs('relat_1/T61')" onmouseout="rh()">RELAT_1:61</a></span>
<br/>.= 
<font color="Maroon" title="c7">Z</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c13">AB</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="ref" href="fdiff_1.html#D7" onmouseover="rs('fdiff_1/D7')" onmouseout="rh()">FDIFF_1:def 7</a></span>
<br/>.= 
<font color="Maroon" title="c13">AB</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="ref" href="xboole_1.html#T28" onmouseover="rs('xboole_1/T28')" onmouseout="rh()">XBOOLE_1:28</a></span>
;<br/>
<a NAME="E3:7_2_1_2"/>
 for <font color="Olive" title="b1">x0</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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"><font color="Maroon" title="c3">x1</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">x2</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x0</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="7_2_1_2_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c14">x0</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c14">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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"><font color="Maroon" title="c3">x1</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">x2</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x0</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:7_2_1_2_3"/><span class="lab"><font color="Green" title="E15">A27</font></span>: 
<font color="Maroon" title="c14">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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"><font color="Maroon" title="c3">x1</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">x2</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x0</font></span><br/>

<a NAME="E2:7_2_1_2_3"/>
<font color="Maroon" title="c14">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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">x1</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7_2_1_2_3"><span class="lab"><font color="Green" title="E15">A27</font></span></a>, <a class="ref" href="relat_1.html#T57" onmouseover="rs('relat_1/T57')" onmouseout="rh()">RELAT_1:57</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c15">x0</font> = <font color="Maroon" title="c14">x0</font> as   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> ;<br/>
<a NAME="E4:7_2_1_2_3"/><span class="lab"><font color="Green" title="E16">A28</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">x2</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">x2</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7_2_1_2_3"><span class="lab"><font color="Green" title="E15">A27</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>;<br/>
<a NAME="E5:7_2_1_2_3"/><span class="lab"><font color="Green" title="E17">A29</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> (<font color="Maroon" title="c3">x1</font>,<font color="Maroon" title="c15">x0</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="txt" href="#E1:7_2_1_2"><span class="lab"><font color="Green" title="E13">A25</font></span></a>, <a class="txt" href="#E1:7_2_1_2_3"><span class="lab"><font color="Green" title="E15">A27</font></span></a>, <a class="ref" href="fdiff_1.html#D7" onmouseover="rs('fdiff_1/D7')" onmouseout="rh()">FDIFF_1:def 7</a></span>;<br/>
<a NAME="E6:7_2_1_2_3"/><span class="lab"><font color="Green" title="E18">A30</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">x2</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> (<font color="Maroon" title="c5">x2</font>,<font color="Maroon" title="c15">x0</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="txt" href="#E1:7_2_1_2"><span class="lab"><font color="Green" title="E13">A25</font></span></a>, <a class="txt" href="#E1:7_2_1_2_3"><span class="lab"><font color="Green" title="E15">A27</font></span></a>, <a class="ref" href="fdiff_1.html#D7" onmouseover="rs('fdiff_1/D7')" onmouseout="rh()">FDIFF_1:def 7</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E19">A31</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="7_2_1_2_3_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c15">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> or <font color="Maroon" title="c15">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font> or <font color="Maroon" title="c15">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b</font> )</span></a><div class="add"><span class="kw">assume </span><a NAME="E1:7_2_1_2_3_1"/><span class="lab"><font color="Green" title="E20">A32</font></span>: 
(  not <font color="Maroon" title="c15">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> &amp;  not <font color="Maroon" title="c15">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font> &amp;  not <font color="Maroon" title="c15">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b</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"> contradiction</span><br/><a NAME="E2:7_2_1_2_3_1"/>
<font color="Maroon" title="c15">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">r</font> where <font color="Olive" title="b1">r</font> is   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> : ( <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font> ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7_2_1_2"><span class="lab"><font color="Green" title="E13">A25</font></span></a>, <a class="txt" href="#E1:7_2_1_2_3"><span class="lab"><font color="Green" title="E15">A27</font></span></a>, <a class="ref" href="rcomp_1.html#D1" onmouseover="rs('rcomp_1/D1')" onmouseout="rh()">RCOMP_1:def 1</a></span>;<br/><a NAME="E3:7_2_1_2_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E21">A33</font></span>: 
 ex <font color="Olive" title="b1">r</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">x0</font> &amp; <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font> )
 ;<br/><div><span class="lab"><font color="Green" title="E22">A34</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="7_2_1_2_3_1_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">x0</font> or <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">x0</font> )</span></a><div class="add"><span class="kw">assume </span><a NAME="E1:7_2_1_2_3_1_1"/><span class="lab"><font color="Green" title="E23">A35</font></span>: 
(  not <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">x0</font> &amp;  not <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">x0</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"> contradiction</span><br/><a NAME="E2:7_2_1_2_3_1_1"/><span class="kw">then </span>
(  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">x0</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c1">a</font> or  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c15">x0</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T55" onmouseover="rs('xreal_1/T55')" onmouseout="rh()">XREAL_1:55</a></span>;<br/><a NAME="E3:7_2_1_2_3_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">x0</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c1">a</font></span>)</span> or  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c15">x0</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T29" onmouseover="rs('xreal_1/T29')" onmouseout="rh()">XREAL_1:29</a></span>;<br/><a NAME="E4:7_2_1_2_3_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">x0</font> or <font color="Maroon" title="c15">x0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">x0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c15">x0</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T29" onmouseover="rs('xreal_1/T29')" onmouseout="rh()">XREAL_1:29</a></span>;<br/><span class="kw">hence </span><a NAME="E5:7_2_1_2_3_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:7_2_1_2_3_1"><span class="lab"><font color="Green" title="E21">A33</font></span></a>, <a class="txt" href="#E1:7_2_1_2_3_1_1"><span class="lab"><font color="Green" title="E23">A35</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><span class="lab"><font color="Green" title="E23">A36</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="7_2_1_2_3_1_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"> ( <font color="Maroon" title="c15">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b</font> or <font color="Maroon" title="c15">x0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">b</font> )</span></a><div class="add"><span class="kw">assume </span><a NAME="E1:7_2_1_2_3_1_2"/><span class="lab"><font color="Green" title="E24">A37</font></span>: 
(  not <font color="Maroon" title="c15">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b</font> &amp;  not <font color="Maroon" title="c15">x0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">b</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"> contradiction</span><br/><a NAME="E2:7_2_1_2_3_1_2"/><span class="kw">then </span>
(  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">x0</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">b</font> or  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c15">x0</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T55" onmouseover="rs('xreal_1/T55')" onmouseout="rh()">XREAL_1:55</a></span>;<br/><a NAME="E3:7_2_1_2_3_1_2"/><span class="kw">then </span>
( <font color="Maroon" title="c2">b</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">x0</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">b</font></span>)</span> or  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c15">x0</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T29" onmouseover="rs('xreal_1/T29')" onmouseout="rh()">XREAL_1:29</a></span>;<br/><a NAME="E4:7_2_1_2_3_1_2"/><span class="kw">then </span>
( <font color="Maroon" title="c2">b</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">x0</font> or <font color="Maroon" title="c15">x0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">x0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c15">x0</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T29" onmouseover="rs('xreal_1/T29')" onmouseout="rh()">XREAL_1:29</a></span>;<br/><span class="kw">hence </span><a NAME="E5:7_2_1_2_3_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:7_2_1_2_3_1"><span class="lab"><font color="Green" title="E21">A33</font></span></a>, <a class="txt" href="#E1:7_2_1_2_3_1_2"><span class="lab"><font color="Green" title="E24">A37</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E6:7_2_1_2_3_1"/>
 not <font color="Maroon" title="c15">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">q</font> where <font color="Olive" title="b1">q</font> is   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> : ( <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">q</font> &amp; <font color="Olive" title="b1">q</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">b</font> ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7_2_1_2_3_1"><span class="lab"><font color="Green" title="E20">A32</font></span></a>, <a class="ref" href="rcomp_1.html#D2" onmouseover="rs('rcomp_1/D2')" onmouseout="rh()">RCOMP_1:def 2</a></span>;<br/><span class="kw">hence </span><a NAME="E7:7_2_1_2_3_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7_2_1_2_3_1"><span class="lab"><font color="Green" title="E20">A32</font></span></a>, <a class="txt" href="#E4:7_2_1_2_3_1"><span class="lab"><font color="Green" title="E22">A34</font></span></a>, <a class="txt" href="#E5:7_2_1_2_3_1"><span class="lab"><font color="Green" title="E23">A36</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E8:7_2_1_2_3"/>
 <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> (<font color="Maroon" title="c3">x1</font>,<font color="Maroon" title="c15">x0</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> (<font color="Maroon" title="c5">x2</font>,<font color="Maroon" title="c15">x0</font>)
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="7_2_1_2_3_2"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_2_1_2_3_2_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:7_2_1_2_3_2_1"/>
( <font color="Maroon" title="c15">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font> or <font color="Maroon" title="c15">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> or <font color="Maroon" title="c15">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:7_2_1_2_3"><span class="lab"><font color="Green" title="E19">A31</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_2_1_2_3_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:7_2_1_2_3_2_1_1"/><span class="lab"><font color="Green" title="E20">A38</font></span>: 
<font color="Maroon" title="c15">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</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">  <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> (<font color="Maroon" title="c3">x1</font>,<font color="Maroon" title="c15">x0</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> (<font color="Maroon" title="c5">x2</font>,<font color="Maroon" title="c15">x0</font>)</span><br/><div class="add"><a NAME="E2:7_2_1_2_3_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c15">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">r</font> where <font color="Olive" title="b1">r</font> is   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> : ( <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font> ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7"><span class="lab"><font color="Green" title="E6">A1</font></span></a></span>;<br/><a NAME="E3:7_2_1_2_3_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E21">A39</font></span>: 
<font color="Maroon" title="c15">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rcomp_1.html#D1" onmouseover="rs('rcomp_1/D1')" onmouseout="rh()">RCOMP_1:def 1</a></span>;<br/><a NAME="E4:7_2_1_2_3_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">A40</font></span>: 
<font color="Maroon" title="c3">x1</font> <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c15">x0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="ref" href="fdiff_1.html#T9" onmouseover="rs('fdiff_1/T9')" onmouseout="rh()">FDIFF_1:9</a></span>;<br/><a NAME="E5:7_2_1_2_3_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A41</font></span>: 
( <font color="Maroon" title="c3">x1</font> <a href="fdiff_3.html#R3" title="FDIFF_3:pred.3">is_right_differentiable_in</a> <font color="Maroon" title="c15">x0</font> &amp;  <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> (<font color="Maroon" title="c3">x1</font>,<font color="Maroon" title="c15">x0</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fdiff_3.html#K2" title="FDIFF_3:func.2">Rdiff</a> (<font color="Maroon" title="c3">x1</font>,<font color="Maroon" title="c15">x0</font>) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fdiff_3.html#T22" onmouseover="rs('fdiff_3/T22')" onmouseout="rh()">FDIFF_3:22</a></span>;<br/><a NAME="E6:7_2_1_2_3_2_1_1"/>
<font color="Maroon" title="c5">x2</font> <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c15">x0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="txt" href="#E3:7_2_1_2_3_2_1_1"><span class="lab"><font color="Green" title="E21">A39</font></span></a>, <a class="ref" href="fdiff_1.html#T9" onmouseover="rs('fdiff_1/T9')" onmouseout="rh()">FDIFF_1:9</a></span>;<br/><a NAME="E7:7_2_1_2_3_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A42</font></span>: 
( <font color="Maroon" title="c5">x2</font> <a href="fdiff_3.html#R3" title="FDIFF_3:pred.3">is_right_differentiable_in</a> <font color="Maroon" title="c15">x0</font> &amp;  <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> (<font color="Maroon" title="c5">x2</font>,<font color="Maroon" title="c15">x0</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fdiff_3.html#K2" title="FDIFF_3:func.2">Rdiff</a> (<font color="Maroon" title="c5">x2</font>,<font color="Maroon" title="c15">x0</font>) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fdiff_3.html#T22" onmouseover="rs('fdiff_3/T22')" onmouseout="rh()">FDIFF_3:22</a></span>;<br/><a NAME="E8:7_2_1_2_3_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A43</font></span>: 
<font color="Maroon" title="c3">x1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c5">x2</font> <a href="fdiff_3.html#R3" title="FDIFF_3:pred.3">is_right_differentiable_in</a> <font color="Maroon" title="c15">x0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:7_2_1_2_3_2_1_1"><span class="lab"><font color="Green" title="E23">A41</font></span></a>, <a class="ref" href="fdiff_3.html#T17" onmouseover="rs('fdiff_3/T17')" onmouseout="rh()">FDIFF_3:17</a></span>;<br/><a NAME="E9:7_2_1_2_3_2_1_1"/>
 <a href="fdiff_3.html#K2" title="FDIFF_3:func.2">Rdiff</a> (<font color="Maroon" title="c3">x1</font>,<font color="Maroon" title="c15">x0</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fdiff_3.html#K2" title="FDIFF_3:func.2">Rdiff</a> (<font color="Maroon" title="c5">x2</font>,<font color="Maroon" title="c15">x0</font>)
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="7_2_1_2_3_2_1_1_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:7_2_1_2_3_2_1_1_1"/><span class="lab"><font color="Green" title="E26">A44</font></span>: 
<span class="p1">(<span class="default"><a href="fdiff_3.html#K2" title="FDIFF_3:func.2">Rdiff</a> (<font color="Maroon" title="c3">x1</font>,<font color="Maroon" title="c15">x0</font>)</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="fdiff_3.html#K2" title="FDIFF_3:func.2">Rdiff</a> (<font color="Maroon" title="c5">x2</font>,<font color="Maroon" title="c15">x0</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fdiff_3.html#K2" title="FDIFF_3:func.2">Rdiff</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c5">x2</font></span>)</span>,<font color="Maroon" title="c15">x0</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:7_2_1_2_3_2_1_1"><span class="lab"><font color="Green" title="E23">A41</font></span></a>, <a class="txt" href="#E7:7_2_1_2_3_2_1_1"><span class="lab"><font color="Green" title="E24">A42</font></span></a>, <a class="ref" href="fdiff_3.html#T17" onmouseover="rs('fdiff_3/T17')" onmouseout="rh()">FDIFF_3:17</a></span>;<br/>
<a NAME="E2:7_2_1_2_3_2_1_1_1"/>
 for <font color="Olive" title="b1">h</font> being   <a href="ordinal1.html#V9" title="ORDINAL1:attr.9">non-zero</a>   <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="fdiff_1.html#V1" title="FDIFF_1:attr.1">-convergent</a>  <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a><br/>  for <font color="Olive" title="b2">c1</font> being   <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a>  <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>  st  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Olive" title="b2">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c15">x0</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Olive" title="b2">c1</font></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"><font color="Maroon" title="c3">x1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c5">x2</font></span>)</span> &amp; (  for <font color="Olive" title="b3">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  holds <font color="Olive" title="b1">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  ) holds <br/> <a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p4">(<span class="default"><font color="Olive" title="b1">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Olive" title="b2">c1</font></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Olive" title="b2">c1</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="7_2_1_2_3_2_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c16">h</font> be   <a href="ordinal1.html#V9" title="ORDINAL1:attr.9">non-zero</a>   <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="fdiff_1.html#V1" title="FDIFF_1:attr.1">-convergent</a>  <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</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">c1</font> being   <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a>  <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>  st  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Olive" title="b1">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c15">x0</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Olive" title="b1">c1</font></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"><font color="Maroon" title="c3">x1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c5">x2</font></span>)</span> &amp; (  for <font color="Olive" title="b2">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  holds <font color="Maroon" title="c16">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  ) holds <br/> <a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Olive" title="b1">c1</font></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Olive" title="b1">c1</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c17">c1</font> be   <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a>  <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</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="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c17">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c15">x0</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c17">c1</font></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"><font color="Maroon" title="c3">x1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c5">x2</font></span>)</span> &amp; (  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  holds <font color="Maroon" title="c16">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  ) implies  <a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )</span><br/>



<span class="kw">assume </span><a NAME="E1:7_2_1_2_3_2_1_1_1_1"/><span class="lab"><font color="Green" title="E27">A45</font></span>: 
(  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c17">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c15">x0</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c17">c1</font></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"><font color="Maroon" title="c3">x1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c5">x2</font></span>)</span> &amp; (  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  holds <font color="Maroon" title="c16">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</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="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> </span><br/>

<a NAME="E2:7_2_1_2_3_2_1_1_1_1"/><span class="lab"><font color="Green" title="E28">A46</font></span>: 
( <font color="Maroon" title="c16">h</font> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a>  &amp;  <a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c16">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 
;<br/>
<a NAME="E3:7_2_1_2_3_2_1_1_1_1"/><span class="lab"><font color="Green" title="E29">A47</font></span>: 
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c15">x0</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7_2_1"><span class="lab"><font color="Green" title="E10">A6</font></span></a>, <a class="txt" href="#E1:7_2_1_2_3_2_1_1"><span class="lab"><font color="Green" title="E20">A38</font></span></a>, <a class="ref" href="xreal_1.html#T50" onmouseover="rs('xreal_1/T50')" onmouseout="rh()">XREAL_1:50</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c18">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E5:7_2_1_2_3_2_1_1_1_1"/><span class="lab"><font color="Green" title="E30">A48</font></span>: 
 for <font color="Olive" title="b1">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon" title="c18">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> holds <br/><span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c15">x0</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:7_2_1_2_3_2_1_1_1_1"><span class="lab"><font color="Green" title="E28">A46</font></span></a>, <a class="ref" href="seq_2.html#D7" onmouseover="rs('seq_2/D7')" onmouseout="rh()">SEQ_2:def 7</a></span>;<br/>
<a NAME="E6:7_2_1_2_3_2_1_1_1_1"/><span class="lab"><font color="Green" title="E31">A49</font></span>: 
 for <font color="Olive" title="b1">p1</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">p1</font> holds <br/> ex <font color="Olive" title="b2">n1</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/> for <font color="Olive" title="b3">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b2">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">m</font> holds <br/><span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">m</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">p1</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="7_2_1_2_3_2_1_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c19">p1</font> be   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</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="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c19">p1</font> implies  ex <font color="Olive" title="b1">n1</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/> for <font color="Olive" title="b2">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">m</font> holds <br/><span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">m</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c19">p1</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:7_2_1_2_3_2_1_1_1_1_1"/><span class="lab"><font color="Green" title="E32">A50</font></span>: 
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c19">p1</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">  ex <font color="Olive" title="b1">n1</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/> for <font color="Olive" title="b2">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">m</font> holds <br/><span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">m</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c19">p1</font></span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c18">n</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon" title="c18">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> holds <br/><span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c19">p1</font></span><br/>

<a NAME="E2:7_2_1_2_3_2_1_1_1_1_1"/>
 for <font color="Olive" title="b1">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon" title="c18">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="7_2_1_2_3_2_1_1_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c20">m</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c18">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">m</font> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:7_2_1_2_3_2_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E33">A51</font></span>: 
<font color="Maroon" title="c18">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">m</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> </span><br/>

<a NAME="E2:7_2_1_2_3_2_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E34">A52</font></span>: 
<font color="Maroon" title="c20">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" onmouseover="rs('ordinal1/D12')" onmouseout="rh()">ORDINAL1:def 12</a></span>;<br/>
<a NAME="E3:7_2_1_2_3_2_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E35">A53</font></span>: 
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c15">x0</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:7_2_1_2_3_2_1_1_1_1"><span class="lab"><font color="Green" title="E30">A48</font></span></a>, <a class="txt" href="#E1:7_2_1_2_3_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E33">A51</font></span></a></span>;<br/>
<a NAME="E4:7_2_1_2_3_2_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E36">A54</font></span>: 
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7_2_1_2_3_2_1_1_1_1"><span class="lab"><font color="Green" title="E27">A45</font></span></a></span>;<br/>
<a NAME="E5:7_2_1_2_3_2_1_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A55</font></span>: 
<font color="Maroon" title="c16">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c15">x0</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:7_2_1_2_3_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E35">A53</font></span></a>, <a class="ref" href="absvalue.html#D1" onmouseover="rs('absvalue/D1')" onmouseout="rh()">ABSVALUE:def 1</a></span>;<br/>
<a NAME="E6:7_2_1_2_3_2_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E38">A56</font></span>: 
<font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">x0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7_2_1_2_3_2_1_1"><span class="lab"><font color="Green" title="E20">A38</font></span></a>, <a class="txt" href="#E4:7_2_1_2_3_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E36">A54</font></span></a>, <a class="ref" href="xreal_1.html#T7" onmouseover="rs('xreal_1/T7')" onmouseout="rh()">XREAL_1:7</a></span>;<br/>
<a NAME="E7:7_2_1_2_3_2_1_1_1_1_1_1"/>
<font color="Maroon" title="c15">x0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">x0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c15">x0</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:7_2_1_2_3_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E37">A55</font></span></a>, <a class="ref" href="xreal_1.html#T7" onmouseover="rs('xreal_1/T7')" onmouseout="rh()">XREAL_1:7</a></span>;<br/>
<a NAME="E8:7_2_1_2_3_2_1_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A57</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c15">x0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c15">x0</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c15">x0</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:7_2_1_2_3_2_1_1_1_1"><span class="lab"><font color="Green" title="E29">A47</font></span></a>, <a class="ref" href="xreal_1.html#T7" onmouseover="rs('xreal_1/T7')" onmouseout="rh()">XREAL_1:7</a></span>;<br/>
<a NAME="E9:7_2_1_2_3_2_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E40">A58</font></span>: 
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">x1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">x2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="ref" href="xboole_1.html#T19" onmouseover="rs('xboole_1/T19')" onmouseout="rh()">XBOOLE_1:19</a></span>;<br/>
<a NAME="E10:7_2_1_2_3_2_1_1_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c15">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">x1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">x2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:7_2_1_2_3_2_1_1"><span class="lab"><font color="Green" title="E21">A39</font></span></a></span>;<br/>
<a NAME="E11:7_2_1_2_3_2_1_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A59</font></span>: 
<font color="Maroon" title="c15">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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">x1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c5">x2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#T12" onmouseover="rs('valued_1/T12')" onmouseout="rh()">VALUED_1:12</a></span>;<br/>
<a NAME="E12:7_2_1_2_3_2_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E42">A60</font></span>: 
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></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"><font color="Maroon" title="c3">x1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c5">x2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:7_2_1_2_3_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E40">A58</font></span></a>, <a class="ref" href="valued_1.html#T12" onmouseover="rs('valued_1/T12')" onmouseout="rh()">VALUED_1:12</a></span>;<br/>
<a NAME="E13:7_2_1_2_3_2_1_1_1_1_1_1"/>
<font color="Maroon" title="c15">x0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">r</font> where <font color="Olive" title="b1">r</font> is   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> : ( <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font> ) </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:7_2_1_2_3_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E38">A56</font></span></a>, <a class="txt" href="#E8:7_2_1_2_3_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E39">A57</font></span></a></span>;<br/>
<a NAME="E14:7_2_1_2_3_2_1_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A61</font></span>: 
<font color="Maroon" title="c15">x0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rcomp_1.html#D1" onmouseover="rs('rcomp_1/D1')" onmouseout="rh()">RCOMP_1:def 1</a></span>;<br/>
<a NAME="E15:7_2_1_2_3_2_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E44">A62</font></span>: 
<font color="Maroon" title="c15">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c17">c1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7_2_1_2_3_2_1_1_1_1"><span class="lab"><font color="Green" title="E27">A45</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/>
<a NAME="E16:7_2_1_2_3_2_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E45">A63</font></span>: 
 <a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c17">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">c1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_4.html#T26" onmouseover="rs('seq_4/T26')" onmouseout="rh()">SEQ_4:26</a></span>;<br/>
<span class="lab"><font color="Green" title="E46">A64</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c17">c1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">c1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_1.html#T7" onmouseover="rs('seq_1/T7')" onmouseout="rh()">SEQ_1:7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c15">x0</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:7_2_1_2_3_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E44">A62</font></span></a>, <a class="txt" href="#E16:7_2_1_2_3_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E45">A63</font></span></a>, <a class="ref" href="seq_4.html#T25" onmouseover="rs('seq_4/T25')" onmouseout="rh()">SEQ_4:25</a></span>
;<br/>
<a NAME="E18:7_2_1_2_3_2_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E47">A65</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c17">c1</font> <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"><font color="Maroon" title="c3">x1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c5">x2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7_2_1_2_3_2_1_1_1_1"><span class="lab"><font color="Green" title="E27">A45</font></span></a>, <a class="txt" href="#E11:7_2_1_2_3_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E41">A59</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/>
<span class="lab"><font color="Green" title="E48">A66</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_1.html#T8" onmouseover="rs('seq_1/T8')" onmouseout="rh()">SEQ_1:8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c17">c1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rfunct_2.html#T1" onmouseover="rs('rfunct_2/T1')" onmouseout="rh()">RFUNCT_2:1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c17">c1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c17">c1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7_2_1_2_3_2_1_1_1_1"><span class="lab"><font color="Green" title="E27">A45</font></span></a>, <a class="ref" href="funct_2.html#T108" onmouseover="rs('funct_2/T108')" onmouseout="rh()">FUNCT_2:108</a>, <a class="txt" href="#E2:7_2_1_2_3_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E34">A52</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c17">c1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">c1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:7_2_1_2_3_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E47">A65</font></span></a>, <a class="ref" href="funct_2.html#T108" onmouseover="rs('funct_2/T108')" onmouseout="rh()">FUNCT_2:108</a>, <a class="txt" href="#E2:7_2_1_2_3_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E34">A52</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c15">x0</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:7_2_1_2_3_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E44">A62</font></span></a>, <a class="txt" href="#E16:7_2_1_2_3_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E45">A63</font></span></a>, <a class="txt" href="#E17:7_2_1_2_3_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E46">A64</font></span></a>, <a class="ref" href="seq_4.html#T25" onmouseover="rs('seq_4/T25')" onmouseout="rh()">SEQ_4:25</a></span>
;<br/>
<span class="lab"><font color="Green" title="E49">A67</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c15">x0</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c15">x0</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">x2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c15">x0</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:7_2_1_2_3_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E42">A60</font></span></a>, <a class="txt" href="#E14:7_2_1_2_3_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E43">A61</font></span></a>, <a class="ref" href="valued_1.html#T13" onmouseover="rs('valued_1/T13')" onmouseout="rh()">VALUED_1:13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c15">x0</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">x2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c15">x0</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:7_2_1_2_3_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E43">A61</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c15">x0</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">x2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c15">x0</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:7_2_1_2_3_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E43">A61</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>
<br/>.= 
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7"><span class="lab"><font color="Green" title="E6">A1</font></span></a></span>
;<br/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">x2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:7_2_1_2_3_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E41">A59</font></span></a>, <a class="ref" href="valued_1.html#T13" onmouseover="rs('valued_1/T13')" onmouseout="rh()">VALUED_1:13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">x2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:7_2_1_2_3_2_1_1"><span class="lab"><font color="Green" title="E21">A39</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">x2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:7_2_1_2_3_2_1_1"><span class="lab"><font color="Green" title="E21">A39</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>
<br/>.= 
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7"><span class="lab"><font color="Green" title="E6">A1</font></span></a></span>
;<br/>
<span class="kw">hence </span><a NAME="E22:7_2_1_2_3_2_1_1_1_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:7_2_1_2_3_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E48">A66</font></span></a>, <a class="txt" href="#E20:7_2_1_2_3_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E49">A67</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:7_2_1_2_3_2_1_1_1_1_1"/>
 for <font color="Olive" title="b1">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon" title="c18">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> holds <br/><span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c19">p1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7_2_1_2_3_2_1_1_1_1_1"><span class="lab"><font color="Green" title="E32">A50</font></span></a>, <a class="ref" href="complex1.html#T44" onmouseover="rs('complex1/T44')" onmouseout="rh()">COMPLEX1:44</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E7:7_2_1_2_3_2_1_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_2.html#D6" onmouseover="rs('seq_2/D6')" onmouseout="rh()">SEQ_2:def 6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:7_2_1_2_3_2_1_1_1_1"/>
 <a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:7_2_1_2_3_2_1_1_1_1"><span class="lab"><font color="Green" title="E31">A49</font></span></a>, <a class="ref" href="seq_2.html#D7" onmouseover="rs('seq_2/D7')" onmouseout="rh()">SEQ_2:def 7</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E3:7_2_1_2_3_2_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="fdiff_3.html#K2" title="FDIFF_3:func.2">Rdiff</a> (<font color="Maroon" title="c3">x1</font>,<font color="Maroon" title="c15">x0</font>)</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="fdiff_3.html#K2" title="FDIFF_3:func.2">Rdiff</a> (<font color="Maroon" title="c5">x2</font>,<font color="Maroon" title="c15">x0</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:7_2_1_2_3_2_1_1"><span class="lab"><font color="Green" title="E25">A43</font></span></a>, <a class="txt" href="#E1:7_2_1_2_3_2_1_1_1"><span class="lab"><font color="Green" title="E26">A44</font></span></a>, <a class="ref" href="fdiff_3.html#D6" onmouseover="rs('fdiff_3/D6')" onmouseout="rh()">FDIFF_3:def 6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:7_2_1_2_3_2_1_1_1"/>
 <a href="fdiff_3.html#K2" title="FDIFF_3:func.2">Rdiff</a> (<font color="Maroon" title="c3">x1</font>,<font color="Maroon" title="c15">x0</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fdiff_3.html#K2" title="FDIFF_3:func.2">Rdiff</a> (<font color="Maroon" title="c5">x2</font>,<font color="Maroon" title="c15">x0</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"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E10:7_2_1_2_3_2_1_1"/>
 <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> (<font color="Maroon" title="c3">x1</font>,<font color="Maroon" title="c15">x0</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> (<font color="Maroon" title="c5">x2</font>,<font color="Maroon" title="c15">x0</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:7_2_1_2_3_2_1_1"><span class="lab"><font color="Green" title="E22">A40</font></span></a>, <a class="txt" href="#E7:7_2_1_2_3_2_1_1"><span class="lab"><font color="Green" title="E24">A42</font></span></a>, <a class="ref" href="fdiff_3.html#T22" onmouseover="rs('fdiff_3/T22')" onmouseout="rh()">FDIFF_3:22</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_2_1_2_3_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:7_2_1_2_3_2_1_2"/><span class="lab"><font color="Green" title="E20">A68</font></span>: 
<font color="Maroon" title="c15">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> (<font color="Maroon" title="c3">x1</font>,<font color="Maroon" title="c15">x0</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> (<font color="Maroon" title="c5">x2</font>,<font color="Maroon" title="c15">x0</font>)</span><br/><div class="add"><a NAME="E2:7_2_1_2_3_2_1_2"/><span class="lab"><font color="Green" title="E21">A69</font></span>: 
<span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> is  <a href="rcomp_1.html#V3" title="RCOMP_1:attr.3">open</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rcomp_1.html#T7" onmouseover="rs('rcomp_1/T7')" onmouseout="rh()">RCOMP_1:7</a></span>;<br/><a NAME="E3:7_2_1_2_3_2_1_2"/><span class="lab"><font color="Green" title="E22">A70</font></span>: 
<span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="7_2_1_2_3_2_1_2_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:7_2_1_2_3_2_1_2_1"/>
 for <font color="Olive" title="b1">x0</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> holds <br/><font color="Olive" title="b1">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="7_2_1_2_3_2_1_2_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c16">x0</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c16">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> implies <font color="Maroon" title="c16">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:7_2_1_2_3_2_1_2_1_1"/><span class="lab"><font color="Green" title="E23">A71</font></span>: 
<font color="Maroon" title="c16">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c16">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span><br/>

<a NAME="E2:7_2_1_2_3_2_1_2_1_1"/>
<font color="Maroon" title="c16">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">r</font> where <font color="Olive" title="b1">r</font> is   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> : ( <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">b</font> ) </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7_2_1_2_3_2_1_2_1_1"><span class="lab"><font color="Green" title="E23">A71</font></span></a>, <a class="ref" href="rcomp_1.html#D2" onmouseover="rs('rcomp_1/D2')" onmouseout="rh()">RCOMP_1:def 2</a></span>;<br/>
<a NAME="E3:7_2_1_2_3_2_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A72</font></span>: 
 ex <font color="Olive" title="b1">r</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">x0</font> &amp; <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">b</font> )
 
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c17">x0</font> = <font color="Maroon" title="c16">x0</font> as   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> ;<br/>
<a NAME="E5:7_2_1_2_3_2_1_2_1_1"/>
<font color="Maroon" title="c17">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">r</font> where <font color="Olive" title="b1">r</font> is   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> : ( <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font> ) </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:7_2_1_2_3_2_1_2_1_1"><span class="lab"><font color="Green" title="E24">A72</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:7_2_1_2_3_2_1_2_1_1"/>
<font color="Maroon" title="c16">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rcomp_1.html#D1" onmouseover="rs('rcomp_1/D1')" onmouseout="rh()">RCOMP_1:def 1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:7_2_1_2_3_2_1_2_1"/>
<span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E4:7_2_1_2_3_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A73</font></span>: 
<font color="Maroon" title="c3">x1</font> <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="txt" href="#E2:7_2_1_2_3_2_1_2"><span class="lab"><font color="Green" title="E21">A69</font></span></a>, <a class="ref" href="fdiff_1.html#T26" onmouseover="rs('fdiff_1/T26')" onmouseout="rh()">FDIFF_1:26</a>, <a class="ref" href="xboole_1.html#T1" onmouseover="rs('xboole_1/T1')" onmouseout="rh()">XBOOLE_1:1</a></span>;<br/><a NAME="E5:7_2_1_2_3_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A74</font></span>: 
( <font color="Maroon" title="c3">x1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c3">x1</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fdiff_2.html#T16" onmouseover="rs('fdiff_2/T16')" onmouseout="rh()">FDIFF_2:16</a></span>;<br/><a NAME="E6:7_2_1_2_3_2_1_2"/><span class="lab"><font color="Green" title="E25">A75</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <span class="p2"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> (<font color="Maroon" title="c3">x1</font>,<font color="Maroon" title="c15">x0</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7_2_1_2_3_2_1_2"><span class="lab"><font color="Green" title="E20">A68</font></span></a>, <a class="txt" href="#E4:7_2_1_2_3_2_1_2"><span class="lab"><font color="Green" title="E23">A73</font></span></a>, <a class="ref" href="fdiff_1.html#D7" onmouseover="rs('fdiff_1/D7')" onmouseout="rh()">FDIFF_1:def 7</a></span>;<br/><a NAME="E7:7_2_1_2_3_2_1_2"/><span class="lab"><font color="Green" title="E26">A76</font></span>: 
<font color="Maroon" title="c5">x2</font> <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:7_2_1_2_3_2_1_2"><span class="lab"><font color="Green" title="E21">A69</font></span></a>, <a class="txt" href="#E3:7_2_1_2_3_2_1_2"><span class="lab"><font color="Green" title="E22">A70</font></span></a>, <a class="txt" href="#E1:7"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="ref" href="fdiff_1.html#T26" onmouseover="rs('fdiff_1/T26')" onmouseout="rh()">FDIFF_1:26</a>, <a class="ref" href="xboole_1.html#T1" onmouseover="rs('xboole_1/T1')" onmouseout="rh()">XBOOLE_1:1</a></span>;<br/><a NAME="E8:7_2_1_2_3_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A77</font></span>: 
( <font color="Maroon" title="c5">x2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">x2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c5">x2</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fdiff_2.html#T16" onmouseover="rs('fdiff_2/T16')" onmouseout="rh()">FDIFF_2:16</a></span>;<br/><a NAME="E9:7_2_1_2_3_2_1_2"/><span class="lab"><font color="Green" title="E28">A78</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">x2</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <span class="p2"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> (<font color="Maroon" title="c5">x2</font>,<font color="Maroon" title="c15">x0</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7_2_1_2_3_2_1_2"><span class="lab"><font color="Green" title="E20">A68</font></span></a>, <a class="txt" href="#E7:7_2_1_2_3_2_1_2"><span class="lab"><font color="Green" title="E26">A76</font></span></a>, <a class="ref" href="fdiff_1.html#D7" onmouseover="rs('fdiff_1/D7')" onmouseout="rh()">FDIFF_1:def 7</a></span>;<br/><span class="lab"><font color="Green" title="E29">A79</font></span>:  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">x1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T61" onmouseover="rs('relat_1/T61')" onmouseout="rh()">RELAT_1:61</a></span>
<br/>.= 
<span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="txt" href="#E3:7_2_1_2_3_2_1_2"><span class="lab"><font color="Green" title="E22">A70</font></span></a>, <a class="ref" href="xboole_1.html#T1" onmouseover="rs('xboole_1/T1')" onmouseout="rh()">XBOOLE_1:1</a>, <a class="ref" href="xboole_1.html#T28" onmouseover="rs('xboole_1/T28')" onmouseout="rh()">XBOOLE_1:28</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="txt" href="#E3:7_2_1_2_3_2_1_2"><span class="lab"><font color="Green" title="E22">A70</font></span></a>, <a class="ref" href="xboole_1.html#T1" onmouseover="rs('xboole_1/T1')" onmouseout="rh()">XBOOLE_1:1</a>, <a class="ref" href="xboole_1.html#T28" onmouseover="rs('xboole_1/T28')" onmouseout="rh()">XBOOLE_1:28</a></span>
<br/>.= 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">x2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T61" onmouseover="rs('relat_1/T61')" onmouseout="rh()">RELAT_1:61</a></span>
;<br/><a NAME="E11:7_2_1_2_3_2_1_2"/>
 for <font color="Olive" title="b1">x0</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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">x1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">x2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x0</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="7_2_1_2_3_2_1_2_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c16">x0</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c16">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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">x1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">x2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x0</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:7_2_1_2_3_2_1_2_3"/><span class="lab"><font color="Green" title="E30">A80</font></span>: 
<font color="Maroon" title="c16">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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">x1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></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"><font color="Maroon" title="c3">x1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">x2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x0</font></span><br/>

<span class="lab"><font color="Green" title="E31">A81</font></span>:  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">x1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T61" onmouseover="rs('relat_1/T61')" onmouseout="rh()">RELAT_1:61</a></span>
<br/>.= 
<span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="txt" href="#E3:7_2_1_2_3_2_1_2"><span class="lab"><font color="Green" title="E22">A70</font></span></a>, <a class="ref" href="xboole_1.html#T1" onmouseover="rs('xboole_1/T1')" onmouseout="rh()">XBOOLE_1:1</a>, <a class="ref" href="xboole_1.html#T28" onmouseover="rs('xboole_1/T28')" onmouseout="rh()">XBOOLE_1:28</a></span>
;<br/>
<span class="kw">then </span><span class="p1">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x0</font> = 
<font color="Maroon" title="c3">x1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x0</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7_2_1_2_3_2_1_2_3"><span class="lab"><font color="Green" title="E30">A80</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x0</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:7_2_1_2_3_2_1_2"><span class="lab"><font color="Green" title="E22">A70</font></span></a>, <a class="txt" href="#E1:7_2_1_2_3_2_1_2_3"><span class="lab"><font color="Green" title="E30">A80</font></span></a>, <a class="txt" href="#E2:7_2_1_2_3_2_1_2_3"><span class="lab"><font color="Green" title="E31">A81</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>
<br/>.= 
<font color="Maroon" title="c5">x2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x0</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:7_2_1_2_3_2_1_2"><span class="lab"><font color="Green" title="E22">A70</font></span></a>, <a class="txt" href="#E1:7_2_1_2_3_2_1_2_3"><span class="lab"><font color="Green" title="E30">A80</font></span></a>, <a class="txt" href="#E2:7_2_1_2_3_2_1_2_3"><span class="lab"><font color="Green" title="E31">A81</font></span></a>, <a class="txt" href="#E1:7"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">x2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x0</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7_2_1_2_3_2_1_2_3"><span class="lab"><font color="Green" title="E30">A80</font></span></a>, <a class="txt" href="#E2:7_2_1_2_3_2_1_2_3"><span class="lab"><font color="Green" title="E31">A81</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E4:7_2_1_2_3_2_1_2_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">x2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x0</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"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E12:7_2_1_2_3_2_1_2"/>
 <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> (<font color="Maroon" title="c3">x1</font>,<font color="Maroon" title="c15">x0</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> (<font color="Maroon" title="c5">x2</font>,<font color="Maroon" title="c15">x0</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:7_2_1_2_3_2_1_2"><span class="lab"><font color="Green" title="E24">A74</font></span></a>, <a class="txt" href="#E6:7_2_1_2_3_2_1_2"><span class="lab"><font color="Green" title="E25">A75</font></span></a>, <a class="txt" href="#E8:7_2_1_2_3_2_1_2"><span class="lab"><font color="Green" title="E27">A77</font></span></a>, <a class="txt" href="#E9:7_2_1_2_3_2_1_2"><span class="lab"><font color="Green" title="E28">A78</font></span></a>, <a class="txt" href="#E10:7_2_1_2_3_2_1_2"><span class="lab"><font color="Green" title="E29">A79</font></span></a>, <a class="ref" href="funct_1.html#T2" onmouseover="rs('funct_1/T2')" onmouseout="rh()">FUNCT_1:2</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_2_1_2_3_2_1_3"><span class="kw">suppose </span></a><a NAME="E1:7_2_1_2_3_2_1_3"/><span class="lab"><font color="Green" title="E20">A82</font></span>: 
<font color="Maroon" title="c15">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b</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">  <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> (<font color="Maroon" title="c3">x1</font>,<font color="Maroon" title="c15">x0</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> (<font color="Maroon" title="c5">x2</font>,<font color="Maroon" title="c15">x0</font>)</span><br/><div class="add"><a NAME="E2:7_2_1_2_3_2_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c15">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">r</font> where <font color="Olive" title="b1">r</font> is   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> : ( <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font> ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7"><span class="lab"><font color="Green" title="E6">A1</font></span></a></span>;<br/><a NAME="E3:7_2_1_2_3_2_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E21">A83</font></span>: 
<font color="Maroon" title="c15">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rcomp_1.html#D1" onmouseover="rs('rcomp_1/D1')" onmouseout="rh()">RCOMP_1:def 1</a></span>;<br/><a NAME="E4:7_2_1_2_3_2_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">A84</font></span>: 
<font color="Maroon" title="c3">x1</font> <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c15">x0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="ref" href="fdiff_1.html#T9" onmouseover="rs('fdiff_1/T9')" onmouseout="rh()">FDIFF_1:9</a></span>;<br/><a NAME="E5:7_2_1_2_3_2_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A85</font></span>: 
( <font color="Maroon" title="c3">x1</font> <a href="fdiff_3.html#R4" title="FDIFF_3:pred.4">is_left_differentiable_in</a> <font color="Maroon" title="c15">x0</font> &amp;  <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> (<font color="Maroon" title="c3">x1</font>,<font color="Maroon" title="c15">x0</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fdiff_3.html#K1" title="FDIFF_3:func.1">Ldiff</a> (<font color="Maroon" title="c3">x1</font>,<font color="Maroon" title="c15">x0</font>) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fdiff_3.html#T22" onmouseover="rs('fdiff_3/T22')" onmouseout="rh()">FDIFF_3:22</a></span>;<br/><a NAME="E6:7_2_1_2_3_2_1_3"/>
<font color="Maroon" title="c5">x2</font> <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c15">x0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="txt" href="#E3:7_2_1_2_3_2_1_3"><span class="lab"><font color="Green" title="E21">A83</font></span></a>, <a class="ref" href="fdiff_1.html#T9" onmouseover="rs('fdiff_1/T9')" onmouseout="rh()">FDIFF_1:9</a></span>;<br/><a NAME="E7:7_2_1_2_3_2_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A86</font></span>: 
( <font color="Maroon" title="c5">x2</font> <a href="fdiff_3.html#R4" title="FDIFF_3:pred.4">is_left_differentiable_in</a> <font color="Maroon" title="c15">x0</font> &amp;  <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> (<font color="Maroon" title="c5">x2</font>,<font color="Maroon" title="c15">x0</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fdiff_3.html#K1" title="FDIFF_3:func.1">Ldiff</a> (<font color="Maroon" title="c5">x2</font>,<font color="Maroon" title="c15">x0</font>) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fdiff_3.html#T22" onmouseover="rs('fdiff_3/T22')" onmouseout="rh()">FDIFF_3:22</a></span>;<br/><a NAME="E8:7_2_1_2_3_2_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A87</font></span>: 
<font color="Maroon" title="c3">x1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c5">x2</font> <a href="fdiff_3.html#R4" title="FDIFF_3:pred.4">is_left_differentiable_in</a> <font color="Maroon" title="c15">x0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:7_2_1_2_3_2_1_3"><span class="lab"><font color="Green" title="E23">A85</font></span></a>, <a class="ref" href="fdiff_3.html#T11" onmouseover="rs('fdiff_3/T11')" onmouseout="rh()">FDIFF_3:11</a></span>;<br/><a NAME="E9:7_2_1_2_3_2_1_3"/>
 <a href="fdiff_3.html#K1" title="FDIFF_3:func.1">Ldiff</a> (<font color="Maroon" title="c3">x1</font>,<font color="Maroon" title="c15">x0</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fdiff_3.html#K1" title="FDIFF_3:func.1">Ldiff</a> (<font color="Maroon" title="c5">x2</font>,<font color="Maroon" title="c15">x0</font>)
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="7_2_1_2_3_2_1_3_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:7_2_1_2_3_2_1_3_1"/><span class="lab"><font color="Green" title="E26">A88</font></span>: 
<span class="p1">(<span class="default"><a href="fdiff_3.html#K1" title="FDIFF_3:func.1">Ldiff</a> (<font color="Maroon" title="c3">x1</font>,<font color="Maroon" title="c15">x0</font>)</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="fdiff_3.html#K1" title="FDIFF_3:func.1">Ldiff</a> (<font color="Maroon" title="c5">x2</font>,<font color="Maroon" title="c15">x0</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fdiff_3.html#K1" title="FDIFF_3:func.1">Ldiff</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c5">x2</font></span>)</span>,<font color="Maroon" title="c15">x0</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:7_2_1_2_3_2_1_3"><span class="lab"><font color="Green" title="E23">A85</font></span></a>, <a class="txt" href="#E7:7_2_1_2_3_2_1_3"><span class="lab"><font color="Green" title="E24">A86</font></span></a>, <a class="ref" href="fdiff_3.html#T11" onmouseover="rs('fdiff_3/T11')" onmouseout="rh()">FDIFF_3:11</a></span>;<br/>
<a NAME="E2:7_2_1_2_3_2_1_3_1"/>
 for <font color="Olive" title="b1">h</font> being   <a href="ordinal1.html#V9" title="ORDINAL1:attr.9">non-zero</a>   <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="fdiff_1.html#V1" title="FDIFF_1:attr.1">-convergent</a>  <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a><br/>  for <font color="Olive" title="b2">c1</font> being   <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a>  <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>  st  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Olive" title="b2">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c15">x0</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Olive" title="b2">c1</font></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"><font color="Maroon" title="c3">x1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c5">x2</font></span>)</span> &amp; (  for <font color="Olive" title="b3">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  holds <font color="Olive" title="b1">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  ) holds <br/> <a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p4">(<span class="default"><font color="Olive" title="b1">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Olive" title="b2">c1</font></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Olive" title="b2">c1</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="7_2_1_2_3_2_1_3_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c16">h</font> be   <a href="ordinal1.html#V9" title="ORDINAL1:attr.9">non-zero</a>   <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="fdiff_1.html#V1" title="FDIFF_1:attr.1">-convergent</a>  <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</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">c1</font> being   <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a>  <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>  st  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Olive" title="b1">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c15">x0</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Olive" title="b1">c1</font></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"><font color="Maroon" title="c3">x1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c5">x2</font></span>)</span> &amp; (  for <font color="Olive" title="b2">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  holds <font color="Maroon" title="c16">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  ) holds <br/> <a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Olive" title="b1">c1</font></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Olive" title="b1">c1</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c17">c1</font> be   <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a>  <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</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="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c17">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c15">x0</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c17">c1</font></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"><font color="Maroon" title="c3">x1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c5">x2</font></span>)</span> &amp; (  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  holds <font color="Maroon" title="c16">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  ) implies  <a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )</span><br/>



<span class="kw">assume </span><a NAME="E1:7_2_1_2_3_2_1_3_1_1"/><span class="lab"><font color="Green" title="E27">A89</font></span>: 
(  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c17">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c15">x0</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c17">c1</font></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"><font color="Maroon" title="c3">x1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c5">x2</font></span>)</span> &amp; (  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  holds <font color="Maroon" title="c16">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</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="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> </span><br/>

<a NAME="E2:7_2_1_2_3_2_1_3_1_1"/><span class="lab"><font color="Green" title="E28">A90</font></span>: 
( <font color="Maroon" title="c16">h</font> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a>  &amp;  <a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c16">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 
;<br/>
<a NAME="E3:7_2_1_2_3_2_1_3_1_1"/><span class="lab"><font color="Green" title="E29">A91</font></span>: 
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c1">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7_2_1"><span class="lab"><font color="Green" title="E10">A6</font></span></a>, <a class="ref" href="xreal_1.html#T50" onmouseover="rs('xreal_1/T50')" onmouseout="rh()">XREAL_1:50</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c18">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E5:7_2_1_2_3_2_1_3_1_1"/><span class="lab"><font color="Green" title="E30">A92</font></span>: 
 for <font color="Olive" title="b1">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon" title="c18">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> holds <br/><span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">x0</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7_2_1_2_3_2_1_3"><span class="lab"><font color="Green" title="E20">A82</font></span></a>, <a class="txt" href="#E2:7_2_1_2_3_2_1_3_1_1"><span class="lab"><font color="Green" title="E28">A90</font></span></a>, <a class="ref" href="seq_2.html#D7" onmouseover="rs('seq_2/D7')" onmouseout="rh()">SEQ_2:def 7</a></span>;<br/>
<a NAME="E6:7_2_1_2_3_2_1_3_1_1"/><span class="lab"><font color="Green" title="E31">A93</font></span>: 
 for <font color="Olive" title="b1">p1</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">p1</font> holds <br/> ex <font color="Olive" title="b2">n1</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/> for <font color="Olive" title="b3">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b2">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">m</font> holds <br/><span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">m</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">p1</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="7_2_1_2_3_2_1_3_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c19">p1</font> be   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</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="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c19">p1</font> implies  ex <font color="Olive" title="b1">n1</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/> for <font color="Olive" title="b2">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">m</font> holds <br/><span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">m</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c19">p1</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:7_2_1_2_3_2_1_3_1_1_1"/><span class="lab"><font color="Green" title="E32">A94</font></span>: 
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c19">p1</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">  ex <font color="Olive" title="b1">n1</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/> for <font color="Olive" title="b2">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">m</font> holds <br/><span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">m</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c19">p1</font></span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c18">n</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon" title="c18">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> holds <br/><span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c19">p1</font></span><br/>

<a NAME="E2:7_2_1_2_3_2_1_3_1_1_1"/>
 for <font color="Olive" title="b1">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon" title="c18">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="7_2_1_2_3_2_1_3_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c20">m</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c18">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">m</font> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:7_2_1_2_3_2_1_3_1_1_1_1"/><span class="lab"><font color="Green" title="E33">A95</font></span>: 
<font color="Maroon" title="c18">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">m</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> </span><br/>

<a NAME="E2:7_2_1_2_3_2_1_3_1_1_1_1"/><span class="lab"><font color="Green" title="E34">A96</font></span>: 
<font color="Maroon" title="c20">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" onmouseover="rs('ordinal1/D12')" onmouseout="rh()">ORDINAL1:def 12</a></span>;<br/>
<a NAME="E3:7_2_1_2_3_2_1_3_1_1_1_1"/><span class="lab"><font color="Green" title="E35">A97</font></span>: 
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">x0</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:7_2_1_2_3_2_1_3_1_1"><span class="lab"><font color="Green" title="E30">A92</font></span></a>, <a class="txt" href="#E1:7_2_1_2_3_2_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E33">A95</font></span></a></span>;<br/>
<a NAME="E4:7_2_1_2_3_2_1_3_1_1_1_1"/><span class="lab"><font color="Green" title="E36">A98</font></span>: 
<font color="Maroon" title="c16">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7_2_1_2_3_2_1_3_1_1"><span class="lab"><font color="Green" title="E27">A89</font></span></a></span>;<br/>
<a NAME="E5:7_2_1_2_3_2_1_3_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A99</font></span>: 
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">x0</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:7_2_1_2_3_2_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E35">A97</font></span></a>, <a class="ref" href="absvalue.html#D1" onmouseover="rs('absvalue/D1')" onmouseout="rh()">ABSVALUE:def 1</a></span>;<br/>
<a NAME="E6:7_2_1_2_3_2_1_3_1_1_1_1"/><span class="lab"><font color="Green" title="E38">A100</font></span>: 
<font color="Maroon" title="c15">x0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7_2_1_2_3_2_1_3"><span class="lab"><font color="Green" title="E20">A82</font></span></a>, <a class="txt" href="#E4:7_2_1_2_3_2_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E36">A98</font></span></a>, <a class="ref" href="xreal_1.html#T7" onmouseover="rs('xreal_1/T7')" onmouseout="rh()">XREAL_1:7</a></span>;<br/>
<a NAME="E7:7_2_1_2_3_2_1_3_1_1_1_1"/>
<font color="Maroon" title="c15">x0</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">x0</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">x0</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:7_2_1_2_3_2_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E37">A99</font></span></a>, <a class="ref" href="xreal_1.html#T13" onmouseover="rs('xreal_1/T13')" onmouseout="rh()">XREAL_1:13</a></span>;<br/>
<a NAME="E8:7_2_1_2_3_2_1_3_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A101</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c15">x0</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">x0</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">x0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7_2_1_2_3_2_1_3"><span class="lab"><font color="Green" title="E20">A82</font></span></a>, <a class="txt" href="#E3:7_2_1_2_3_2_1_3_1_1"><span class="lab"><font color="Green" title="E29">A91</font></span></a>, <a class="ref" href="xreal_1.html#T13" onmouseover="rs('xreal_1/T13')" onmouseout="rh()">XREAL_1:13</a></span>;<br/>
<a NAME="E9:7_2_1_2_3_2_1_3_1_1_1_1"/><span class="lab"><font color="Green" title="E40">A102</font></span>: 
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">x1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">x2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="ref" href="xboole_1.html#T19" onmouseover="rs('xboole_1/T19')" onmouseout="rh()">XBOOLE_1:19</a></span>;<br/>
<a NAME="E10:7_2_1_2_3_2_1_3_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c15">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">x1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">x2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:7_2_1_2_3_2_1_3"><span class="lab"><font color="Green" title="E21">A83</font></span></a></span>;<br/>
<a NAME="E11:7_2_1_2_3_2_1_3_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A103</font></span>: 
<font color="Maroon" title="c15">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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">x1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c5">x2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#T12" onmouseover="rs('valued_1/T12')" onmouseout="rh()">VALUED_1:12</a></span>;<br/>
<a NAME="E12:7_2_1_2_3_2_1_3_1_1_1_1"/><span class="lab"><font color="Green" title="E42">A104</font></span>: 
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></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"><font color="Maroon" title="c3">x1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c5">x2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:7_2_1_2_3_2_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E40">A102</font></span></a>, <a class="ref" href="valued_1.html#T12" onmouseover="rs('valued_1/T12')" onmouseout="rh()">VALUED_1:12</a></span>;<br/>
<a NAME="E13:7_2_1_2_3_2_1_3_1_1_1_1"/>
<font color="Maroon" title="c15">x0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">r</font> where <font color="Olive" title="b1">r</font> is   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> : ( <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font> ) </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:7_2_1_2_3_2_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E38">A100</font></span></a>, <a class="txt" href="#E8:7_2_1_2_3_2_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E39">A101</font></span></a></span>;<br/>
<a NAME="E14:7_2_1_2_3_2_1_3_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A105</font></span>: 
<font color="Maroon" title="c15">x0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rcomp_1.html#D1" onmouseover="rs('rcomp_1/D1')" onmouseout="rh()">RCOMP_1:def 1</a></span>;<br/>
<a NAME="E15:7_2_1_2_3_2_1_3_1_1_1_1"/>
<font color="Maroon" title="c15">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c17">c1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7_2_1_2_3_2_1_3_1_1"><span class="lab"><font color="Green" title="E27">A89</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/>
<a NAME="E16:7_2_1_2_3_2_1_3_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A106</font></span>: 
 <a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c17">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">x0</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_4.html#T25" onmouseover="rs('seq_4/T25')" onmouseout="rh()">SEQ_4:25</a></span>;<br/>
<span class="lab"><font color="Green" title="E45">A107</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c17">c1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">c1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_1.html#T7" onmouseover="rs('seq_1/T7')" onmouseout="rh()">SEQ_1:7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c15">x0</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:7_2_1_2_3_2_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E44">A106</font></span></a>, <a class="ref" href="seq_4.html#T26" onmouseover="rs('seq_4/T26')" onmouseout="rh()">SEQ_4:26</a></span>
;<br/>
<a NAME="E18:7_2_1_2_3_2_1_3_1_1_1_1"/><span class="lab"><font color="Green" title="E46">A108</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c17">c1</font> <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"><font color="Maroon" title="c3">x1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c5">x2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7_2_1_2_3_2_1_3_1_1"><span class="lab"><font color="Green" title="E27">A89</font></span></a>, <a class="txt" href="#E11:7_2_1_2_3_2_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E41">A103</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/>
<span class="lab"><font color="Green" title="E47">A109</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_1.html#T8" onmouseover="rs('seq_1/T8')" onmouseout="rh()">SEQ_1:8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c17">c1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rfunct_2.html#T1" onmouseover="rs('rfunct_2/T1')" onmouseout="rh()">RFUNCT_2:1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c17">c1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c17">c1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7_2_1_2_3_2_1_3_1_1"><span class="lab"><font color="Green" title="E27">A89</font></span></a>, <a class="ref" href="funct_2.html#T108" onmouseover="rs('funct_2/T108')" onmouseout="rh()">FUNCT_2:108</a>, <a class="txt" href="#E2:7_2_1_2_3_2_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E34">A96</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c17">c1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">c1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:7_2_1_2_3_2_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E46">A108</font></span></a>, <a class="ref" href="funct_2.html#T108" onmouseover="rs('funct_2/T108')" onmouseout="rh()">FUNCT_2:108</a>, <a class="txt" href="#E2:7_2_1_2_3_2_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E34">A96</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c15">x0</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:7_2_1_2_3_2_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E44">A106</font></span></a>, <a class="txt" href="#E17:7_2_1_2_3_2_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E45">A107</font></span></a>, <a class="ref" href="seq_4.html#T26" onmouseover="rs('seq_4/T26')" onmouseout="rh()">SEQ_4:26</a></span>
;<br/>
<span class="lab"><font color="Green" title="E48">A110</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c15">x0</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c15">x0</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">x2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c15">x0</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:7_2_1_2_3_2_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E42">A104</font></span></a>, <a class="txt" href="#E14:7_2_1_2_3_2_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E43">A105</font></span></a>, <a class="ref" href="valued_1.html#T13" onmouseover="rs('valued_1/T13')" onmouseout="rh()">VALUED_1:13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c15">x0</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">x2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c15">x0</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:7_2_1_2_3_2_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E43">A105</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c15">x0</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">x2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c15">x0</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:7_2_1_2_3_2_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E43">A105</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>
<br/>.= 
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7"><span class="lab"><font color="Green" title="E6">A1</font></span></a></span>
;<br/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">x2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:7_2_1_2_3_2_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E41">A103</font></span></a>, <a class="ref" href="valued_1.html#T13" onmouseover="rs('valued_1/T13')" onmouseout="rh()">VALUED_1:13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">x2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:7_2_1_2_3_2_1_3"><span class="lab"><font color="Green" title="E21">A83</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">x2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:7_2_1_2_3_2_1_3"><span class="lab"><font color="Green" title="E21">A83</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>
<br/>.= 
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7"><span class="lab"><font color="Green" title="E6">A1</font></span></a></span>
;<br/>
<span class="kw">hence </span><a NAME="E22:7_2_1_2_3_2_1_3_1_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:7_2_1_2_3_2_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E47">A109</font></span></a>, <a class="txt" href="#E20:7_2_1_2_3_2_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E48">A110</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:7_2_1_2_3_2_1_3_1_1_1"/>
 for <font color="Olive" title="b1">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon" title="c18">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> holds <br/><span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c19">p1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7_2_1_2_3_2_1_3_1_1_1"><span class="lab"><font color="Green" title="E32">A94</font></span></a>, <a class="ref" href="complex1.html#T44" onmouseover="rs('complex1/T44')" onmouseout="rh()">COMPLEX1:44</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E7:7_2_1_2_3_2_1_3_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_2.html#D6" onmouseover="rs('seq_2/D6')" onmouseout="rh()">SEQ_2:def 6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:7_2_1_2_3_2_1_3_1_1"/>
 <a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:7_2_1_2_3_2_1_3_1_1"><span class="lab"><font color="Green" title="E31">A93</font></span></a>, <a class="ref" href="seq_2.html#D7" onmouseover="rs('seq_2/D7')" onmouseout="rh()">SEQ_2:def 7</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E3:7_2_1_2_3_2_1_3_1"/><span class="kw">then </span>
 <a href="fdiff_3.html#K1" title="FDIFF_3:func.1">Ldiff</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c5">x2</font></span>)</span>,<font color="Maroon" title="c15">x0</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:7_2_1_2_3_2_1_3"><span class="lab"><font color="Green" title="E25">A87</font></span></a>, <a class="ref" href="fdiff_3.html#D5" onmouseover="rs('fdiff_3/D5')" onmouseout="rh()">FDIFF_3:def 5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:7_2_1_2_3_2_1_3_1"/>
 <a href="fdiff_3.html#K1" title="FDIFF_3:func.1">Ldiff</a> (<font color="Maroon" title="c3">x1</font>,<font color="Maroon" title="c15">x0</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fdiff_3.html#K1" title="FDIFF_3:func.1">Ldiff</a> (<font color="Maroon" title="c5">x2</font>,<font color="Maroon" title="c15">x0</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7_2_1_2_3_2_1_3_1"><span class="lab"><font color="Green" title="E26">A88</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E10:7_2_1_2_3_2_1_3"/>
 <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> (<font color="Maroon" title="c3">x1</font>,<font color="Maroon" title="c15">x0</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> (<font color="Maroon" title="c5">x2</font>,<font color="Maroon" title="c15">x0</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:7_2_1_2_3_2_1_3"><span class="lab"><font color="Green" title="E22">A84</font></span></a>, <a class="txt" href="#E7:7_2_1_2_3_2_1_3"><span class="lab"><font color="Green" title="E24">A86</font></span></a>, <a class="ref" href="fdiff_3.html#T22" onmouseover="rs('fdiff_3/T22')" onmouseout="rh()">FDIFF_3:22</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></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E9:7_2_1_2_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">x2</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7_2_1_2_3"><span class="lab"><font color="Green" title="E15">A27</font></span></a>, <a class="txt" href="#E4:7_2_1_2_3"><span class="lab"><font color="Green" title="E16">A28</font></span></a>, <a class="txt" href="#E5:7_2_1_2_3"><span class="lab"><font color="Green" title="E17">A29</font></span></a>, <a class="txt" href="#E6:7_2_1_2_3"><span class="lab"><font color="Green" title="E18">A30</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</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="E4:7_2_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">x2</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7_2_1_2"><span class="lab"><font color="Green" title="E13">A25</font></span></a>, <a class="txt" href="#E2:7_2_1_2"><span class="lab"><font color="Green" title="E14">A26</font></span></a>, <a class="ref" href="funct_1.html#D11" onmouseover="rs('funct_1/D11')" onmouseout="rh()">FUNCT_1:def 11</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E4:7_2_1"/><span class="lab"><font color="Green" title="E13">A111</font></span>: 
<font color="Maroon" title="c10">v01</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c12">v02</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="7_2_1_3"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:7_2_1_3"/><span class="lab"><font color="Green" title="E14">A112</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">v01</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</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> <span class="p1">(<span class="default"><font color="Maroon" title="c12">v02</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="7_2_1_3_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:7_2_1_3_1"/><span class="lab"><font color="Green" title="E15">A113</font></span>: 
 for <font color="Olive" title="b1">x0</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">v01</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span> holds <br/><font color="Olive" title="b1">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">v02</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="7_2_1_3_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c14">x0</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c14">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">v01</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span> implies <font color="Maroon" title="c14">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">v02</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:7_2_1_3_1_1"/><span class="lab"><font color="Green" title="E16">A114</font></span>: 
<font color="Maroon" title="c14">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">v01</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c14">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">v02</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span></span><br/>

<a NAME="E2:7_2_1_3_1_1"/>
<font color="Maroon" title="c14">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c10">v01</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c13">AB</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7_2_1_3_1_1"><span class="lab"><font color="Green" title="E16">A114</font></span></a>, <a class="ref" href="relat_1.html#T61" onmouseover="rs('relat_1/T61')" onmouseout="rh()">RELAT_1:61</a></span>;<br/>
<a NAME="E3:7_2_1_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E17">A115</font></span>: 
( <font color="Maroon" title="c14">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c10">v01</font> &amp; <font color="Maroon" title="c14">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">AB</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E4:7_2_1_3_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c14">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c3">x1</font>,<font color="Maroon" title="c4">y1</font></span><a href="funct_3.html#K13" title="FUNCT_3:func.13">:&gt;</a></span> &amp; <span class="p1"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c3">x1</font>,<font color="Maroon" title="c4">y1</font></span><a href="funct_3.html#K13" title="FUNCT_3:func.13">:&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x0</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"><span class="p2">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Im</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="integr1c.html#K1" title="INTEGR1C:func.1">R2-to-C</a></span>)</span> &amp; <font color="Maroon" title="c14">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">AB</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:7"><span class="lab"><font color="Green" title="E7">A2</font></span></a>, <a class="ref" href="funct_1.html#T11" onmouseover="rs('funct_1/T11')" onmouseout="rh()">FUNCT_1:11</a></span>;<br/>
<a NAME="E5:7_2_1_3_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c14">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">AB</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x0</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x0</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="comseq_3.html#K6" title="COMSEQ_3:func.6">Im</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="integr1c.html#K1" title="INTEGR1C:func.1">R2-to-C</a></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_3.html#D7" onmouseover="rs('funct_3/D7')" onmouseout="rh()">FUNCT_3:def 7</a></span>;<br/>
<a NAME="E6:7_2_1_3_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c14">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">AB</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x0</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x0</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="comseq_3.html#K6" title="COMSEQ_3:func.6">Im</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="integr1c.html#K1" title="INTEGR1C:func.1">R2-to-C</a></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>;<br/>
<a NAME="E7:7_2_1_3_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c14">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">AB</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">x2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x0</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">y2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x0</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="comseq_3.html#K6" title="COMSEQ_3:func.6">Im</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="integr1c.html#K1" title="INTEGR1C:func.1">R2-to-C</a></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>;<br/>
<a NAME="E8:7_2_1_3_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c14">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">AB</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">x2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x0</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">y2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x0</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="comseq_3.html#K6" title="COMSEQ_3:func.6">Im</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="integr1c.html#K1" title="INTEGR1C:func.1">R2-to-C</a></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>;<br/>
<a NAME="E9:7_2_1_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E18">A116</font></span>: 
( <font color="Maroon" title="c14">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">AB</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">x2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x0</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">y2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x0</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="comseq_3.html#K6" title="COMSEQ_3:func.6">Im</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="integr1c.html#K1" title="INTEGR1C:func.1">R2-to-C</a></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>;<br/>
<a NAME="E10:7_2_1_3_1_1"/>
<font color="Maroon" title="c14">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c6">y2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="txt" href="#E3:7_2_1_3_1_1"><span class="lab"><font color="Green" title="E17">A115</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E11:7_2_1_3_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c5">x2</font>,<font color="Maroon" title="c6">y2</font></span><a href="funct_3.html#K13" title="FUNCT_3:func.13">:&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_3.html#D7" onmouseover="rs('funct_3/D7')" onmouseout="rh()">FUNCT_3:def 7</a></span>;<br/>
<a NAME="E12:7_2_1_3_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c14">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">AB</font> &amp; <font color="Maroon" title="c14">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c5">x2</font>,<font color="Maroon" title="c6">y2</font></span><a href="funct_3.html#K13" title="FUNCT_3:func.13">:&gt;</a></span> &amp; <span class="p1"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c5">x2</font>,<font color="Maroon" title="c6">y2</font></span><a href="funct_3.html#K13" title="FUNCT_3:func.13">:&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x0</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"><span class="p2">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Im</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="integr1c.html#K1" title="INTEGR1C:func.1">R2-to-C</a></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:7_2_1_3_1_1"><span class="lab"><font color="Green" title="E18">A116</font></span></a>, <a class="ref" href="funct_3.html#D7" onmouseover="rs('funct_3/D7')" onmouseout="rh()">FUNCT_3:def 7</a></span>;<br/>
<a NAME="E13:7_2_1_3_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c14">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">AB</font> &amp; <font color="Maroon" title="c14">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c12">v02</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:7"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="ref" href="funct_1.html#T11" onmouseover="rs('funct_1/T11')" onmouseout="rh()">FUNCT_1:11</a></span>;<br/>
<a NAME="E14:7_2_1_3_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c12">v02</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c13">AB</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E15:7_2_1_3_1_1"/>
<font color="Maroon" title="c14">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">v02</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T61" onmouseover="rs('relat_1/T61')" onmouseout="rh()">RELAT_1:61</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:7_2_1_3_1"/>
 for <font color="Olive" title="b1">x0</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">v02</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span> holds <br/><font color="Olive" title="b1">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">v01</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="7_2_1_3_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c14">x0</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c14">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">v02</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span> implies <font color="Maroon" title="c14">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">v01</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:7_2_1_3_1_2"/><span class="lab"><font color="Green" title="E16">A117</font></span>: 
<font color="Maroon" title="c14">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">v02</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c14">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">v01</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span></span><br/>

<a NAME="E2:7_2_1_3_1_2"/>
<font color="Maroon" title="c14">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c12">v02</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c13">AB</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7_2_1_3_1_2"><span class="lab"><font color="Green" title="E16">A117</font></span></a>, <a class="ref" href="relat_1.html#T61" onmouseover="rs('relat_1/T61')" onmouseout="rh()">RELAT_1:61</a></span>;<br/>
<a NAME="E3:7_2_1_3_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E17">A118</font></span>: 
( <font color="Maroon" title="c14">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c12">v02</font> &amp; <font color="Maroon" title="c14">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">AB</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E4:7_2_1_3_1_2"/><span class="kw">then </span>
( <font color="Maroon" title="c14">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c5">x2</font>,<font color="Maroon" title="c6">y2</font></span><a href="funct_3.html#K13" title="FUNCT_3:func.13">:&gt;</a></span> &amp; <span class="p1"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c5">x2</font>,<font color="Maroon" title="c6">y2</font></span><a href="funct_3.html#K13" title="FUNCT_3:func.13">:&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x0</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"><span class="p2">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Im</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="integr1c.html#K1" title="INTEGR1C:func.1">R2-to-C</a></span>)</span> &amp; <font color="Maroon" title="c14">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">AB</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:7"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="ref" href="funct_1.html#T11" onmouseover="rs('funct_1/T11')" onmouseout="rh()">FUNCT_1:11</a></span>;<br/>
<a NAME="E5:7_2_1_3_1_2"/><span class="kw">then </span>
( <font color="Maroon" title="c14">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">AB</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">x2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x0</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">y2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x0</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="comseq_3.html#K6" title="COMSEQ_3:func.6">Im</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="integr1c.html#K1" title="INTEGR1C:func.1">R2-to-C</a></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_3.html#D7" onmouseover="rs('funct_3/D7')" onmouseout="rh()">FUNCT_3:def 7</a></span>;<br/>
<a NAME="E6:7_2_1_3_1_2"/><span class="kw">then </span>
( <font color="Maroon" title="c14">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">AB</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">x2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x0</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">y2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x0</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="comseq_3.html#K6" title="COMSEQ_3:func.6">Im</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="integr1c.html#K1" title="INTEGR1C:func.1">R2-to-C</a></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>;<br/>
<a NAME="E7:7_2_1_3_1_2"/><span class="kw">then </span>
( <font color="Maroon" title="c14">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">AB</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x0</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x0</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="comseq_3.html#K6" title="COMSEQ_3:func.6">Im</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="integr1c.html#K1" title="INTEGR1C:func.1">R2-to-C</a></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>;<br/>
<a NAME="E8:7_2_1_3_1_2"/><span class="kw">then </span>
( <font color="Maroon" title="c14">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">AB</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x0</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x0</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="comseq_3.html#K6" title="COMSEQ_3:func.6">Im</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="integr1c.html#K1" title="INTEGR1C:func.1">R2-to-C</a></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>;<br/>
<a NAME="E9:7_2_1_3_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E18">A119</font></span>: 
( <font color="Maroon" title="c14">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">AB</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x0</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x0</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="comseq_3.html#K6" title="COMSEQ_3:func.6">Im</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="integr1c.html#K1" title="INTEGR1C:func.1">R2-to-C</a></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>;<br/>
<a NAME="E10:7_2_1_3_1_2"/>
<font color="Maroon" title="c14">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">x1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">y1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="txt" href="#E3:7_2_1_3_1_2"><span class="lab"><font color="Green" title="E17">A118</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E11:7_2_1_3_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c14">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c3">x1</font>,<font color="Maroon" title="c4">y1</font></span><a href="funct_3.html#K13" title="FUNCT_3:func.13">:&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_3.html#D7" onmouseover="rs('funct_3/D7')" onmouseout="rh()">FUNCT_3:def 7</a></span>;<br/>
<a NAME="E12:7_2_1_3_1_2"/><span class="kw">then </span>
( <font color="Maroon" title="c14">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">AB</font> &amp; <font color="Maroon" title="c14">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c3">x1</font>,<font color="Maroon" title="c4">y1</font></span><a href="funct_3.html#K13" title="FUNCT_3:func.13">:&gt;</a></span> &amp; <span class="p1"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c3">x1</font>,<font color="Maroon" title="c4">y1</font></span><a href="funct_3.html#K13" title="FUNCT_3:func.13">:&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x0</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"><span class="p2">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Im</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="integr1c.html#K1" title="INTEGR1C:func.1">R2-to-C</a></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:7_2_1_3_1_2"><span class="lab"><font color="Green" title="E18">A119</font></span></a>, <a class="ref" href="funct_3.html#D7" onmouseover="rs('funct_3/D7')" onmouseout="rh()">FUNCT_3:def 7</a></span>;<br/>
<a NAME="E13:7_2_1_3_1_2"/><span class="kw">then </span>
( <font color="Maroon" title="c14">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">AB</font> &amp; <font color="Maroon" title="c14">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c10">v01</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:7"><span class="lab"><font color="Green" title="E7">A2</font></span></a>, <a class="ref" href="funct_1.html#T11" onmouseover="rs('funct_1/T11')" onmouseout="rh()">FUNCT_1:11</a></span>;<br/>
<a NAME="E14:7_2_1_3_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c14">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c10">v01</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c13">AB</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E15:7_2_1_3_1_2"/>
<font color="Maroon" title="c14">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">v01</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T61" onmouseover="rs('relat_1/T61')" onmouseout="rh()">RELAT_1:61</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="E3:7_2_1_3_1"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">v01</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</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> <span class="p1">(<span class="default"><font color="Maroon" title="c12">v02</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7_2_1_3_1"><span class="lab"><font color="Green" title="E15">A113</font></span></a>, <a class="ref" href="tarski.html#T2" onmouseover="rs('tarski/T2')" onmouseout="rh()">TARSKI:2</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:7_2_1_3"/>
 for <font color="Olive" title="b1">x0</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">v01</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c10">v01</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">v02</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x0</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="7_2_1_3_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c14">x0</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c14">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">v01</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c10">v01</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">v02</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x0</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:7_2_1_3_2"/><span class="lab"><font color="Green" title="E15">A120</font></span>: 
<font color="Maroon" title="c14">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">v01</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c10">v01</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">v02</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x0</font></span><br/>

<a NAME="E2:7_2_1_3_2"/>
<font color="Maroon" title="c14">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c10">v01</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c13">AB</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7_2_1_3_2"><span class="lab"><font color="Green" title="E15">A120</font></span></a>, <a class="ref" href="relat_1.html#T61" onmouseover="rs('relat_1/T61')" onmouseout="rh()">RELAT_1:61</a></span>;<br/>
<a NAME="E3:7_2_1_3_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E16">A121</font></span>: 
( <font color="Maroon" title="c14">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">AB</font> &amp; <font color="Maroon" title="c14">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c10">v01</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c15">x0</font> = <font color="Maroon" title="c14">x0</font> as   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> ;<br/>
<a NAME="E5:7_2_1_3_2"/><span class="lab"><font color="Green" title="E17">A122</font></span>: 
<font color="Maroon" title="c13">AB</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">x1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">y1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="ref" href="xboole_1.html#T19" onmouseover="rs('xboole_1/T19')" onmouseout="rh()">XBOOLE_1:19</a></span>;<br/>
<a NAME="E6:7_2_1_3_2"/><span class="kw">then </span>
<font color="Maroon" title="c15">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">x1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">y1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:7_2_1_3_2"><span class="lab"><font color="Green" title="E16">A121</font></span></a></span>;<br/>
<a NAME="E7:7_2_1_3_2"/><span class="kw">then </span>
<font color="Maroon" title="c15">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c3">x1</font>,<font color="Maroon" title="c4">y1</font></span><a href="funct_3.html#K13" title="FUNCT_3:func.13">:&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_3.html#D7" onmouseover="rs('funct_3/D7')" onmouseout="rh()">FUNCT_3:def 7</a></span>;<br/>
<a NAME="E8:7_2_1_3_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E18">A123</font></span>: 
<font color="Maroon" title="c10">v01</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</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="comseq_3.html#K6" title="COMSEQ_3:func.6">Im</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="integr1c.html#K1" title="INTEGR1C:func.1">R2-to-C</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c3">x1</font>,<font color="Maroon" title="c4">y1</font></span><a href="funct_3.html#K13" title="FUNCT_3:func.13">:&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:7"><span class="lab"><font color="Green" title="E7">A2</font></span></a>, <a class="ref" href="funct_1.html#T13" onmouseover="rs('funct_1/T13')" onmouseout="rh()">FUNCT_1:13</a></span>;<br/>
<a NAME="E9:7_2_1_3_2"/><span class="lab"><font color="Green" title="E19">A124</font></span>: 
<font color="Maroon" title="c13">AB</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c6">y2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="ref" href="xboole_1.html#T19" onmouseover="rs('xboole_1/T19')" onmouseout="rh()">XBOOLE_1:19</a></span>;<br/>
<a NAME="E10:7_2_1_3_2"/><span class="kw">then </span>
<font color="Maroon" title="c15">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c6">y2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:7_2_1_3_2"><span class="lab"><font color="Green" title="E16">A121</font></span></a></span>;<br/>
<a NAME="E11:7_2_1_3_2"/><span class="kw">then </span>
<font color="Maroon" title="c15">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c5">x2</font>,<font color="Maroon" title="c6">y2</font></span><a href="funct_3.html#K13" title="FUNCT_3:func.13">:&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_3.html#D7" onmouseover="rs('funct_3/D7')" onmouseout="rh()">FUNCT_3:def 7</a></span>;<br/>
<a NAME="E12:7_2_1_3_2"/><span class="kw">then </span>
<font color="Maroon" title="c12">v02</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</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="comseq_3.html#K6" title="COMSEQ_3:func.6">Im</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="integr1c.html#K1" title="INTEGR1C:func.1">R2-to-C</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c5">x2</font>,<font color="Maroon" title="c6">y2</font></span><a href="funct_3.html#K13" title="FUNCT_3:func.13">:&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:7"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="ref" href="funct_1.html#T13" onmouseover="rs('funct_1/T13')" onmouseout="rh()">FUNCT_1:13</a></span>;<br/>
<a NAME="E13:7_2_1_3_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A125</font></span>: 
<font color="Maroon" title="c12">v02</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</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="comseq_3.html#K6" title="COMSEQ_3:func.6">Im</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="integr1c.html#K1" title="INTEGR1C:func.1">R2-to-C</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">x2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">y2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:7_2_1_3_2"><span class="lab"><font color="Green" title="E16">A121</font></span></a>, <a class="txt" href="#E9:7_2_1_3_2"><span class="lab"><font color="Green" title="E19">A124</font></span></a>, <a class="ref" href="funct_3.html#T48" onmouseover="rs('funct_3/T48')" onmouseout="rh()">FUNCT_3:48</a></span>;<br/>
<span class="lab"><font color="Green" title="E21">A126</font></span>: <font color="Maroon" title="c3">x1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:7_2_1_3_2"><span class="lab"><font color="Green" title="E16">A121</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>
<br/>.= 
<font color="Maroon" title="c5">x2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:7_2_1_3_2"><span class="lab"><font color="Green" title="E16">A121</font></span></a>, <a class="txt" href="#E1:7"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>
;<br/>
<span class="lab"><font color="Green" title="E22">A127</font></span>: <font color="Maroon" title="c4">y1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:7_2_1_3_2"><span class="lab"><font color="Green" title="E16">A121</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>
<br/>.= 
<font color="Maroon" title="c6">y2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:7_2_1_3_2"><span class="lab"><font color="Green" title="E16">A121</font></span></a>, <a class="txt" href="#E1:7"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>
;<br/>
<a NAME="E16:7_2_1_3_2"/>
( <font color="Maroon" title="c10">v01</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">v01</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font> &amp; <font color="Maroon" title="c12">v02</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">v02</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:7_2_1_3_2"><span class="lab"><font color="Green" title="E16">A121</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>;<br/>
<span class="kw">hence </span><a NAME="E17:7_2_1_3_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c10">v01</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">v02</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:7_2_1_3_2"><span class="lab"><font color="Green" title="E21">A126</font></span></a>, <a class="txt" href="#E15:7_2_1_3_2"><span class="lab"><font color="Green" title="E22">A127</font></span></a>, <a class="txt" href="#E8:7_2_1_3_2"><span class="lab"><font color="Green" title="E18">A123</font></span></a>, <a class="txt" href="#E3:7_2_1_3_2"><span class="lab"><font color="Green" title="E16">A121</font></span></a>, <a class="txt" href="#E5:7_2_1_3_2"><span class="lab"><font color="Green" title="E17">A122</font></span></a>, <a class="txt" href="#E13:7_2_1_3_2"><span class="lab"><font color="Green" title="E20">A125</font></span></a>, <a class="ref" href="funct_3.html#T48" onmouseover="rs('funct_3/T48')" onmouseout="rh()">FUNCT_3:48</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="E3:7_2_1_3"/>
<font color="Maroon" title="c10">v01</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c12">v02</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7_2_1_3"><span class="lab"><font color="Green" title="E14">A112</font></span></a>, <a class="ref" href="funct_1.html#D11" onmouseover="rs('funct_1/D11')" onmouseout="rh()">FUNCT_1:def 11</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E5:7_2_1"/><span class="lab"><font color="Green" title="E14">A128</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">y2</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="7_2_1_4"><span class="kw">proof </span></a><div class="add">

<span class="lab"><font color="Green" title="E15">A129</font></span>:  <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"><font color="Maroon" title="c4">y1</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c13">AB</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T61" onmouseover="rs('relat_1/T61')" onmouseout="rh()">RELAT_1:61</a></span>
<br/>.= 
<font color="Maroon" title="c7">Z</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c13">AB</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="ref" href="fdiff_1.html#D7" onmouseover="rs('fdiff_1/D7')" onmouseout="rh()">FDIFF_1:def 7</a></span>
<br/>.= 
<font color="Maroon" title="c13">AB</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="ref" href="xboole_1.html#T28" onmouseover="rs('xboole_1/T28')" onmouseout="rh()">XBOOLE_1:28</a></span>
;<br/>
<span class="lab"><font color="Green" title="E16">A130</font></span>:  <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"><font color="Maroon" title="c6">y2</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">y2</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c13">AB</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T61" onmouseover="rs('relat_1/T61')" onmouseout="rh()">RELAT_1:61</a></span>
<br/>.= 
<font color="Maroon" title="c7">Z</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c13">AB</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="ref" href="fdiff_1.html#D7" onmouseover="rs('fdiff_1/D7')" onmouseout="rh()">FDIFF_1:def 7</a></span>
<br/>.= 
<font color="Maroon" title="c13">AB</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="ref" href="xboole_1.html#T28" onmouseover="rs('xboole_1/T28')" onmouseout="rh()">XBOOLE_1:28</a></span>
;<br/>
<a NAME="E3:7_2_1_4"/>
 for <font color="Olive" title="b1">x0</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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"><font color="Maroon" title="c4">y1</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">y2</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x0</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="7_2_1_4_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c14">x0</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c14">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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"><font color="Maroon" title="c4">y1</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">y2</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x0</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:7_2_1_4_3"/><span class="lab"><font color="Green" title="E17">A131</font></span>: 
<font color="Maroon" title="c14">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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"><font color="Maroon" title="c4">y1</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">y2</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x0</font></span><br/>

<a NAME="E2:7_2_1_4_3"/>
<font color="Maroon" title="c14">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7_2_1_4_3"><span class="lab"><font color="Green" title="E17">A131</font></span></a>, <a class="ref" href="relat_1.html#T57" onmouseover="rs('relat_1/T57')" onmouseout="rh()">RELAT_1:57</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c15">x0</font> = <font color="Maroon" title="c14">x0</font> as   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> ;<br/>
<a NAME="E4:7_2_1_4_3"/><span class="lab"><font color="Green" title="E18">A132</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">y2</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">y2</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7_2_1_4_3"><span class="lab"><font color="Green" title="E17">A131</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>;<br/>
<a NAME="E5:7_2_1_4_3"/><span class="lab"><font color="Green" title="E19">A133</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> (<font color="Maroon" title="c4">y1</font>,<font color="Maroon" title="c15">x0</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="txt" href="#E1:7_2_1_4_3"><span class="lab"><font color="Green" title="E17">A131</font></span></a>, <a class="txt" href="#E1:7_2_1_4"><span class="lab"><font color="Green" title="E15">A129</font></span></a>, <a class="ref" href="fdiff_1.html#D7" onmouseover="rs('fdiff_1/D7')" onmouseout="rh()">FDIFF_1:def 7</a></span>;<br/>
<a NAME="E6:7_2_1_4_3"/><span class="lab"><font color="Green" title="E20">A134</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">y2</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> (<font color="Maroon" title="c6">y2</font>,<font color="Maroon" title="c15">x0</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="txt" href="#E1:7_2_1_4_3"><span class="lab"><font color="Green" title="E17">A131</font></span></a>, <a class="txt" href="#E1:7_2_1_4"><span class="lab"><font color="Green" title="E15">A129</font></span></a>, <a class="ref" href="fdiff_1.html#D7" onmouseover="rs('fdiff_1/D7')" onmouseout="rh()">FDIFF_1:def 7</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E21">A135</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="7_2_1_4_3_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c15">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> or <font color="Maroon" title="c15">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font> or <font color="Maroon" title="c15">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b</font> )</span></a><div class="add"><span class="kw">assume </span><a NAME="E1:7_2_1_4_3_1"/><span class="lab"><font color="Green" title="E22">A136</font></span>: 
(  not <font color="Maroon" title="c15">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> &amp;  not <font color="Maroon" title="c15">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font> &amp;  not <font color="Maroon" title="c15">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b</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"> contradiction</span><br/><a NAME="E2:7_2_1_4_3_1"/>
<font color="Maroon" title="c15">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">r</font> where <font color="Olive" title="b1">r</font> is   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> : ( <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font> ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7_2_1_4"><span class="lab"><font color="Green" title="E15">A129</font></span></a>, <a class="txt" href="#E1:7_2_1_4_3"><span class="lab"><font color="Green" title="E17">A131</font></span></a>, <a class="ref" href="rcomp_1.html#D1" onmouseover="rs('rcomp_1/D1')" onmouseout="rh()">RCOMP_1:def 1</a></span>;<br/><a NAME="E3:7_2_1_4_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A137</font></span>: 
 ex <font color="Olive" title="b1">r</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">x0</font> &amp; <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font> )
 ;<br/><div><span class="lab"><font color="Green" title="E24">A138</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="7_2_1_4_3_1_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">x0</font> or <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">x0</font> )</span></a><div class="add"><span class="kw">assume </span><a NAME="E1:7_2_1_4_3_1_1"/><span class="lab"><font color="Green" title="E25">A139</font></span>: 
(  not <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">x0</font> &amp;  not <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">x0</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"> contradiction</span><br/><a NAME="E2:7_2_1_4_3_1_1"/><span class="kw">then </span>
(  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">x0</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c1">a</font> or  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c15">x0</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T55" onmouseover="rs('xreal_1/T55')" onmouseout="rh()">XREAL_1:55</a></span>;<br/><a NAME="E3:7_2_1_4_3_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">x0</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c1">a</font></span>)</span> or  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c15">x0</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T29" onmouseover="rs('xreal_1/T29')" onmouseout="rh()">XREAL_1:29</a></span>;<br/><a NAME="E4:7_2_1_4_3_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">x0</font> or <font color="Maroon" title="c15">x0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">x0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c15">x0</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T29" onmouseover="rs('xreal_1/T29')" onmouseout="rh()">XREAL_1:29</a></span>;<br/><span class="kw">hence </span><a NAME="E5:7_2_1_4_3_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:7_2_1_4_3_1"><span class="lab"><font color="Green" title="E23">A137</font></span></a>, <a class="txt" href="#E1:7_2_1_4_3_1_1"><span class="lab"><font color="Green" title="E25">A139</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><span class="lab"><font color="Green" title="E25">A140</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="7_2_1_4_3_1_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"> ( <font color="Maroon" title="c15">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b</font> or <font color="Maroon" title="c15">x0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">b</font> )</span></a><div class="add"><span class="kw">assume </span><a NAME="E1:7_2_1_4_3_1_2"/><span class="lab"><font color="Green" title="E26">A141</font></span>: 
(  not <font color="Maroon" title="c15">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b</font> &amp;  not <font color="Maroon" title="c15">x0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">b</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"> contradiction</span><br/><a NAME="E2:7_2_1_4_3_1_2"/><span class="kw">then </span>
(  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">x0</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">b</font> or  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c15">x0</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T55" onmouseover="rs('xreal_1/T55')" onmouseout="rh()">XREAL_1:55</a></span>;<br/><a NAME="E3:7_2_1_4_3_1_2"/><span class="kw">then </span>
( <font color="Maroon" title="c2">b</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">x0</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">b</font></span>)</span> or  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c15">x0</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T29" onmouseover="rs('xreal_1/T29')" onmouseout="rh()">XREAL_1:29</a></span>;<br/><a NAME="E4:7_2_1_4_3_1_2"/><span class="kw">then </span>
( <font color="Maroon" title="c2">b</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">x0</font> or <font color="Maroon" title="c15">x0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">x0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c15">x0</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T29" onmouseover="rs('xreal_1/T29')" onmouseout="rh()">XREAL_1:29</a></span>;<br/><span class="kw">hence </span><a NAME="E5:7_2_1_4_3_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:7_2_1_4_3_1"><span class="lab"><font color="Green" title="E23">A137</font></span></a>, <a class="txt" href="#E1:7_2_1_4_3_1_2"><span class="lab"><font color="Green" title="E26">A141</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E6:7_2_1_4_3_1"/>
 not <font color="Maroon" title="c15">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">q</font> where <font color="Olive" title="b1">q</font> is   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> : ( <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">q</font> &amp; <font color="Olive" title="b1">q</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">b</font> ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7_2_1_4_3_1"><span class="lab"><font color="Green" title="E22">A136</font></span></a>, <a class="ref" href="rcomp_1.html#D2" onmouseover="rs('rcomp_1/D2')" onmouseout="rh()">RCOMP_1:def 2</a></span>;<br/><span class="kw">hence </span><a NAME="E7:7_2_1_4_3_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7_2_1_4_3_1"><span class="lab"><font color="Green" title="E22">A136</font></span></a>, <a class="txt" href="#E4:7_2_1_4_3_1"><span class="lab"><font color="Green" title="E24">A138</font></span></a>, <a class="txt" href="#E5:7_2_1_4_3_1"><span class="lab"><font color="Green" title="E25">A140</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E8:7_2_1_4_3"/>
 <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> (<font color="Maroon" title="c4">y1</font>,<font color="Maroon" title="c15">x0</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> (<font color="Maroon" title="c6">y2</font>,<font color="Maroon" title="c15">x0</font>)
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="7_2_1_4_3_2"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_2_1_4_3_2_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:7_2_1_4_3_2_1"/>
( <font color="Maroon" title="c15">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font> or <font color="Maroon" title="c15">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> or <font color="Maroon" title="c15">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:7_2_1_4_3"><span class="lab"><font color="Green" title="E21">A135</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_2_1_4_3_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:7_2_1_4_3_2_1_1"/><span class="lab"><font color="Green" title="E22">A142</font></span>: 
<font color="Maroon" title="c15">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</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">  <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> (<font color="Maroon" title="c4">y1</font>,<font color="Maroon" title="c15">x0</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> (<font color="Maroon" title="c6">y2</font>,<font color="Maroon" title="c15">x0</font>)</span><br/><div class="add"><a NAME="E2:7_2_1_4_3_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c15">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">r</font> where <font color="Olive" title="b1">r</font> is   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> : ( <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font> ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7"><span class="lab"><font color="Green" title="E6">A1</font></span></a></span>;<br/><a NAME="E3:7_2_1_4_3_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A143</font></span>: 
<font color="Maroon" title="c15">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rcomp_1.html#D1" onmouseover="rs('rcomp_1/D1')" onmouseout="rh()">RCOMP_1:def 1</a></span>;<br/><a NAME="E4:7_2_1_4_3_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A144</font></span>: 
<font color="Maroon" title="c4">y1</font> <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c15">x0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="ref" href="fdiff_1.html#T9" onmouseover="rs('fdiff_1/T9')" onmouseout="rh()">FDIFF_1:9</a></span>;<br/><a NAME="E5:7_2_1_4_3_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A145</font></span>: 
( <font color="Maroon" title="c4">y1</font> <a href="fdiff_3.html#R3" title="FDIFF_3:pred.3">is_right_differentiable_in</a> <font color="Maroon" title="c15">x0</font> &amp;  <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> (<font color="Maroon" title="c4">y1</font>,<font color="Maroon" title="c15">x0</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fdiff_3.html#K2" title="FDIFF_3:func.2">Rdiff</a> (<font color="Maroon" title="c4">y1</font>,<font color="Maroon" title="c15">x0</font>) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fdiff_3.html#T22" onmouseover="rs('fdiff_3/T22')" onmouseout="rh()">FDIFF_3:22</a></span>;<br/><a NAME="E6:7_2_1_4_3_2_1_1"/>
<font color="Maroon" title="c6">y2</font> <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c15">x0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="txt" href="#E3:7_2_1_4_3_2_1_1"><span class="lab"><font color="Green" title="E23">A143</font></span></a>, <a class="ref" href="fdiff_1.html#T9" onmouseover="rs('fdiff_1/T9')" onmouseout="rh()">FDIFF_1:9</a></span>;<br/><a NAME="E7:7_2_1_4_3_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A146</font></span>: 
( <font color="Maroon" title="c6">y2</font> <a href="fdiff_3.html#R3" title="FDIFF_3:pred.3">is_right_differentiable_in</a> <font color="Maroon" title="c15">x0</font> &amp;  <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> (<font color="Maroon" title="c6">y2</font>,<font color="Maroon" title="c15">x0</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fdiff_3.html#K2" title="FDIFF_3:func.2">Rdiff</a> (<font color="Maroon" title="c6">y2</font>,<font color="Maroon" title="c15">x0</font>) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fdiff_3.html#T22" onmouseover="rs('fdiff_3/T22')" onmouseout="rh()">FDIFF_3:22</a></span>;<br/><a NAME="E8:7_2_1_4_3_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A147</font></span>: 
<font color="Maroon" title="c4">y1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c6">y2</font> <a href="fdiff_3.html#R3" title="FDIFF_3:pred.3">is_right_differentiable_in</a> <font color="Maroon" title="c15">x0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:7_2_1_4_3_2_1_1"><span class="lab"><font color="Green" title="E25">A145</font></span></a>, <a class="ref" href="fdiff_3.html#T17" onmouseover="rs('fdiff_3/T17')" onmouseout="rh()">FDIFF_3:17</a></span>;<br/><a NAME="E9:7_2_1_4_3_2_1_1"/>
 <a href="fdiff_3.html#K2" title="FDIFF_3:func.2">Rdiff</a> (<font color="Maroon" title="c4">y1</font>,<font color="Maroon" title="c15">x0</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fdiff_3.html#K2" title="FDIFF_3:func.2">Rdiff</a> (<font color="Maroon" title="c6">y2</font>,<font color="Maroon" title="c15">x0</font>)
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="7_2_1_4_3_2_1_1_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:7_2_1_4_3_2_1_1_1"/><span class="lab"><font color="Green" title="E28">A148</font></span>: 
<span class="p1">(<span class="default"><a href="fdiff_3.html#K2" title="FDIFF_3:func.2">Rdiff</a> (<font color="Maroon" title="c4">y1</font>,<font color="Maroon" title="c15">x0</font>)</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="fdiff_3.html#K2" title="FDIFF_3:func.2">Rdiff</a> (<font color="Maroon" title="c6">y2</font>,<font color="Maroon" title="c15">x0</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fdiff_3.html#K2" title="FDIFF_3:func.2">Rdiff</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c6">y2</font></span>)</span>,<font color="Maroon" title="c15">x0</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:7_2_1_4_3_2_1_1"><span class="lab"><font color="Green" title="E25">A145</font></span></a>, <a class="txt" href="#E7:7_2_1_4_3_2_1_1"><span class="lab"><font color="Green" title="E26">A146</font></span></a>, <a class="ref" href="fdiff_3.html#T17" onmouseover="rs('fdiff_3/T17')" onmouseout="rh()">FDIFF_3:17</a></span>;<br/>
<a NAME="E2:7_2_1_4_3_2_1_1_1"/>
 for <font color="Olive" title="b1">h</font> being   <a href="ordinal1.html#V9" title="ORDINAL1:attr.9">non-zero</a>   <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="fdiff_1.html#V1" title="FDIFF_1:attr.1">-convergent</a>  <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a><br/>  for <font color="Olive" title="b2">c1</font> being   <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a>  <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>  st  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Olive" title="b2">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c15">x0</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Olive" title="b2">c1</font></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"><font color="Maroon" title="c4">y1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c6">y2</font></span>)</span> &amp; (  for <font color="Olive" title="b3">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  holds <font color="Olive" title="b1">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  ) holds <br/> <a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c6">y2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p4">(<span class="default"><font color="Olive" title="b1">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Olive" title="b2">c1</font></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c6">y2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Olive" title="b2">c1</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="7_2_1_4_3_2_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c16">h</font> be   <a href="ordinal1.html#V9" title="ORDINAL1:attr.9">non-zero</a>   <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="fdiff_1.html#V1" title="FDIFF_1:attr.1">-convergent</a>  <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</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">c1</font> being   <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a>  <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>  st  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Olive" title="b1">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c15">x0</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Olive" title="b1">c1</font></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"><font color="Maroon" title="c4">y1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c6">y2</font></span>)</span> &amp; (  for <font color="Olive" title="b2">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  holds <font color="Maroon" title="c16">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  ) holds <br/> <a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c6">y2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Olive" title="b1">c1</font></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c6">y2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Olive" title="b1">c1</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c17">c1</font> be   <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a>  <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</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="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c17">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c15">x0</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c17">c1</font></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"><font color="Maroon" title="c4">y1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c6">y2</font></span>)</span> &amp; (  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  holds <font color="Maroon" title="c16">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  ) implies  <a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c6">y2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c6">y2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )</span><br/>



<span class="kw">assume </span><a NAME="E1:7_2_1_4_3_2_1_1_1_1"/><span class="lab"><font color="Green" title="E29">A149</font></span>: 
(  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c17">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c15">x0</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c17">c1</font></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"><font color="Maroon" title="c4">y1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c6">y2</font></span>)</span> &amp; (  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  holds <font color="Maroon" title="c16">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</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="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c6">y2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c6">y2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> </span><br/>

<a NAME="E2:7_2_1_4_3_2_1_1_1_1"/><span class="lab"><font color="Green" title="E30">A150</font></span>: 
( <font color="Maroon" title="c16">h</font> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a>  &amp;  <a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c16">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 
;<br/>
<a NAME="E3:7_2_1_4_3_2_1_1_1_1"/><span class="lab"><font color="Green" title="E31">A151</font></span>: 
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c15">x0</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7_2_1"><span class="lab"><font color="Green" title="E10">A6</font></span></a>, <a class="txt" href="#E1:7_2_1_4_3_2_1_1"><span class="lab"><font color="Green" title="E22">A142</font></span></a>, <a class="ref" href="xreal_1.html#T50" onmouseover="rs('xreal_1/T50')" onmouseout="rh()">XREAL_1:50</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c18">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E5:7_2_1_4_3_2_1_1_1_1"/><span class="lab"><font color="Green" title="E32">A152</font></span>: 
 for <font color="Olive" title="b1">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon" title="c18">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> holds <br/><span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c15">x0</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:7_2_1_4_3_2_1_1_1_1"><span class="lab"><font color="Green" title="E30">A150</font></span></a>, <a class="ref" href="seq_2.html#D7" onmouseover="rs('seq_2/D7')" onmouseout="rh()">SEQ_2:def 7</a></span>;<br/>
<a NAME="E6:7_2_1_4_3_2_1_1_1_1"/><span class="lab"><font color="Green" title="E33">A153</font></span>: 
 for <font color="Olive" title="b1">p1</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">p1</font> holds <br/> ex <font color="Olive" title="b2">n1</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/> for <font color="Olive" title="b3">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b2">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">m</font> holds <br/><span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c6">y2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c6">y2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">m</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">p1</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="7_2_1_4_3_2_1_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c19">p1</font> be   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</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="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c19">p1</font> implies  ex <font color="Olive" title="b1">n1</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/> for <font color="Olive" title="b2">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">m</font> holds <br/><span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c6">y2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c6">y2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">m</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c19">p1</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:7_2_1_4_3_2_1_1_1_1_1"/><span class="lab"><font color="Green" title="E34">A154</font></span>: 
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c19">p1</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">  ex <font color="Olive" title="b1">n1</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/> for <font color="Olive" title="b2">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">m</font> holds <br/><span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c6">y2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c6">y2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">m</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c19">p1</font></span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c18">n</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon" title="c18">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> holds <br/><span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c6">y2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c6">y2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c19">p1</font></span><br/>

<a NAME="E2:7_2_1_4_3_2_1_1_1_1_1"/>
 for <font color="Olive" title="b1">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon" title="c18">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c6">y2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c6">y2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="7_2_1_4_3_2_1_1_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c20">m</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c18">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">m</font> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c6">y2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c6">y2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:7_2_1_4_3_2_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E35">A155</font></span>: 
<font color="Maroon" title="c18">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">m</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c6">y2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c6">y2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> </span><br/>

<a NAME="E2:7_2_1_4_3_2_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E36">A156</font></span>: 
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c15">x0</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:7_2_1_4_3_2_1_1_1_1"><span class="lab"><font color="Green" title="E32">A152</font></span></a>, <a class="txt" href="#E1:7_2_1_4_3_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E35">A155</font></span></a></span>;<br/>
<a NAME="E3:7_2_1_4_3_2_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E37">A157</font></span>: 
<font color="Maroon" title="c20">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" onmouseover="rs('ordinal1/D12')" onmouseout="rh()">ORDINAL1:def 12</a></span>;<br/>
<a NAME="E4:7_2_1_4_3_2_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E38">A158</font></span>: 
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7_2_1_4_3_2_1_1_1_1"><span class="lab"><font color="Green" title="E29">A149</font></span></a></span>;<br/>
<a NAME="E5:7_2_1_4_3_2_1_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A159</font></span>: 
<font color="Maroon" title="c16">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c15">x0</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:7_2_1_4_3_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E36">A156</font></span></a>, <a class="ref" href="absvalue.html#D1" onmouseover="rs('absvalue/D1')" onmouseout="rh()">ABSVALUE:def 1</a></span>;<br/>
<a NAME="E6:7_2_1_4_3_2_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E40">A160</font></span>: 
<font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">x0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7_2_1_4_3_2_1_1"><span class="lab"><font color="Green" title="E22">A142</font></span></a>, <a class="txt" href="#E4:7_2_1_4_3_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E38">A158</font></span></a>, <a class="ref" href="xreal_1.html#T7" onmouseover="rs('xreal_1/T7')" onmouseout="rh()">XREAL_1:7</a></span>;<br/>
<a NAME="E7:7_2_1_4_3_2_1_1_1_1_1_1"/>
<font color="Maroon" title="c15">x0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">x0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c15">x0</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:7_2_1_4_3_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E39">A159</font></span></a>, <a class="ref" href="xreal_1.html#T7" onmouseover="rs('xreal_1/T7')" onmouseout="rh()">XREAL_1:7</a></span>;<br/>
<a NAME="E8:7_2_1_4_3_2_1_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A161</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c15">x0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c15">x0</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c15">x0</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:7_2_1_4_3_2_1_1_1_1"><span class="lab"><font color="Green" title="E31">A151</font></span></a>, <a class="ref" href="xreal_1.html#T7" onmouseover="rs('xreal_1/T7')" onmouseout="rh()">XREAL_1:7</a></span>;<br/>
<a NAME="E9:7_2_1_4_3_2_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E42">A162</font></span>: 
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">y1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c6">y2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="ref" href="xboole_1.html#T19" onmouseover="rs('xboole_1/T19')" onmouseout="rh()">XBOOLE_1:19</a></span>;<br/>
<a NAME="E10:7_2_1_4_3_2_1_1_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c15">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">y1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c6">y2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:7_2_1_4_3_2_1_1"><span class="lab"><font color="Green" title="E23">A143</font></span></a></span>;<br/>
<a NAME="E11:7_2_1_4_3_2_1_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A163</font></span>: 
<font color="Maroon" title="c15">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c6">y2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#T12" onmouseover="rs('valued_1/T12')" onmouseout="rh()">VALUED_1:12</a></span>;<br/>
<a NAME="E12:7_2_1_4_3_2_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E44">A164</font></span>: 
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></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"><font color="Maroon" title="c4">y1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c6">y2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:7_2_1_4_3_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E42">A162</font></span></a>, <a class="ref" href="valued_1.html#T12" onmouseover="rs('valued_1/T12')" onmouseout="rh()">VALUED_1:12</a></span>;<br/>
<a NAME="E13:7_2_1_4_3_2_1_1_1_1_1_1"/>
<font color="Maroon" title="c15">x0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">r</font> where <font color="Olive" title="b1">r</font> is   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> : ( <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font> ) </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:7_2_1_4_3_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E40">A160</font></span></a>, <a class="txt" href="#E8:7_2_1_4_3_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E41">A161</font></span></a></span>;<br/>
<a NAME="E14:7_2_1_4_3_2_1_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A165</font></span>: 
<font color="Maroon" title="c15">x0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rcomp_1.html#D1" onmouseover="rs('rcomp_1/D1')" onmouseout="rh()">RCOMP_1:def 1</a></span>;<br/>
<a NAME="E15:7_2_1_4_3_2_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E46">A166</font></span>: 
<font color="Maroon" title="c15">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c17">c1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7_2_1_4_3_2_1_1_1_1"><span class="lab"><font color="Green" title="E29">A149</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/>
<a NAME="E16:7_2_1_4_3_2_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E47">A167</font></span>: 
 <a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c17">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">c1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_4.html#T26" onmouseover="rs('seq_4/T26')" onmouseout="rh()">SEQ_4:26</a></span>;<br/>
<span class="lab"><font color="Green" title="E48">A168</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c17">c1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">c1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_1.html#T7" onmouseover="rs('seq_1/T7')" onmouseout="rh()">SEQ_1:7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c15">x0</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:7_2_1_4_3_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E47">A167</font></span></a>, <a class="txt" href="#E15:7_2_1_4_3_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E46">A166</font></span></a>, <a class="ref" href="seq_4.html#T25" onmouseover="rs('seq_4/T25')" onmouseout="rh()">SEQ_4:25</a></span>
;<br/>
<a NAME="E18:7_2_1_4_3_2_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E49">A169</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c17">c1</font> <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"><font color="Maroon" title="c4">y1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c6">y2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:7_2_1_4_3_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E43">A163</font></span></a>, <a class="txt" href="#E1:7_2_1_4_3_2_1_1_1_1"><span class="lab"><font color="Green" title="E29">A149</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/>
<span class="lab"><font color="Green" title="E50">A170</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c6">y2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c6">y2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c6">y2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c6">y2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_1.html#T8" onmouseover="rs('seq_1/T8')" onmouseout="rh()">SEQ_1:8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c6">y2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c6">y2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c17">c1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rfunct_2.html#T1" onmouseover="rs('rfunct_2/T1')" onmouseout="rh()">RFUNCT_2:1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c6">y2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c17">c1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c6">y2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c17">c1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7_2_1_4_3_2_1_1_1_1"><span class="lab"><font color="Green" title="E29">A149</font></span></a>, <a class="ref" href="funct_2.html#T108" onmouseover="rs('funct_2/T108')" onmouseout="rh()">FUNCT_2:108</a>, <a class="txt" href="#E3:7_2_1_4_3_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E37">A157</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c6">y2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c17">c1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c6">y2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">c1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:7_2_1_4_3_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E49">A169</font></span></a>, <a class="ref" href="funct_2.html#T108" onmouseover="rs('funct_2/T108')" onmouseout="rh()">FUNCT_2:108</a>, <a class="txt" href="#E3:7_2_1_4_3_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E37">A157</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c6">y2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c15">x0</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c6">y2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:7_2_1_4_3_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E46">A166</font></span></a>, <a class="txt" href="#E16:7_2_1_4_3_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E47">A167</font></span></a>, <a class="txt" href="#E17:7_2_1_4_3_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E48">A168</font></span></a>, <a class="ref" href="seq_4.html#T25" onmouseover="rs('seq_4/T25')" onmouseout="rh()">SEQ_4:25</a></span>
;<br/>
<span class="lab"><font color="Green" title="E51">A171</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c6">y2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c15">x0</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c15">x0</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">y2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c15">x0</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:7_2_1_4_3_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E44">A164</font></span></a>, <a class="txt" href="#E14:7_2_1_4_3_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E45">A165</font></span></a>, <a class="ref" href="valued_1.html#T13" onmouseover="rs('valued_1/T13')" onmouseout="rh()">VALUED_1:13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c15">x0</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">y2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c15">x0</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:7_2_1_4_3_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E45">A165</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c15">x0</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">y2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c15">x0</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:7_2_1_4_3_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E45">A165</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>
<br/>.= 
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7"><span class="lab"><font color="Green" title="E6">A1</font></span></a></span>
;<br/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c6">y2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">y2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:7_2_1_4_3_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E43">A163</font></span></a>, <a class="ref" href="valued_1.html#T13" onmouseover="rs('valued_1/T13')" onmouseout="rh()">VALUED_1:13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">y2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:7_2_1_4_3_2_1_1"><span class="lab"><font color="Green" title="E23">A143</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">y2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:7_2_1_4_3_2_1_1"><span class="lab"><font color="Green" title="E23">A143</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>
<br/>.= 
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7"><span class="lab"><font color="Green" title="E6">A1</font></span></a></span>
;<br/>
<span class="kw">hence </span><a NAME="E22:7_2_1_4_3_2_1_1_1_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c6">y2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c6">y2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:7_2_1_4_3_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E50">A170</font></span></a>, <a class="txt" href="#E20:7_2_1_4_3_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E51">A171</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:7_2_1_4_3_2_1_1_1_1_1"/>
 for <font color="Olive" title="b1">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon" title="c18">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> holds <br/><span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c6">y2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c6">y2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c19">p1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7_2_1_4_3_2_1_1_1_1_1"><span class="lab"><font color="Green" title="E34">A154</font></span></a>, <a class="ref" href="complex1.html#T44" onmouseover="rs('complex1/T44')" onmouseout="rh()">COMPLEX1:44</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E7:7_2_1_4_3_2_1_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c6">y2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c6">y2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_2.html#D6" onmouseover="rs('seq_2/D6')" onmouseout="rh()">SEQ_2:def 6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:7_2_1_4_3_2_1_1_1_1"/>
 <a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c6">y2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c6">y2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:7_2_1_4_3_2_1_1_1_1"><span class="lab"><font color="Green" title="E33">A153</font></span></a>, <a class="ref" href="seq_2.html#D7" onmouseover="rs('seq_2/D7')" onmouseout="rh()">SEQ_2:def 7</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E3:7_2_1_4_3_2_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="fdiff_3.html#K2" title="FDIFF_3:func.2">Rdiff</a> (<font color="Maroon" title="c4">y1</font>,<font color="Maroon" title="c15">x0</font>)</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="fdiff_3.html#K2" title="FDIFF_3:func.2">Rdiff</a> (<font color="Maroon" title="c6">y2</font>,<font color="Maroon" title="c15">x0</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:7_2_1_4_3_2_1_1"><span class="lab"><font color="Green" title="E27">A147</font></span></a>, <a class="txt" href="#E1:7_2_1_4_3_2_1_1_1"><span class="lab"><font color="Green" title="E28">A148</font></span></a>, <a class="ref" href="fdiff_3.html#D6" onmouseover="rs('fdiff_3/D6')" onmouseout="rh()">FDIFF_3:def 6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:7_2_1_4_3_2_1_1_1"/>
 <a href="fdiff_3.html#K2" title="FDIFF_3:func.2">Rdiff</a> (<font color="Maroon" title="c4">y1</font>,<font color="Maroon" title="c15">x0</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fdiff_3.html#K2" title="FDIFF_3:func.2">Rdiff</a> (<font color="Maroon" title="c6">y2</font>,<font color="Maroon" title="c15">x0</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"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E10:7_2_1_4_3_2_1_1"/>
 <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> (<font color="Maroon" title="c4">y1</font>,<font color="Maroon" title="c15">x0</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> (<font color="Maroon" title="c6">y2</font>,<font color="Maroon" title="c15">x0</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:7_2_1_4_3_2_1_1"><span class="lab"><font color="Green" title="E24">A144</font></span></a>, <a class="txt" href="#E7:7_2_1_4_3_2_1_1"><span class="lab"><font color="Green" title="E26">A146</font></span></a>, <a class="ref" href="fdiff_3.html#T22" onmouseover="rs('fdiff_3/T22')" onmouseout="rh()">FDIFF_3:22</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_2_1_4_3_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:7_2_1_4_3_2_1_2"/><span class="lab"><font color="Green" title="E22">A172</font></span>: 
<font color="Maroon" title="c15">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> (<font color="Maroon" title="c4">y1</font>,<font color="Maroon" title="c15">x0</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> (<font color="Maroon" title="c6">y2</font>,<font color="Maroon" title="c15">x0</font>)</span><br/><div class="add"><a NAME="E2:7_2_1_4_3_2_1_2"/><span class="lab"><font color="Green" title="E23">A173</font></span>: 
<span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> is  <a href="rcomp_1.html#V3" title="RCOMP_1:attr.3">open</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rcomp_1.html#T7" onmouseover="rs('rcomp_1/T7')" onmouseout="rh()">RCOMP_1:7</a></span>;<br/><a NAME="E3:7_2_1_4_3_2_1_2"/><span class="lab"><font color="Green" title="E24">A174</font></span>: 
<span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="7_2_1_4_3_2_1_2_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:7_2_1_4_3_2_1_2_1"/>
 for <font color="Olive" title="b1">x0</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> holds <br/><font color="Olive" title="b1">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="7_2_1_4_3_2_1_2_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c16">x0</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c16">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> implies <font color="Maroon" title="c16">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:7_2_1_4_3_2_1_2_1_1"/><span class="lab"><font color="Green" title="E25">A175</font></span>: 
<font color="Maroon" title="c16">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c16">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span><br/>

<a NAME="E2:7_2_1_4_3_2_1_2_1_1"/>
<font color="Maroon" title="c16">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">r</font> where <font color="Olive" title="b1">r</font> is   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> : ( <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">b</font> ) </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7_2_1_4_3_2_1_2_1_1"><span class="lab"><font color="Green" title="E25">A175</font></span></a>, <a class="ref" href="rcomp_1.html#D2" onmouseover="rs('rcomp_1/D2')" onmouseout="rh()">RCOMP_1:def 2</a></span>;<br/>
<a NAME="E3:7_2_1_4_3_2_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A176</font></span>: 
 ex <font color="Olive" title="b1">r</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">x0</font> &amp; <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">b</font> )
 
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c17">x0</font> = <font color="Maroon" title="c16">x0</font> as   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> ;<br/>
<a NAME="E5:7_2_1_4_3_2_1_2_1_1"/>
<font color="Maroon" title="c17">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">r</font> where <font color="Olive" title="b1">r</font> is   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> : ( <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font> ) </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:7_2_1_4_3_2_1_2_1_1"><span class="lab"><font color="Green" title="E26">A176</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:7_2_1_4_3_2_1_2_1_1"/>
<font color="Maroon" title="c16">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rcomp_1.html#D1" onmouseover="rs('rcomp_1/D1')" onmouseout="rh()">RCOMP_1:def 1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:7_2_1_4_3_2_1_2_1"/>
<span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E4:7_2_1_4_3_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A177</font></span>: 
<font color="Maroon" title="c4">y1</font> <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="txt" href="#E2:7_2_1_4_3_2_1_2"><span class="lab"><font color="Green" title="E23">A173</font></span></a>, <a class="ref" href="fdiff_1.html#T26" onmouseover="rs('fdiff_1/T26')" onmouseout="rh()">FDIFF_1:26</a>, <a class="ref" href="xboole_1.html#T1" onmouseover="rs('xboole_1/T1')" onmouseout="rh()">XBOOLE_1:1</a></span>;<br/><a NAME="E5:7_2_1_4_3_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A178</font></span>: 
( <font color="Maroon" title="c4">y1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c4">y1</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fdiff_2.html#T16" onmouseover="rs('fdiff_2/T16')" onmouseout="rh()">FDIFF_2:16</a></span>;<br/><a NAME="E6:7_2_1_4_3_2_1_2"/><span class="lab"><font color="Green" title="E27">A179</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <span class="p2"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> (<font color="Maroon" title="c4">y1</font>,<font color="Maroon" title="c15">x0</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7_2_1_4_3_2_1_2"><span class="lab"><font color="Green" title="E22">A172</font></span></a>, <a class="txt" href="#E4:7_2_1_4_3_2_1_2"><span class="lab"><font color="Green" title="E25">A177</font></span></a>, <a class="ref" href="fdiff_1.html#D7" onmouseover="rs('fdiff_1/D7')" onmouseout="rh()">FDIFF_1:def 7</a></span>;<br/><a NAME="E7:7_2_1_4_3_2_1_2"/><span class="lab"><font color="Green" title="E28">A180</font></span>: 
<font color="Maroon" title="c6">y2</font> <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="txt" href="#E2:7_2_1_4_3_2_1_2"><span class="lab"><font color="Green" title="E23">A173</font></span></a>, <a class="txt" href="#E3:7_2_1_4_3_2_1_2"><span class="lab"><font color="Green" title="E24">A174</font></span></a>, <a class="ref" href="fdiff_1.html#T26" onmouseover="rs('fdiff_1/T26')" onmouseout="rh()">FDIFF_1:26</a>, <a class="ref" href="xboole_1.html#T1" onmouseover="rs('xboole_1/T1')" onmouseout="rh()">XBOOLE_1:1</a></span>;<br/><a NAME="E8:7_2_1_4_3_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A181</font></span>: 
( <font color="Maroon" title="c6">y2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c6">y2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c6">y2</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fdiff_2.html#T16" onmouseover="rs('fdiff_2/T16')" onmouseout="rh()">FDIFF_2:16</a></span>;<br/><a NAME="E9:7_2_1_4_3_2_1_2"/><span class="lab"><font color="Green" title="E30">A182</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">y2</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <span class="p2"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> (<font color="Maroon" title="c6">y2</font>,<font color="Maroon" title="c15">x0</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7_2_1_4_3_2_1_2"><span class="lab"><font color="Green" title="E22">A172</font></span></a>, <a class="txt" href="#E7:7_2_1_4_3_2_1_2"><span class="lab"><font color="Green" title="E28">A180</font></span></a>, <a class="ref" href="fdiff_1.html#D7" onmouseover="rs('fdiff_1/D7')" onmouseout="rh()">FDIFF_1:def 7</a></span>;<br/><span class="lab"><font color="Green" title="E31">A183</font></span>:  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">y1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T61" onmouseover="rs('relat_1/T61')" onmouseout="rh()">RELAT_1:61</a></span>
<br/>.= 
<span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="txt" href="#E3:7_2_1_4_3_2_1_2"><span class="lab"><font color="Green" title="E24">A174</font></span></a>, <a class="ref" href="xboole_1.html#T1" onmouseover="rs('xboole_1/T1')" onmouseout="rh()">XBOOLE_1:1</a>, <a class="ref" href="xboole_1.html#T28" onmouseover="rs('xboole_1/T28')" onmouseout="rh()">XBOOLE_1:28</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c6">y2</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="txt" href="#E3:7_2_1_4_3_2_1_2"><span class="lab"><font color="Green" title="E24">A174</font></span></a>, <a class="ref" href="xboole_1.html#T1" onmouseover="rs('xboole_1/T1')" onmouseout="rh()">XBOOLE_1:1</a>, <a class="ref" href="xboole_1.html#T28" onmouseover="rs('xboole_1/T28')" onmouseout="rh()">XBOOLE_1:28</a></span>
<br/>.= 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">y2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T61" onmouseover="rs('relat_1/T61')" onmouseout="rh()">RELAT_1:61</a></span>
;<br/><a NAME="E11:7_2_1_4_3_2_1_2"/>
 for <font color="Olive" title="b1">x0</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">y2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x0</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="7_2_1_4_3_2_1_2_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c16">x0</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c16">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">y2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x0</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:7_2_1_4_3_2_1_2_3"/><span class="lab"><font color="Green" title="E32">A184</font></span>: 
<font color="Maroon" title="c16">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></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"><font color="Maroon" title="c4">y1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">y2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x0</font></span><br/>

<span class="lab"><font color="Green" title="E33">A185</font></span>:  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">y1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T61" onmouseover="rs('relat_1/T61')" onmouseout="rh()">RELAT_1:61</a></span>
<br/>.= 
<span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="txt" href="#E3:7_2_1_4_3_2_1_2"><span class="lab"><font color="Green" title="E24">A174</font></span></a>, <a class="ref" href="xboole_1.html#T1" onmouseover="rs('xboole_1/T1')" onmouseout="rh()">XBOOLE_1:1</a>, <a class="ref" href="xboole_1.html#T28" onmouseover="rs('xboole_1/T28')" onmouseout="rh()">XBOOLE_1:28</a></span>
;<br/>
<span class="kw">then </span><span class="p1">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x0</font> = 
<font color="Maroon" title="c4">y1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x0</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7_2_1_4_3_2_1_2_3"><span class="lab"><font color="Green" title="E32">A184</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x0</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:7_2_1_4_3_2_1_2"><span class="lab"><font color="Green" title="E24">A174</font></span></a>, <a class="txt" href="#E1:7_2_1_4_3_2_1_2_3"><span class="lab"><font color="Green" title="E32">A184</font></span></a>, <a class="txt" href="#E2:7_2_1_4_3_2_1_2_3"><span class="lab"><font color="Green" title="E33">A185</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>
<br/>.= 
<font color="Maroon" title="c6">y2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x0</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:7_2_1_4_3_2_1_2"><span class="lab"><font color="Green" title="E24">A174</font></span></a>, <a class="txt" href="#E1:7_2_1_4_3_2_1_2_3"><span class="lab"><font color="Green" title="E32">A184</font></span></a>, <a class="txt" href="#E2:7_2_1_4_3_2_1_2_3"><span class="lab"><font color="Green" title="E33">A185</font></span></a>, <a class="txt" href="#E1:7"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">y2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x0</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7_2_1_4_3_2_1_2_3"><span class="lab"><font color="Green" title="E32">A184</font></span></a>, <a class="txt" href="#E2:7_2_1_4_3_2_1_2_3"><span class="lab"><font color="Green" title="E33">A185</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E4:7_2_1_4_3_2_1_2_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">y2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x0</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"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E12:7_2_1_4_3_2_1_2"/>
 <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> (<font color="Maroon" title="c4">y1</font>,<font color="Maroon" title="c15">x0</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> (<font color="Maroon" title="c6">y2</font>,<font color="Maroon" title="c15">x0</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:7_2_1_4_3_2_1_2"><span class="lab"><font color="Green" title="E26">A178</font></span></a>, <a class="txt" href="#E6:7_2_1_4_3_2_1_2"><span class="lab"><font color="Green" title="E27">A179</font></span></a>, <a class="txt" href="#E8:7_2_1_4_3_2_1_2"><span class="lab"><font color="Green" title="E29">A181</font></span></a>, <a class="txt" href="#E9:7_2_1_4_3_2_1_2"><span class="lab"><font color="Green" title="E30">A182</font></span></a>, <a class="txt" href="#E10:7_2_1_4_3_2_1_2"><span class="lab"><font color="Green" title="E31">A183</font></span></a>, <a class="ref" href="funct_1.html#T2" onmouseover="rs('funct_1/T2')" onmouseout="rh()">FUNCT_1:2</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_2_1_4_3_2_1_3"><span class="kw">suppose </span></a><a NAME="E1:7_2_1_4_3_2_1_3"/><span class="lab"><font color="Green" title="E22">A186</font></span>: 
<font color="Maroon" title="c15">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b</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">  <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> (<font color="Maroon" title="c4">y1</font>,<font color="Maroon" title="c15">x0</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> (<font color="Maroon" title="c6">y2</font>,<font color="Maroon" title="c15">x0</font>)</span><br/><div class="add"><a NAME="E2:7_2_1_4_3_2_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c15">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">r</font> where <font color="Olive" title="b1">r</font> is   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> : ( <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font> ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7"><span class="lab"><font color="Green" title="E6">A1</font></span></a></span>;<br/><a NAME="E3:7_2_1_4_3_2_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A187</font></span>: 
<font color="Maroon" title="c15">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rcomp_1.html#D1" onmouseover="rs('rcomp_1/D1')" onmouseout="rh()">RCOMP_1:def 1</a></span>;<br/><a NAME="E4:7_2_1_4_3_2_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A188</font></span>: 
<font color="Maroon" title="c4">y1</font> <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c15">x0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="ref" href="fdiff_1.html#T9" onmouseover="rs('fdiff_1/T9')" onmouseout="rh()">FDIFF_1:9</a></span>;<br/><a NAME="E5:7_2_1_4_3_2_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A189</font></span>: 
( <font color="Maroon" title="c4">y1</font> <a href="fdiff_3.html#R4" title="FDIFF_3:pred.4">is_left_differentiable_in</a> <font color="Maroon" title="c15">x0</font> &amp;  <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> (<font color="Maroon" title="c4">y1</font>,<font color="Maroon" title="c15">x0</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fdiff_3.html#K1" title="FDIFF_3:func.1">Ldiff</a> (<font color="Maroon" title="c4">y1</font>,<font color="Maroon" title="c15">x0</font>) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fdiff_3.html#T22" onmouseover="rs('fdiff_3/T22')" onmouseout="rh()">FDIFF_3:22</a></span>;<br/><a NAME="E6:7_2_1_4_3_2_1_3"/>
<font color="Maroon" title="c6">y2</font> <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c15">x0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="txt" href="#E3:7_2_1_4_3_2_1_3"><span class="lab"><font color="Green" title="E23">A187</font></span></a>, <a class="ref" href="fdiff_1.html#T9" onmouseover="rs('fdiff_1/T9')" onmouseout="rh()">FDIFF_1:9</a></span>;<br/><a NAME="E7:7_2_1_4_3_2_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A190</font></span>: 
( <font color="Maroon" title="c6">y2</font> <a href="fdiff_3.html#R4" title="FDIFF_3:pred.4">is_left_differentiable_in</a> <font color="Maroon" title="c15">x0</font> &amp;  <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> (<font color="Maroon" title="c6">y2</font>,<font color="Maroon" title="c15">x0</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fdiff_3.html#K1" title="FDIFF_3:func.1">Ldiff</a> (<font color="Maroon" title="c6">y2</font>,<font color="Maroon" title="c15">x0</font>) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fdiff_3.html#T22" onmouseover="rs('fdiff_3/T22')" onmouseout="rh()">FDIFF_3:22</a></span>;<br/><a NAME="E8:7_2_1_4_3_2_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A191</font></span>: 
<font color="Maroon" title="c4">y1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c6">y2</font> <a href="fdiff_3.html#R4" title="FDIFF_3:pred.4">is_left_differentiable_in</a> <font color="Maroon" title="c15">x0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:7_2_1_4_3_2_1_3"><span class="lab"><font color="Green" title="E25">A189</font></span></a>, <a class="ref" href="fdiff_3.html#T11" onmouseover="rs('fdiff_3/T11')" onmouseout="rh()">FDIFF_3:11</a></span>;<br/><a NAME="E9:7_2_1_4_3_2_1_3"/>
 <a href="fdiff_3.html#K1" title="FDIFF_3:func.1">Ldiff</a> (<font color="Maroon" title="c4">y1</font>,<font color="Maroon" title="c15">x0</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fdiff_3.html#K1" title="FDIFF_3:func.1">Ldiff</a> (<font color="Maroon" title="c6">y2</font>,<font color="Maroon" title="c15">x0</font>)
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="7_2_1_4_3_2_1_3_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:7_2_1_4_3_2_1_3_1"/><span class="lab"><font color="Green" title="E28">A192</font></span>: 
<span class="p1">(<span class="default"><a href="fdiff_3.html#K1" title="FDIFF_3:func.1">Ldiff</a> (<font color="Maroon" title="c4">y1</font>,<font color="Maroon" title="c15">x0</font>)</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="fdiff_3.html#K1" title="FDIFF_3:func.1">Ldiff</a> (<font color="Maroon" title="c6">y2</font>,<font color="Maroon" title="c15">x0</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fdiff_3.html#K1" title="FDIFF_3:func.1">Ldiff</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c6">y2</font></span>)</span>,<font color="Maroon" title="c15">x0</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:7_2_1_4_3_2_1_3"><span class="lab"><font color="Green" title="E25">A189</font></span></a>, <a class="txt" href="#E7:7_2_1_4_3_2_1_3"><span class="lab"><font color="Green" title="E26">A190</font></span></a>, <a class="ref" href="fdiff_3.html#T11" onmouseover="rs('fdiff_3/T11')" onmouseout="rh()">FDIFF_3:11</a></span>;<br/>
<a NAME="E2:7_2_1_4_3_2_1_3_1"/>
 for <font color="Olive" title="b1">h</font> being   <a href="ordinal1.html#V9" title="ORDINAL1:attr.9">non-zero</a>   <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="fdiff_1.html#V1" title="FDIFF_1:attr.1">-convergent</a>  <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a><br/>  for <font color="Olive" title="b2">c1</font> being   <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a>  <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>  st  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Olive" title="b2">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c15">x0</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Olive" title="b2">c1</font></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"><font color="Maroon" title="c4">y1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c6">y2</font></span>)</span> &amp; (  for <font color="Olive" title="b3">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  holds <font color="Olive" title="b1">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  ) holds <br/> <a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c6">y2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p4">(<span class="default"><font color="Olive" title="b1">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Olive" title="b2">c1</font></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c6">y2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Olive" title="b2">c1</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="7_2_1_4_3_2_1_3_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c16">h</font> be   <a href="ordinal1.html#V9" title="ORDINAL1:attr.9">non-zero</a>   <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="fdiff_1.html#V1" title="FDIFF_1:attr.1">-convergent</a>  <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</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">c1</font> being   <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a>  <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>  st  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Olive" title="b1">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c15">x0</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Olive" title="b1">c1</font></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"><font color="Maroon" title="c4">y1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c6">y2</font></span>)</span> &amp; (  for <font color="Olive" title="b2">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  holds <font color="Maroon" title="c16">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  ) holds <br/> <a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c6">y2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Olive" title="b1">c1</font></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c6">y2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Olive" title="b1">c1</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c17">c1</font> be   <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a>  <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</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="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c17">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c15">x0</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c17">c1</font></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"><font color="Maroon" title="c4">y1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c6">y2</font></span>)</span> &amp; (  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  holds <font color="Maroon" title="c16">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  ) implies  <a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c6">y2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c6">y2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )</span><br/>



<span class="kw">assume </span><a NAME="E1:7_2_1_4_3_2_1_3_1_1"/><span class="lab"><font color="Green" title="E29">A193</font></span>: 
(  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c17">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c15">x0</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c17">c1</font></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"><font color="Maroon" title="c4">y1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c6">y2</font></span>)</span> &amp; (  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  holds <font color="Maroon" title="c16">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</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="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c6">y2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c6">y2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> </span><br/>

<a NAME="E2:7_2_1_4_3_2_1_3_1_1"/><span class="lab"><font color="Green" title="E30">A194</font></span>: 
( <font color="Maroon" title="c16">h</font> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a>  &amp;  <a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c16">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 
;<br/>
<a NAME="E3:7_2_1_4_3_2_1_3_1_1"/><span class="lab"><font color="Green" title="E31">A195</font></span>: 
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c1">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7_2_1"><span class="lab"><font color="Green" title="E10">A6</font></span></a>, <a class="ref" href="xreal_1.html#T50" onmouseover="rs('xreal_1/T50')" onmouseout="rh()">XREAL_1:50</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c18">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E5:7_2_1_4_3_2_1_3_1_1"/><span class="lab"><font color="Green" title="E32">A196</font></span>: 
 for <font color="Olive" title="b1">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon" title="c18">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> holds <br/><span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">x0</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7_2_1_4_3_2_1_3"><span class="lab"><font color="Green" title="E22">A186</font></span></a>, <a class="txt" href="#E2:7_2_1_4_3_2_1_3_1_1"><span class="lab"><font color="Green" title="E30">A194</font></span></a>, <a class="ref" href="seq_2.html#D7" onmouseover="rs('seq_2/D7')" onmouseout="rh()">SEQ_2:def 7</a></span>;<br/>
<a NAME="E6:7_2_1_4_3_2_1_3_1_1"/><span class="lab"><font color="Green" title="E33">A197</font></span>: 
 for <font color="Olive" title="b1">p1</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">p1</font> holds <br/> ex <font color="Olive" title="b2">n1</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/> for <font color="Olive" title="b3">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b2">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">m</font> holds <br/><span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c6">y2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c6">y2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">m</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">p1</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="7_2_1_4_3_2_1_3_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c19">p1</font> be   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</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="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c19">p1</font> implies  ex <font color="Olive" title="b1">n1</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/> for <font color="Olive" title="b2">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">m</font> holds <br/><span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c6">y2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c6">y2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">m</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c19">p1</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:7_2_1_4_3_2_1_3_1_1_1"/><span class="lab"><font color="Green" title="E34">A198</font></span>: 
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c19">p1</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">  ex <font color="Olive" title="b1">n1</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/> for <font color="Olive" title="b2">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">m</font> holds <br/><span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c6">y2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c6">y2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">m</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c19">p1</font></span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c18">n</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon" title="c18">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> holds <br/><span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c6">y2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c6">y2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c19">p1</font></span><br/>

<a NAME="E2:7_2_1_4_3_2_1_3_1_1_1"/>
 for <font color="Olive" title="b1">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon" title="c18">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c6">y2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c6">y2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="7_2_1_4_3_2_1_3_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c20">m</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c18">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">m</font> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c6">y2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c6">y2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:7_2_1_4_3_2_1_3_1_1_1_1"/><span class="lab"><font color="Green" title="E35">A199</font></span>: 
<font color="Maroon" title="c18">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">m</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c6">y2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c6">y2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> </span><br/>

<a NAME="E2:7_2_1_4_3_2_1_3_1_1_1_1"/><span class="lab"><font color="Green" title="E36">A200</font></span>: 
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">x0</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:7_2_1_4_3_2_1_3_1_1"><span class="lab"><font color="Green" title="E32">A196</font></span></a>, <a class="txt" href="#E1:7_2_1_4_3_2_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E35">A199</font></span></a></span>;<br/>
<a NAME="E3:7_2_1_4_3_2_1_3_1_1_1_1"/><span class="lab"><font color="Green" title="E37">A201</font></span>: 
<font color="Maroon" title="c20">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" onmouseover="rs('ordinal1/D12')" onmouseout="rh()">ORDINAL1:def 12</a></span>;<br/>
<a NAME="E4:7_2_1_4_3_2_1_3_1_1_1_1"/><span class="lab"><font color="Green" title="E38">A202</font></span>: 
<font color="Maroon" title="c16">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7_2_1_4_3_2_1_3_1_1"><span class="lab"><font color="Green" title="E29">A193</font></span></a></span>;<br/>
<a NAME="E5:7_2_1_4_3_2_1_3_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A203</font></span>: 
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">x0</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:7_2_1_4_3_2_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E36">A200</font></span></a>, <a class="ref" href="absvalue.html#D1" onmouseover="rs('absvalue/D1')" onmouseout="rh()">ABSVALUE:def 1</a></span>;<br/>
<a NAME="E6:7_2_1_4_3_2_1_3_1_1_1_1"/><span class="lab"><font color="Green" title="E40">A204</font></span>: 
<font color="Maroon" title="c15">x0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7_2_1_4_3_2_1_3"><span class="lab"><font color="Green" title="E22">A186</font></span></a>, <a class="txt" href="#E4:7_2_1_4_3_2_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E38">A202</font></span></a>, <a class="ref" href="xreal_1.html#T7" onmouseover="rs('xreal_1/T7')" onmouseout="rh()">XREAL_1:7</a></span>;<br/>
<a NAME="E7:7_2_1_4_3_2_1_3_1_1_1_1"/>
<font color="Maroon" title="c15">x0</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">x0</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">x0</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:7_2_1_4_3_2_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E39">A203</font></span></a>, <a class="ref" href="xreal_1.html#T13" onmouseover="rs('xreal_1/T13')" onmouseout="rh()">XREAL_1:13</a></span>;<br/>
<a NAME="E8:7_2_1_4_3_2_1_3_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A205</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c15">x0</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">x0</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">x0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7_2_1_4_3_2_1_3"><span class="lab"><font color="Green" title="E22">A186</font></span></a>, <a class="txt" href="#E3:7_2_1_4_3_2_1_3_1_1"><span class="lab"><font color="Green" title="E31">A195</font></span></a>, <a class="ref" href="xreal_1.html#T13" onmouseover="rs('xreal_1/T13')" onmouseout="rh()">XREAL_1:13</a></span>;<br/>
<a NAME="E9:7_2_1_4_3_2_1_3_1_1_1_1"/><span class="lab"><font color="Green" title="E42">A206</font></span>: 
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">y1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c6">y2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="ref" href="xboole_1.html#T19" onmouseover="rs('xboole_1/T19')" onmouseout="rh()">XBOOLE_1:19</a></span>;<br/>
<a NAME="E10:7_2_1_4_3_2_1_3_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c15">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">y1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c6">y2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:7_2_1_4_3_2_1_3"><span class="lab"><font color="Green" title="E23">A187</font></span></a></span>;<br/>
<a NAME="E11:7_2_1_4_3_2_1_3_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A207</font></span>: 
<font color="Maroon" title="c15">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c6">y2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#T12" onmouseover="rs('valued_1/T12')" onmouseout="rh()">VALUED_1:12</a></span>;<br/>
<a NAME="E12:7_2_1_4_3_2_1_3_1_1_1_1"/><span class="lab"><font color="Green" title="E44">A208</font></span>: 
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></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"><font color="Maroon" title="c4">y1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c6">y2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:7_2_1_4_3_2_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E42">A206</font></span></a>, <a class="ref" href="valued_1.html#T12" onmouseover="rs('valued_1/T12')" onmouseout="rh()">VALUED_1:12</a></span>;<br/>
<a NAME="E13:7_2_1_4_3_2_1_3_1_1_1_1"/>
<font color="Maroon" title="c15">x0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">r</font> where <font color="Olive" title="b1">r</font> is   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> : ( <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font> ) </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:7_2_1_4_3_2_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E40">A204</font></span></a>, <a class="txt" href="#E8:7_2_1_4_3_2_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E41">A205</font></span></a></span>;<br/>
<a NAME="E14:7_2_1_4_3_2_1_3_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A209</font></span>: 
<font color="Maroon" title="c15">x0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rcomp_1.html#D1" onmouseover="rs('rcomp_1/D1')" onmouseout="rh()">RCOMP_1:def 1</a></span>;<br/>
<a NAME="E15:7_2_1_4_3_2_1_3_1_1_1_1"/>
<font color="Maroon" title="c15">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c17">c1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7_2_1_4_3_2_1_3_1_1"><span class="lab"><font color="Green" title="E29">A193</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/>
<a NAME="E16:7_2_1_4_3_2_1_3_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A210</font></span>: 
 <a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c17">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">x0</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_4.html#T25" onmouseover="rs('seq_4/T25')" onmouseout="rh()">SEQ_4:25</a></span>;<br/>
<span class="lab"><font color="Green" title="E47">A211</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c17">c1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">c1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_1.html#T7" onmouseover="rs('seq_1/T7')" onmouseout="rh()">SEQ_1:7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c15">x0</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:7_2_1_4_3_2_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E46">A210</font></span></a>, <a class="ref" href="seq_4.html#T26" onmouseover="rs('seq_4/T26')" onmouseout="rh()">SEQ_4:26</a></span>
;<br/>
<a NAME="E18:7_2_1_4_3_2_1_3_1_1_1_1"/><span class="lab"><font color="Green" title="E48">A212</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c17">c1</font> <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"><font color="Maroon" title="c4">y1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c6">y2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7_2_1_4_3_2_1_3_1_1"><span class="lab"><font color="Green" title="E29">A193</font></span></a>, <a class="txt" href="#E11:7_2_1_4_3_2_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E43">A207</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/>
<span class="lab"><font color="Green" title="E49">A213</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c6">y2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c6">y2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c6">y2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c6">y2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_1.html#T8" onmouseover="rs('seq_1/T8')" onmouseout="rh()">SEQ_1:8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c6">y2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c6">y2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c17">c1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rfunct_2.html#T1" onmouseover="rs('rfunct_2/T1')" onmouseout="rh()">RFUNCT_2:1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c6">y2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c17">c1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c6">y2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c17">c1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7_2_1_4_3_2_1_3_1_1"><span class="lab"><font color="Green" title="E29">A193</font></span></a>, <a class="ref" href="funct_2.html#T108" onmouseover="rs('funct_2/T108')" onmouseout="rh()">FUNCT_2:108</a>, <a class="txt" href="#E3:7_2_1_4_3_2_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E37">A201</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c6">y2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c17">c1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c6">y2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">c1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:7_2_1_4_3_2_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E48">A212</font></span></a>, <a class="ref" href="funct_2.html#T108" onmouseover="rs('funct_2/T108')" onmouseout="rh()">FUNCT_2:108</a>, <a class="txt" href="#E3:7_2_1_4_3_2_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E37">A201</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c6">y2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c15">x0</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c6">y2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:7_2_1_4_3_2_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E46">A210</font></span></a>, <a class="txt" href="#E17:7_2_1_4_3_2_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E47">A211</font></span></a>, <a class="ref" href="seq_4.html#T26" onmouseover="rs('seq_4/T26')" onmouseout="rh()">SEQ_4:26</a></span>
;<br/>
<span class="lab"><font color="Green" title="E50">A214</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c6">y2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c15">x0</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c15">x0</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">y2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c15">x0</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:7_2_1_4_3_2_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E44">A208</font></span></a>, <a class="txt" href="#E14:7_2_1_4_3_2_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E45">A209</font></span></a>, <a class="ref" href="valued_1.html#T13" onmouseover="rs('valued_1/T13')" onmouseout="rh()">VALUED_1:13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c15">x0</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">y2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c15">x0</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:7_2_1_4_3_2_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E45">A209</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c15">x0</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">y2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c15">x0</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:7_2_1_4_3_2_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E45">A209</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>
<br/>.= 
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7"><span class="lab"><font color="Green" title="E6">A1</font></span></a></span>
;<br/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c6">y2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">y2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:7_2_1_4_3_2_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E43">A207</font></span></a>, <a class="ref" href="valued_1.html#T13" onmouseover="rs('valued_1/T13')" onmouseout="rh()">VALUED_1:13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">y2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:7_2_1_4_3_2_1_3"><span class="lab"><font color="Green" title="E23">A187</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">y2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:7_2_1_4_3_2_1_3"><span class="lab"><font color="Green" title="E23">A187</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>
<br/>.= 
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7"><span class="lab"><font color="Green" title="E6">A1</font></span></a></span>
;<br/>
<span class="kw">hence </span><a NAME="E22:7_2_1_4_3_2_1_3_1_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c6">y2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c6">y2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:7_2_1_4_3_2_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E49">A213</font></span></a>, <a class="txt" href="#E20:7_2_1_4_3_2_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E50">A214</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:7_2_1_4_3_2_1_3_1_1_1"/>
 for <font color="Olive" title="b1">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon" title="c18">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> holds <br/><span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c6">y2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c6">y2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c19">p1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7_2_1_4_3_2_1_3_1_1_1"><span class="lab"><font color="Green" title="E34">A198</font></span></a>, <a class="ref" href="complex1.html#T44" onmouseover="rs('complex1/T44')" onmouseout="rh()">COMPLEX1:44</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E7:7_2_1_4_3_2_1_3_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c6">y2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c6">y2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_2.html#D6" onmouseover="rs('seq_2/D6')" onmouseout="rh()">SEQ_2:def 6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:7_2_1_4_3_2_1_3_1_1"/>
 <a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c6">y2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c6">y2</font></span>)</span> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c17">c1</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:7_2_1_4_3_2_1_3_1_1"><span class="lab"><font color="Green" title="E33">A197</font></span></a>, <a class="ref" href="seq_2.html#D7" onmouseover="rs('seq_2/D7')" onmouseout="rh()">SEQ_2:def 7</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E3:7_2_1_4_3_2_1_3_1"/><span class="kw">then </span>
 <a href="fdiff_3.html#K1" title="FDIFF_3:func.1">Ldiff</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c6">y2</font></span>)</span>,<font color="Maroon" title="c15">x0</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:7_2_1_4_3_2_1_3"><span class="lab"><font color="Green" title="E27">A191</font></span></a>, <a class="ref" href="fdiff_3.html#D5" onmouseover="rs('fdiff_3/D5')" onmouseout="rh()">FDIFF_3:def 5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:7_2_1_4_3_2_1_3_1"/>
 <a href="fdiff_3.html#K1" title="FDIFF_3:func.1">Ldiff</a> (<font color="Maroon" title="c4">y1</font>,<font color="Maroon" title="c15">x0</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fdiff_3.html#K1" title="FDIFF_3:func.1">Ldiff</a> (<font color="Maroon" title="c6">y2</font>,<font color="Maroon" title="c15">x0</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7_2_1_4_3_2_1_3_1"><span class="lab"><font color="Green" title="E28">A192</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E10:7_2_1_4_3_2_1_3"/>
 <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> (<font color="Maroon" title="c4">y1</font>,<font color="Maroon" title="c15">x0</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> (<font color="Maroon" title="c6">y2</font>,<font color="Maroon" title="c15">x0</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:7_2_1_4_3_2_1_3"><span class="lab"><font color="Green" title="E24">A188</font></span></a>, <a class="txt" href="#E7:7_2_1_4_3_2_1_3"><span class="lab"><font color="Green" title="E26">A190</font></span></a>, <a class="ref" href="fdiff_3.html#T22" onmouseover="rs('fdiff_3/T22')" onmouseout="rh()">FDIFF_3:22</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></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E9:7_2_1_4_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">y2</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7_2_1_4_3"><span class="lab"><font color="Green" title="E17">A131</font></span></a>, <a class="txt" href="#E4:7_2_1_4_3"><span class="lab"><font color="Green" title="E18">A132</font></span></a>, <a class="txt" href="#E5:7_2_1_4_3"><span class="lab"><font color="Green" title="E19">A133</font></span></a>, <a class="txt" href="#E6:7_2_1_4_3"><span class="lab"><font color="Green" title="E20">A134</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</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="E4:7_2_1_4"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">y2</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7_2_1_4"><span class="lab"><font color="Green" title="E15">A129</font></span></a>, <a class="txt" href="#E2:7_2_1_4"><span class="lab"><font color="Green" title="E16">A130</font></span></a>, <a class="ref" href="funct_1.html#D11" onmouseover="rs('funct_1/D11')" onmouseout="rh()">FUNCT_1:def 11</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E6:7_2_1"/><span class="lab"><font color="Green" title="E15">A215</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">u01</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">v01</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">u02</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">x2</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">v02</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">y2</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="7_2_1_5"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:7_2_1_5"/><span class="lab"><font color="Green" title="E16">A216</font></span>: 
( <font color="Maroon" title="c3">x1</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font> is   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> &amp; <font color="Maroon" title="c4">y1</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font> is   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="numbers.html#T11" onmouseover="rs('numbers/T11')" onmouseout="rh()">NUMBERS:11</a>, <a class="ref" href="relset_1.html#T7" onmouseover="rs('relset_1/T7')" onmouseout="rh()">RELSET_1:7</a></span>;<br/>
<a NAME="E2:7_2_1_5"/><span class="lab"><font color="Green" title="E17">A217</font></span>: 
( <font color="Maroon" title="c9">u01</font> is   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> &amp; <font color="Maroon" title="c10">v01</font> is   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="numbers.html#T11" onmouseover="rs('numbers/T11')" onmouseout="rh()">NUMBERS:11</a>, <a class="ref" href="relset_1.html#T7" onmouseover="rs('relset_1/T7')" onmouseout="rh()">RELSET_1:7</a></span>;<br/>
<a NAME="E3:7_2_1_5"/>
( <font color="Maroon" title="c9">u01</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span> is   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> &amp; <font color="Maroon" title="c10">v01</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span> is   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="numbers.html#T11" onmouseover="rs('numbers/T11')" onmouseout="rh()">NUMBERS:11</a>, <a class="ref" href="relset_1.html#T7" onmouseover="rs('relset_1/T7')" onmouseout="rh()">RELSET_1:7</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E18">A218</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">u01</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">v01</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">u01</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">v01</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="cfunct_1.html#T55" onmouseover="rs('cfunct_1/T55')" onmouseout="rh()">CFUNCT_1:55</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">u01</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">v01</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7_2_1_5"><span class="lab"><font color="Green" title="E16">A216</font></span></a>, <a class="txt" href="#E2:7_2_1_5"><span class="lab"><font color="Green" title="E17">A217</font></span></a>, <a class="ref" href="cfunct_1.html#T53" onmouseover="rs('cfunct_1/T53')" onmouseout="rh()">CFUNCT_1:53</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">u01</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">v01</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7_2_1_5"><span class="lab"><font color="Green" title="E16">A216</font></span></a>, <a class="txt" href="#E2:7_2_1_5"><span class="lab"><font color="Green" title="E17">A217</font></span></a>, <a class="ref" href="cfunct_1.html#T53" onmouseover="rs('cfunct_1/T53')" onmouseout="rh()">CFUNCT_1:53</a></span>
;<br/>
<a NAME="E5:7_2_1_5"/><span class="lab"><font color="Green" title="E19">A219</font></span>: 
( <font color="Maroon" title="c5">x2</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font> is   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> &amp; <font color="Maroon" title="c6">y2</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font> is   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="numbers.html#T11" onmouseover="rs('numbers/T11')" onmouseout="rh()">NUMBERS:11</a>, <a class="ref" href="relset_1.html#T7" onmouseover="rs('relset_1/T7')" onmouseout="rh()">RELSET_1:7</a></span>;<br/>
<a NAME="E6:7_2_1_5"/><span class="lab"><font color="Green" title="E20">A220</font></span>: 
( <font color="Maroon" title="c11">u02</font> is   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> &amp; <font color="Maroon" title="c12">v02</font> is   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="numbers.html#T11" onmouseover="rs('numbers/T11')" onmouseout="rh()">NUMBERS:11</a>, <a class="ref" href="relset_1.html#T7" onmouseover="rs('relset_1/T7')" onmouseout="rh()">RELSET_1:7</a></span>;<br/>
<a NAME="E7:7_2_1_5"/>
( <font color="Maroon" title="c11">u02</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">x2</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span> is   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> &amp; <font color="Maroon" title="c12">v02</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">y2</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span> is   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="numbers.html#T11" onmouseover="rs('numbers/T11')" onmouseout="rh()">NUMBERS:11</a>, <a class="ref" href="relset_1.html#T7" onmouseover="rs('relset_1/T7')" onmouseout="rh()">RELSET_1:7</a></span>;<br/>
<span class="kw">then </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">u02</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">x2</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">v02</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">y2</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">u02</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">x2</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">v02</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">y2</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="cfunct_1.html#T55" onmouseover="rs('cfunct_1/T55')" onmouseout="rh()">CFUNCT_1:55</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">u02</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">x2</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">v02</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">y2</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:7_2_1_5"><span class="lab"><font color="Green" title="E19">A219</font></span></a>, <a class="txt" href="#E6:7_2_1_5"><span class="lab"><font color="Green" title="E20">A220</font></span></a>, <a class="ref" href="cfunct_1.html#T53" onmouseover="rs('cfunct_1/T53')" onmouseout="rh()">CFUNCT_1:53</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">u02</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">x2</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">v02</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">y2</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:7_2_1_5"><span class="lab"><font color="Green" title="E19">A219</font></span></a>, <a class="txt" href="#E6:7_2_1_5"><span class="lab"><font color="Green" title="E20">A220</font></span></a>, <a class="ref" href="cfunct_1.html#T53" onmouseover="rs('cfunct_1/T53')" onmouseout="rh()">CFUNCT_1:53</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E9:7_2_1_5"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">u01</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">v01</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">u02</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">x2</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">v02</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">y2</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:7_2_1"><span class="lab"><font color="Green" title="E11">A7</font></span></a>, <a class="txt" href="#E3:7_2_1"><span class="lab"><font color="Green" title="E12">A24</font></span></a>, <a class="txt" href="#E4:7_2_1"><span class="lab"><font color="Green" title="E13">A111</font></span></a>, <a class="txt" href="#E5:7_2_1"><span class="lab"><font color="Green" title="E14">A128</font></span></a>, <a class="txt" href="#E4:7_2_1_5"><span class="lab"><font color="Green" title="E18">A218</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="lab"><font color="Green" title="E16">A221</font></span>:  <a href="integra5.html#K4" title="INTEGRA5:func.4">integral</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">u01</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">v01</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>) = 
 <a href="integra5.html#K2" title="INTEGRA5:func.2">integral</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">u01</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">v01</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c13">AB</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra5.html#T19" onmouseover="rs('integra5/T19')" onmouseout="rh()">INTEGRA5:19</a></span>
<br/>.= 
 <a href="integra5.html#K2" title="INTEGRA5:func.2">integral</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">u02</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">x2</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">v02</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">y2</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c13">AB</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:7_2_1"><span class="lab"><font color="Green" title="E15">A215</font></span></a></span>
<br/>.= 
 <a href="integra5.html#K4" title="INTEGRA5:func.4">integral</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">u02</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">x2</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">v02</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">y2</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra5.html#T19" onmouseover="rs('integra5/T19')" onmouseout="rh()">INTEGRA5:19</a></span>
;<br/><a NAME="E8:7_2_1"/><span class="lab"><font color="Green" title="E17">A222</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">v01</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">u01</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">v02</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">x2</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">u02</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">y2</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="7_2_1_7"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:7_2_1_7"/><span class="lab"><font color="Green" title="E18">A223</font></span>: 
( <font color="Maroon" title="c3">x1</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font> is   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> &amp; <font color="Maroon" title="c4">y1</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font> is   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="numbers.html#T11" onmouseover="rs('numbers/T11')" onmouseout="rh()">NUMBERS:11</a>, <a class="ref" href="relset_1.html#T7" onmouseover="rs('relset_1/T7')" onmouseout="rh()">RELSET_1:7</a></span>;<br/>
<a NAME="E2:7_2_1_7"/><span class="lab"><font color="Green" title="E19">A224</font></span>: 
( <font color="Maroon" title="c10">v01</font> is   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> &amp; <font color="Maroon" title="c9">u01</font> is   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="numbers.html#T11" onmouseover="rs('numbers/T11')" onmouseout="rh()">NUMBERS:11</a>, <a class="ref" href="relset_1.html#T7" onmouseover="rs('relset_1/T7')" onmouseout="rh()">RELSET_1:7</a></span>;<br/>
<a NAME="E3:7_2_1_7"/>
( <font color="Maroon" title="c10">v01</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span> is   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> &amp; <font color="Maroon" title="c9">u01</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span> is   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="numbers.html#T11" onmouseover="rs('numbers/T11')" onmouseout="rh()">NUMBERS:11</a>, <a class="ref" href="relset_1.html#T7" onmouseover="rs('relset_1/T7')" onmouseout="rh()">RELSET_1:7</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E20">A225</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">v01</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">u01</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">v01</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">u01</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="cfunct_1.html#T52" onmouseover="rs('cfunct_1/T52')" onmouseout="rh()">CFUNCT_1:52</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">v01</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">u01</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7_2_1_7"><span class="lab"><font color="Green" title="E18">A223</font></span></a>, <a class="txt" href="#E2:7_2_1_7"><span class="lab"><font color="Green" title="E19">A224</font></span></a>, <a class="ref" href="cfunct_1.html#T53" onmouseover="rs('cfunct_1/T53')" onmouseout="rh()">CFUNCT_1:53</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">v01</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">u01</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7_2_1_7"><span class="lab"><font color="Green" title="E18">A223</font></span></a>, <a class="txt" href="#E2:7_2_1_7"><span class="lab"><font color="Green" title="E19">A224</font></span></a>, <a class="ref" href="cfunct_1.html#T53" onmouseover="rs('cfunct_1/T53')" onmouseout="rh()">CFUNCT_1:53</a></span>
;<br/>
<a NAME="E5:7_2_1_7"/><span class="lab"><font color="Green" title="E21">A226</font></span>: 
( <font color="Maroon" title="c5">x2</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font> is   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> &amp; <font color="Maroon" title="c6">y2</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font> is   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="numbers.html#T11" onmouseover="rs('numbers/T11')" onmouseout="rh()">NUMBERS:11</a>, <a class="ref" href="relset_1.html#T7" onmouseover="rs('relset_1/T7')" onmouseout="rh()">RELSET_1:7</a></span>;<br/>
<a NAME="E6:7_2_1_7"/><span class="lab"><font color="Green" title="E22">A227</font></span>: 
( <font color="Maroon" title="c12">v02</font> is   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> &amp; <font color="Maroon" title="c11">u02</font> is   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="numbers.html#T11" onmouseover="rs('numbers/T11')" onmouseout="rh()">NUMBERS:11</a>, <a class="ref" href="relset_1.html#T7" onmouseover="rs('relset_1/T7')" onmouseout="rh()">RELSET_1:7</a></span>;<br/>
<a NAME="E7:7_2_1_7"/>
( <font color="Maroon" title="c12">v02</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">x2</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span> is   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> &amp; <font color="Maroon" title="c11">u02</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">y2</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span> is   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="numbers.html#T11" onmouseover="rs('numbers/T11')" onmouseout="rh()">NUMBERS:11</a>, <a class="ref" href="relset_1.html#T7" onmouseover="rs('relset_1/T7')" onmouseout="rh()">RELSET_1:7</a></span>;<br/>
<span class="kw">then </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">v02</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">x2</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">u02</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">y2</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">v02</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">x2</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">u02</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">y2</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="cfunct_1.html#T52" onmouseover="rs('cfunct_1/T52')" onmouseout="rh()">CFUNCT_1:52</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">v02</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">x2</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">u02</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">y2</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:7_2_1_7"><span class="lab"><font color="Green" title="E21">A226</font></span></a>, <a class="txt" href="#E6:7_2_1_7"><span class="lab"><font color="Green" title="E22">A227</font></span></a>, <a class="ref" href="cfunct_1.html#T53" onmouseover="rs('cfunct_1/T53')" onmouseout="rh()">CFUNCT_1:53</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">v02</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">x2</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">u02</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">y2</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:7_2_1_7"><span class="lab"><font color="Green" title="E21">A226</font></span></a>, <a class="txt" href="#E6:7_2_1_7"><span class="lab"><font color="Green" title="E22">A227</font></span></a>, <a class="ref" href="cfunct_1.html#T53" onmouseover="rs('cfunct_1/T53')" onmouseout="rh()">CFUNCT_1:53</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E9:7_2_1_7"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">v01</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">u01</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">v02</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">x2</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">u02</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">y2</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">AB</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:7_2_1"><span class="lab"><font color="Green" title="E11">A7</font></span></a>, <a class="txt" href="#E3:7_2_1"><span class="lab"><font color="Green" title="E12">A24</font></span></a>, <a class="txt" href="#E4:7_2_1"><span class="lab"><font color="Green" title="E13">A111</font></span></a>, <a class="txt" href="#E5:7_2_1"><span class="lab"><font color="Green" title="E14">A128</font></span></a>, <a class="txt" href="#E4:7_2_1_7"><span class="lab"><font color="Green" title="E20">A225</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div> <a href="integra5.html#K4" title="INTEGRA5:func.4">integral</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">v01</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">u01</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>) = 
 <a href="integra5.html#K2" title="INTEGRA5:func.2">integral</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">v01</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">u01</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c13">AB</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra5.html#T19" onmouseover="rs('integra5/T19')" onmouseout="rh()">INTEGRA5:19</a></span>
<br/>.= 
 <a href="integra5.html#K2" title="INTEGRA5:func.2">integral</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">v02</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">x2</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">u02</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">y2</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c13">AB</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:7_2_1"><span class="lab"><font color="Green" title="E17">A222</font></span></a></span>
<br/>.= 
 <a href="integra5.html#K4" title="INTEGRA5:func.4">integral</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">v02</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">x2</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">u02</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">y2</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra5.html#T19" onmouseover="rs('integra5/T19')" onmouseout="rh()">INTEGRA5:19</a></span>
;<br/><span class="kw">hence </span><a NAME="E10:7_2_1"/>
 <a href="integr1c.html#K2" title="INTEGR1C:func.2">integral</a> (<font color="Maroon" title="c8">f</font>,<font color="Maroon" title="c3">x1</font>,<font color="Maroon" title="c4">y1</font>,<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c7">Z</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integr1c.html#K2" title="INTEGR1C:func.2">integral</a> (<font color="Maroon" title="c8">f</font>,<font color="Maroon" title="c5">x2</font>,<font color="Maroon" title="c6">y2</font>,<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c7">Z</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:7"><span class="lab"><font color="Green" title="E7">A2</font></span></a>, <a class="txt" href="#E5:7"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="txt" href="#E7:7_2_1"><span class="lab"><font color="Green" title="E16">A221</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_2_2"><span class="kw">suppose </span></a><a NAME="E1:7_2_2"/><span class="lab"><font color="Green" title="E10">A228</font></span>: 
<font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b</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">  <a href="integr1c.html#K2" title="INTEGR1C:func.2">integral</a> (<font color="Maroon" title="c8">f</font>,<font color="Maroon" title="c3">x1</font>,<font color="Maroon" title="c4">y1</font>,<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c7">Z</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integr1c.html#K2" title="INTEGR1C:func.2">integral</a> (<font color="Maroon" title="c8">f</font>,<font color="Maroon" title="c5">x2</font>,<font color="Maroon" title="c6">y2</font>,<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c7">Z</font>)</span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c14">BA</font> = <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c1">a</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="measure5.html#V2" title="MEASURE5:attr.2">closed_interval</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="measure5.html#T14" onmouseover="rs('measure5/T14')" onmouseout="rh()">MEASURE5:14</a></span>;<br/><a NAME="E3:7_2_2"/><span class="lab"><font color="Green" title="E11">A229</font></span>: 
 <a href="integra5.html#K4" title="INTEGRA5:func.4">integral</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">u01</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">v01</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra5.html#K2" title="INTEGRA5:func.2">integral</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">u01</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">v01</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c14">BA</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7_2_2"><span class="lab"><font color="Green" title="E10">A228</font></span></a>, <a class="ref" href="integra5.html#T19" onmouseover="rs('integra5/T19')" onmouseout="rh()">INTEGRA5:19</a></span>;<br/><a NAME="E4:7_2_2"/><span class="lab"><font color="Green" title="E12">A230</font></span>: 
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><a href="integra5.html#K2" title="INTEGRA5:func.2">integral</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">u01</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">v01</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c14">BA</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra5.html#K4" title="INTEGRA5:func.4">integral</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">u01</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">v01</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="integra5.html#T20" onmouseover="rs('integra5/T20')" onmouseout="rh()">INTEGRA5:20</a></span>;<br/><a NAME="E5:7_2_2"/><span class="lab"><font color="Green" title="E13">A231</font></span>: 
 <a href="integra5.html#K4" title="INTEGRA5:func.4">integral</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">v01</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">u01</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra5.html#K2" title="INTEGRA5:func.2">integral</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">v01</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">u01</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c14">BA</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7_2_2"><span class="lab"><font color="Green" title="E10">A228</font></span></a>, <a class="ref" href="integra5.html#T19" onmouseover="rs('integra5/T19')" onmouseout="rh()">INTEGRA5:19</a></span>;<br/><a NAME="E6:7_2_2"/><span class="lab"><font color="Green" title="E14">A232</font></span>: 
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><a href="integra5.html#K2" title="INTEGRA5:func.2">integral</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">v01</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">u01</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c14">BA</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra5.html#K4" title="INTEGRA5:func.4">integral</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">v01</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">x1</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">u01</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">y1</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="integra5.html#T20" onmouseover="rs('integra5/T20')" onmouseout="rh()">INTEGRA5:20</a></span>;<br/><a NAME="E7:7_2_2"/>
 <a href="integra5.html#K4" title="INTEGRA5:func.4">integral</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">u02</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">x2</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">v02</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">y2</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra5.html#K2" title="INTEGRA5:func.2">integral</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">u02</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">x2</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">v02</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">y2</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c14">BA</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7_2_2"><span class="lab"><font color="Green" title="E10">A228</font></span></a>, <a class="ref" href="integra5.html#T19" onmouseover="rs('integra5/T19')" onmouseout="rh()">INTEGRA5:19</a></span>;<br/><a NAME="E8:7_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E15">A233</font></span>: 
 <a href="integra5.html#K4" title="INTEGRA5:func.4">integral</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">u02</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">x2</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">v02</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">y2</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><a href="integra5.html#K4" title="INTEGRA5:func.4">integral</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">u02</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">x2</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">v02</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">y2</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="integra5.html#T20" onmouseover="rs('integra5/T20')" onmouseout="rh()">INTEGRA5:20</a></span>;<br/><a NAME="E9:7_2_2"/>
 <a href="integra5.html#K4" title="INTEGRA5:func.4">integral</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">v02</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">x2</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">u02</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">y2</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra5.html#K2" title="INTEGRA5:func.2">integral</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">v02</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">x2</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">u02</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">y2</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c13">AB</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="integra5.html#T19" onmouseover="rs('integra5/T19')" onmouseout="rh()">INTEGRA5:19</a></span>;<br/><a NAME="E10:7_2_2"/><span class="kw">then </span>
 <a href="integra5.html#K4" title="INTEGRA5:func.4">integral</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">v02</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">x2</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">u02</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">y2</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><a href="integra5.html#K4" title="INTEGRA5:func.4">integral</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">v02</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">x2</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">u02</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">y2</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c7">Z</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7_2_2"><span class="lab"><font color="Green" title="E10">A228</font></span></a>, <a class="ref" href="integra5.html#T20" onmouseover="rs('integra5/T20')" onmouseout="rh()">INTEGRA5:20</a></span>;<br/><span class="kw">hence </span><a NAME="E11:7_2_2"/>
 <a href="integr1c.html#K2" title="INTEGR1C:func.2">integral</a> (<font color="Maroon" title="c8">f</font>,<font color="Maroon" title="c3">x1</font>,<font color="Maroon" title="c4">y1</font>,<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c7">Z</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integr1c.html#K2" title="INTEGR1C:func.2">integral</a> (<font color="Maroon" title="c8">f</font>,<font color="Maroon" title="c5">x2</font>,<font color="Maroon" title="c6">y2</font>,<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c7">Z</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:7"><span class="lab"><font color="Green" title="E7">A2</font></span></a>, <a class="txt" href="#E5:7"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="txt" href="#E3:7_2_2"><span class="lab"><font color="Green" title="E11">A229</font></span></a>, <a class="txt" href="#E4:7_2_2"><span class="lab"><font color="Green" title="E12">A230</font></span></a>, <a class="txt" href="#E5:7_2_2"><span class="lab"><font color="Green" title="E13">A231</font></span></a>, <a class="txt" href="#E6:7_2_2"><span class="lab"><font color="Green" title="E14">A232</font></span></a>, <a class="txt" href="#E8:7_2_2"><span class="lab"><font color="Green" title="E15">A233</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

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