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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="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="mesfun6c.html#K1" title="MESFUN6C:func.1">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="mesfun6c.html#K2" title="MESFUN6C:func.2">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" title="INTEGR1C:def.2">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="mesfun6c.html#K1" title="MESFUN6C:func.1">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="mesfun6c.html#K2" title="MESFUN6C:func.2">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" title="INTEGR1C:def.2">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   <a href="integra1.html#V1" title="INTEGRA1:attr.1">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="integra1.html#D1" title="INTEGRA1:def.1">INTEGRA1:def 1</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E9">A4</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1"><span class="kw">now </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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:7_1"/><span class="kw">then </span>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">b</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">a</font> or  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">a</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">b</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T57" title="XREAL_1:th.57">XREAL_1:57</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="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">a</font></span>)</span> or  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">a</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">b</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</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="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</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#T31" title="XREAL_1:th.31">XREAL_1:31</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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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" 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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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" 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" 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">set</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" 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">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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#T90" title="RELAT_1:th.90">RELAT_1:90</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" title="XBOOLE_0:def.4">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#NK2" title="RELAT_1:NK.2">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="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="mesfun6c.html#K1" title="MESFUN6C:func.1">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#T21" title="FUNCT_1:th.21">FUNCT_1:21</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="seq_1.html#K1" title="SEQ_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="seq_1.html#K1" title="SEQ_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="mesfun6c.html#K1" title="MESFUN6C:func.1">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#D8" title="FUNCT_3:def.8">FUNCT_3:def 8</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="seq_1.html#K1" title="SEQ_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="seq_1.html#K1" title="SEQ_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="mesfun6c.html#K1" title="MESFUN6C:func.1">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#T72" title="FUNCT_1:th.72">FUNCT_1:72</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="seq_1.html#K1" title="SEQ_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="seq_1.html#K1" title="SEQ_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="mesfun6c.html#K1" title="MESFUN6C:func.1">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#T72" title="FUNCT_1:th.72">FUNCT_1:72</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="seq_1.html#K1" title="SEQ_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="seq_1.html#K1" title="SEQ_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="mesfun6c.html#K1" title="MESFUN6C:func.1">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#T72" title="FUNCT_1:th.72">FUNCT_1:72</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="seq_1.html#K1" title="SEQ_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="seq_1.html#K1" title="SEQ_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="mesfun6c.html#K1" title="MESFUN6C:func.1">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#T72" title="FUNCT_1:th.72">FUNCT_1:72</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" title="XBOOLE_0:def.4">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#NK2" title="RELAT_1:NK.2">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#D8" title="FUNCT_3:def.8">FUNCT_3:def 8</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#NK2" title="RELAT_1:NK.2">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="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="mesfun6c.html#K1" title="MESFUN6C:func.1">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#D8" title="FUNCT_3:def.8">FUNCT_3:def 8</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#T21" title="FUNCT_1:th.21">FUNCT_1:21</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" title="XBOOLE_0:def.4">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#T90" title="RELAT_1:th.90">RELAT_1:90</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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">set</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" 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">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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#T90" title="RELAT_1:th.90">RELAT_1:90</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" title="XBOOLE_0:def.4">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#NK2" title="RELAT_1:NK.2">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="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="mesfun6c.html#K1" title="MESFUN6C:func.1">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#T21" title="FUNCT_1:th.21">FUNCT_1:21</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="seq_1.html#K1" title="SEQ_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="seq_1.html#K1" title="SEQ_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="mesfun6c.html#K1" title="MESFUN6C:func.1">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#D8" title="FUNCT_3:def.8">FUNCT_3:def 8</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="seq_1.html#K1" title="SEQ_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="seq_1.html#K1" title="SEQ_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="mesfun6c.html#K1" title="MESFUN6C:func.1">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#T72" title="FUNCT_1:th.72">FUNCT_1:72</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="seq_1.html#K1" title="SEQ_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="seq_1.html#K1" title="SEQ_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="mesfun6c.html#K1" title="MESFUN6C:func.1">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#T72" title="FUNCT_1:th.72">FUNCT_1:72</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="seq_1.html#K1" title="SEQ_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="seq_1.html#K1" title="SEQ_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="mesfun6c.html#K1" title="MESFUN6C:func.1">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#T72" title="FUNCT_1:th.72">FUNCT_1:72</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="seq_1.html#K1" title="SEQ_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="seq_1.html#K1" title="SEQ_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="mesfun6c.html#K1" title="MESFUN6C:func.1">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#T72" title="FUNCT_1:th.72">FUNCT_1:72</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" title="XBOOLE_0:def.4">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#NK2" title="RELAT_1:NK.2">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#D8" title="FUNCT_3:def.8">FUNCT_3:def 8</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#NK2" title="RELAT_1:NK.2">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="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="mesfun6c.html#K1" title="MESFUN6C:func.1">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#D8" title="FUNCT_3:def.8">FUNCT_3:def 8</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#T21" title="FUNCT_1:th.21">FUNCT_1:21</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" title="XBOOLE_0:def.4">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#T90" title="RELAT_1:th.90">RELAT_1:90</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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" title="TARSKI:th.2">TARSKI:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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">set</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="seq_1.html#K1" title="SEQ_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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x0</font>
 
<div><a class="txt" 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">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="seq_1.html#K1" title="SEQ_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="seq_1.html#K1" title="SEQ_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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="seq_1.html#K1" title="SEQ_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="seq_1.html#K1" title="SEQ_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#T90" title="RELAT_1:th.90">RELAT_1:90</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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c15">x0</font> = <font color="Maroon" title="c14">x0</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <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></span>;<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" title="XBOOLE_1:th.19">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#NK2" title="RELAT_1:NK.2">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#D8" title="FUNCT_3:def.8">FUNCT_3:def 8</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="seq_1.html#K1" title="SEQ_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="mesfun6c.html#K1" title="MESFUN6C:func.1">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="seq_1.html#K1" title="SEQ_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#T23" title="FUNCT_1:th.23">FUNCT_1:23</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" title="XBOOLE_1:th.19">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#NK2" title="RELAT_1:NK.2">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#D8" title="FUNCT_3:def.8">FUNCT_3:def 8</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="seq_1.html#K1" title="SEQ_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="mesfun6c.html#K1" title="MESFUN6C:func.1">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="seq_1.html#K1" title="SEQ_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#T23" title="FUNCT_1:th.23">FUNCT_1:23</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="seq_1.html#K1" title="SEQ_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="mesfun6c.html#K1" title="MESFUN6C:func.1">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="seq_1.html#K1" title="SEQ_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="seq_1.html#K1" title="SEQ_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="seq_1.html#K1" title="SEQ_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#T68" title="FUNCT_3:th.68">FUNCT_3:68</a></span>;<br/>
<span class="lab"><font color="Green" title="E19">A22</font></span>: <font color="Maroon" title="c3">x1</font> <a href="seq_1.html#K1" title="SEQ_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="seq_1.html#K1" title="SEQ_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#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></span>
<br/>.= 
<font color="Maroon" title="c5">x2</font> <a href="seq_1.html#K1" title="SEQ_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#T72" title="FUNCT_1:th.72">FUNCT_1:72</a>, <a class="txt" href="#E1:7"><span class="lab"><font color="Green" title="E6">A1</font></span></a></span>
;<br/>
<span class="lab"><font color="Green" title="E20">A23</font></span>: <font color="Maroon" title="c4">y1</font> <a href="seq_1.html#K1" title="SEQ_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="seq_1.html#K1" title="SEQ_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#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></span>
<br/>.= 
<font color="Maroon" title="c6">y2</font> <a href="seq_1.html#K1" title="SEQ_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#T72" title="FUNCT_1:th.72">FUNCT_1:72</a>, <a class="txt" href="#E1:7"><span class="lab"><font color="Green" title="E6">A1</font></span></a></span>
;<br/>
<a NAME="E16:7_2_1_1_2"/>
( <font color="Maroon" title="c9">u01</font> <a href="seq_1.html#K1" title="SEQ_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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">x0</font> &amp; <font color="Maroon" title="c11">u02</font> <a href="seq_1.html#K1" title="SEQ_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="seq_1.html#K1" title="SEQ_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#T72" title="FUNCT_1:th.72">FUNCT_1:72</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="seq_1.html#K1" title="SEQ_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="seq_1.html#K1" title="SEQ_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#T68" title="FUNCT_3:th.68">FUNCT_3:68</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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#D17" title="FUNCT_1:def.17">FUNCT_1:def 17</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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" 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#T90" title="RELAT_1:th.90">RELAT_1:90</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#D8" title="FDIFF_1:def.8">FDIFF_1:def 8</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" title="XBOOLE_1:th.28">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#T90" title="RELAT_1:th.90">RELAT_1:90</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#D8" title="FDIFF_1:def.8">FDIFF_1:def 8</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" title="XBOOLE_1:th.28">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">set</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="seq_1.html#K1" title="SEQ_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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x0</font>
 
<div><a class="txt" 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">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="seq_1.html#K1" title="SEQ_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="seq_1.html#K1" title="SEQ_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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="seq_1.html#K1" title="SEQ_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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">x0</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c15">x0</font> = <font color="Maroon" title="c14">x0</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <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></span>;<br/>
<a NAME="E3:7_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="seq_1.html#K1" title="SEQ_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="seq_1.html#K1" title="SEQ_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"><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="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></span>;<br/>
<a NAME="E4:7_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="seq_1.html#K1" title="SEQ_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#D8" title="FDIFF_1:def.8">FDIFF_1:def 8</a></span>;<br/>
<a NAME="E5:7_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="seq_1.html#K1" title="SEQ_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#D8" title="FDIFF_1:def.8">FDIFF_1:def 8</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E19">A31</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="7_2_1_2_3_1"><span class="kw">now </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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="real_1.html#NM1" title="REAL_1: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" title="RCOMP_1:def.1">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="real_1.html#NM1" title="REAL_1: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" onclick="hs2(this)" href="javascript:()" title="7_2_1_2_3_1_1"><span class="kw">now </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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">x0</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">a</font> or  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">a</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c15">x0</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T57" title="XREAL_1:th.57">XREAL_1:57</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="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">x0</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">a</font></span>)</span> or  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">a</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c15">x0</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</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="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</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#T31" title="XREAL_1:th.31">XREAL_1:31</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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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" onclick="hs2(this)" href="javascript:()" title="7_2_1_2_3_1_2"><span class="kw">now </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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">x0</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">b</font> or  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">b</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c15">x0</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T57" title="XREAL_1:th.57">XREAL_1:57</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="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">x0</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">b</font></span>)</span> or  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">b</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c15">x0</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</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="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</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#T31" title="XREAL_1:th.31">XREAL_1:31</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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="real_1.html#NM1" title="REAL_1: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" title="RCOMP_1:def.2">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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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"/>
 <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" 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" 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="#E6:7_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">A39</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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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"/>
<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="real_1.html#NM1" title="REAL_1: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>, <a class="txt" href="#E1:7_2_1_2_3_2_1_1"><span class="lab"><font color="Green" title="E20">A39</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">A40</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" title="RCOMP_1:def.1">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">A41</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#T16" title="FDIFF_1:th.16">FDIFF_1:16</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">A42</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" title="FDIFF_3:th.22">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">A40</font></span></a>, <a class="ref" href="fdiff_1.html#T16" title="FDIFF_1:th.16">FDIFF_1:16</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">A43</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" title="FDIFF_3:th.22">FDIFF_3:22</a></span>;<br/><a NAME="E8:7_2_1_2_3_2_1_1"/><span class="lab"><font color="Green" title="E25">A44</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">A42</font></span></a>, <a class="txt" href="#E7:7_2_1_2_3_2_1_1"><span class="lab"><font color="Green" title="E24">A43</font></span></a>, <a class="ref" href="fdiff_3.html#T17" title="FDIFF_3:th.17">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" 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">A45</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="real_1.html#K9" title="REAL_1:func.9">-</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">A42</font></span></a>, <a class="txt" href="#E7:7_2_1_2_3_2_1_1"><span class="lab"><font color="Green" title="E24">A43</font></span></a>, <a class="ref" href="fdiff_3.html#T17" title="FDIFF_3:th.17">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="fdiff_1.html#V1" title="FDIFF_1:attr.1">convergent_to_0</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="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   holds <font color="Olive" title="b1">h</font> <a href="seq_1.html#K1" title="SEQ_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#K6" title="NUMBERS:func.6">0</a>  ) holds <br/> <a href="seq_2.html#K2" title="SEQ_2:func.2">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#K6" title="NUMBERS:func.6">0</a> 
 
<div><a class="txt" 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="fdiff_1.html#V1" title="FDIFF_1:attr.1">convergent_to_0</a>  <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   holds <font color="Maroon" title="c16">h</font> <a href="seq_1.html#K1" title="SEQ_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#K6" title="NUMBERS:func.6">0</a>  ) holds <br/> <a href="seq_2.html#K2" title="SEQ_2:func.2">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#K6" title="NUMBERS:func.6">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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   holds <font color="Maroon" title="c16">h</font> <a href="seq_1.html#K1" title="SEQ_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#K6" title="NUMBERS:func.6">0</a>  ) implies  <a href="seq_2.html#K2" title="SEQ_2:func.2">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#K6" title="NUMBERS:func.6">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">A46</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="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   holds <font color="Maroon" title="c16">h</font> <a href="seq_1.html#K1" title="SEQ_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#K6" title="NUMBERS:func.6">0</a>  ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="seq_2.html#K2" title="SEQ_2:func.2">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#K6" title="NUMBERS:func.6">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">A47</font></span>: 
( <font color="Maroon" title="c16">h</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c16">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="fdiff_1.html#D1" title="FDIFF_1:def.1">FDIFF_1:def 1</a></span>;<br/>
<a NAME="E3:7_2_1_2_3_2_1_1_1_1"/><span class="lab"><font color="Green" title="E29">A48</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">b</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</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">A39</font></span></a>, <a class="ref" href="xreal_1.html#T52" title="XREAL_1:th.52">XREAL_1:52</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c18">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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">A49</font></span>: 
for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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/> <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </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="c2">b</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c15">x0</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</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">A47</font></span></a>, <a class="txt" href="#E3:7_2_1_2_3_2_1_1_1_1"><span class="lab"><font color="Green" title="E29">A48</font></span></a>, <a class="ref" href="seq_2.html#D7" title="SEQ_2:def.7">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">A50</font></span>: 
for <font color="Olive" title="b1">p1</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st  <a href="numbers.html#K6" title="NUMBERS:func.6">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="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>for <font color="Olive" title="b3">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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/> <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</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="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="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<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="p0">(<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="p5">(<span class="default"><span class="p0">(<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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b3">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</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" 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#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="numbers.html#K6" title="NUMBERS:func.6">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="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>for <font color="Olive" title="b2">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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/> <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</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="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="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<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="p0">(<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="p5">(<span class="default"><span class="p0">(<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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</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">A51</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">n1</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>for <font color="Olive" title="b2">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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/> <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</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="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="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<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="p0">(<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="p5">(<span class="default"><span class="p0">(<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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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/> <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</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="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="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<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="p0">(<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="p5">(<span class="default"><span class="p0">(<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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</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="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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="seq_1.html#K1" title="SEQ_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#K6" title="NUMBERS:func.6">0</a> 
 
<div><a class="txt" 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="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="seq_1.html#K1" title="SEQ_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#K6" title="NUMBERS:func.6">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">A52</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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="seq_1.html#K1" title="SEQ_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#K6" title="NUMBERS:func.6">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">A53</font></span>: 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </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="c2">b</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c15">x0</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</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">A49</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">A52</font></span></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">A54</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">h</font> <a href="seq_1.html#K1" title="SEQ_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">A46</font></span></a></span>;<br/>
<a NAME="E4:7_2_1_2_3_2_1_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A55</font></span>: 
<font color="Maroon" title="c16">h</font> <a href="seq_1.html#K1" title="SEQ_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="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c15">x0</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</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_1_1"><span class="lab"><font color="Green" title="E34">A53</font></span></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>;<br/>
<a NAME="E5:7_2_1_2_3_2_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E37">A56</font></span>: 
<font color="Maroon" title="c1">a</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">x0</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="seq_1.html#K1" title="SEQ_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">A39</font></span></a>, <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">A54</font></span></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/>
<a NAME="E6:7_2_1_2_3_2_1_1_1_1_1_1"/>
<font color="Maroon" title="c15">x0</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="seq_1.html#K1" title="SEQ_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="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c15">x0</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>
 
<span class="kw">by</span> <span class="lab"><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">A55</font></span></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/>
<a NAME="E7:7_2_1_2_3_2_1_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A57</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c15">x0</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">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="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c15">x0</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c15">x0</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</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">A48</font></span></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/>
<a NAME="E8:7_2_1_2_3_2_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E39">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" title="XBOOLE_1:th.19">XBOOLE_1:19</a></span>;<br/>
<a NAME="E9:7_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">A40</font></span></a></span>;<br/>
<a NAME="E10:7_2_1_2_3_2_1_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">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" title="VALUED_1:th.12">VALUED_1:12</a></span>;<br/>
<a NAME="E11:7_2_1_2_3_2_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E41">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="#E8:7_2_1_2_3_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E39">A58</font></span></a>, <a class="ref" href="valued_1.html#T12" title="VALUED_1:th.12">VALUED_1:12</a></span>;<br/>
<a NAME="E12:7_2_1_2_3_2_1_1_1_1_1_1"/>
<font color="Maroon" title="c15">x0</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <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="real_1.html#NM1" title="REAL_1: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="#E5:7_2_1_2_3_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E37">A56</font></span></a>, <a class="txt" href="#E7:7_2_1_2_3_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E38">A57</font></span></a></span>;<br/>
<a NAME="E13:7_2_1_2_3_2_1_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A61</font></span>: 
<font color="Maroon" title="c15">x0</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <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" title="RCOMP_1:def.1">RCOMP_1:def 1</a></span>;<br/>
<a NAME="E14:7_2_1_2_3_2_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E43">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">A46</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI: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">A63</font></span>: 
 <a href="seq_2.html#K2" title="SEQ_2:func.2">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="seq_1.html#K1" title="SEQ_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#T41" title="SEQ_4:th.41">SEQ_4:41</a></span>;<br/>
<span class="lab"><font color="Green" title="E45">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="seq_1.html#K1" title="SEQ_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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">c1</font> <a href="seq_1.html#K1" title="SEQ_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#T11" title="SEQ_1:th.11">SEQ_1:11</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c15">x0</font>
<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">A62</font></span></a>, <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">A63</font></span></a>, <a class="ref" href="seq_4.html#T40" title="SEQ_4:th.40">SEQ_4:40</a></span>
;<br/>
<a NAME="E17:7_2_1_2_3_2_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E46">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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_2_1_2_3_2_1_1_1_1_1_1_2"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:7_2_1_2_3_2_1_1_1_1_1_1_2"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</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> holds <br/><font color="Olive" title="b1">x</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="txt" href="#E1:7_2_1_2_3_2_1_1_1_1"><span class="lab"><font color="Green" title="E27">A46</font></span></a>, <a class="txt" href="#E10:7_2_1_2_3_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E40">A59</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:7_2_1_2_3_2_1_1_1_1_1_1_2"/>
 <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="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="lab"><font color="Green" title="E47">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="seq_1.html#K1" title="SEQ_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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</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="seq_1.html#K1" title="SEQ_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#T12" title="SEQ_1:th.12">SEQ_1:12</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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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="seq_1.html#K1" title="SEQ_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#T6" title="RFUNCT_2:th.6">RFUNCT_2:6</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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</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="seq_1.html#K1" title="SEQ_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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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="seq_1.html#K1" title="SEQ_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">A46</font></span></a>, <a class="ref" href="funct_2.html#T185" title="FUNCT_2:th.185">FUNCT_2:185</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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</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="seq_1.html#K1" title="SEQ_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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">c1</font> <a href="seq_1.html#K1" title="SEQ_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="#E17:7_2_1_2_3_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E46">A65</font></span></a>, <a class="ref" href="funct_2.html#T185" title="FUNCT_2:th.185">FUNCT_2:185</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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</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="seq_1.html#K1" title="SEQ_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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c15">x0</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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="seq_1.html#K1" title="SEQ_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="#E14:7_2_1_2_3_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E43">A62</font></span></a>, <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">A63</font></span></a>, <a class="ref" href="seq_4.html#T40" title="SEQ_4:th.40">SEQ_4:40</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">A64</font></span></a></span>
;<br/>
<span class="lab"><font color="Green" title="E48">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="seq_1.html#K1" title="SEQ_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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</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="seq_1.html#K1" title="SEQ_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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c15">x0</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">x2</font> <a href="seq_1.html#K1" title="SEQ_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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c15">x0</font></span>)</span></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">A60</font></span></a>, <a class="txt" href="#E13:7_2_1_2_3_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E42">A61</font></span></a>, <a class="ref" href="valued_1.html#T13" title="VALUED_1:th.13">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="seq_1.html#K1" title="SEQ_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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c15">x0</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">x2</font> <a href="seq_1.html#K1" title="SEQ_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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c15">x0</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:7_2_1_2_3_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E42">A61</font></span></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</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="seq_1.html#K1" title="SEQ_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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c15">x0</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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="seq_1.html#K1" title="SEQ_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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c15">x0</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:7_2_1_2_3_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E42">A61</font></span></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></span>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">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="seq_1.html#K1" title="SEQ_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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">x0</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">x2</font> <a href="seq_1.html#K1" title="SEQ_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="#E10:7_2_1_2_3_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E40">A59</font></span></a>, <a class="ref" href="valued_1.html#T13" title="VALUED_1:th.13">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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">x0</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">x2</font> <a href="seq_1.html#K1" title="SEQ_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">A40</font></span></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">x0</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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="seq_1.html#K1" title="SEQ_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">A40</font></span></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></span>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">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="E21:7_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="seq_1.html#K1" title="SEQ_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#K6" title="NUMBERS:func.6">0</a> 
 <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">A66</font></span></a>, <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">A67</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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/> <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</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="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="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<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="p0">(<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="p5">(<span class="default"><span class="p0">(<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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</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">A51</font></span></a>, <a class="ref" href="complex1.html#T130" title="COMPLEX1:th.130">COMPLEX1:130</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="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="seq_2.html#V4" title="SEQ_2:attr.4">convergent</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">A50</font></span></a>, <a class="ref" href="seq_2.html#D6" title="SEQ_2:def.6">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#K2" title="SEQ_2:func.2">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#K6" title="NUMBERS:func.6">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">A50</font></span></a>, <a class="ref" href="seq_2.html#D7" title="SEQ_2:def.7">SEQ_2:def 7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="real_1.html#K9" title="REAL_1:func.9">-</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#K6" title="NUMBERS:func.6">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">A44</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">A45</font></span></a>, <a class="ref" href="fdiff_3.html#D6" title="FDIFF_3:def.6">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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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">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">A43</font></span></a>, <a class="ref" href="fdiff_3.html#T22" title="FDIFF_3:th.22">FDIFF_3:22</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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#T25" title="RCOMP_1:th.25">RCOMP_1:25</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" 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">set</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" 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">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="real_1.html#NM1" title="REAL_1: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" title="RCOMP_1:def.2">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="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.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="real_1.html#NM1" title="REAL_1: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="real_1.html#NM1" title="REAL_1: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" title="RCOMP_1:def.1">RCOMP_1:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a>, <a class="ref" href="fdiff_1.html#T34" title="FDIFF_1:th.34">FDIFF_1:34</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="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_2.html#T16" title="FDIFF_2:th.16">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="seq_1.html#K1" title="SEQ_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#D8" title="FDIFF_1:def.8">FDIFF_1:def 8</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="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a>, <a class="ref" href="fdiff_1.html#T34" title="FDIFF_1:th.34">FDIFF_1:34</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="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_2.html#T16" title="FDIFF_2:th.16">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="seq_1.html#K1" title="SEQ_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#D8" title="FDIFF_1:def.8">FDIFF_1:def 8</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#T90" title="RELAT_1:th.90">RELAT_1:90</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" title="XBOOLE_1:th.1">XBOOLE_1:1</a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">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" title="XBOOLE_1:th.1">XBOOLE_1:1</a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">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#T90" title="RELAT_1:th.90">RELAT_1:90</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">set</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="seq_1.html#K1" title="SEQ_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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x0</font>
 <div><a class="txt" 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">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="seq_1.html#K1" title="SEQ_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="seq_1.html#K1" title="SEQ_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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="seq_1.html#K1" title="SEQ_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="seq_1.html#K1" title="SEQ_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#T90" title="RELAT_1:th.90">RELAT_1:90</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" title="XBOOLE_1:th.1">XBOOLE_1:1</a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</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#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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c16">x0</font> = 
<font color="Maroon" title="c3">x1</font> <a href="seq_1.html#K1" title="SEQ_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#T72" title="FUNCT_1:th.72">FUNCT_1:72</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="seq_1.html#K1" title="SEQ_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#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></span>
<br/>.= 
<font color="Maroon" title="c5">x2</font> <a href="seq_1.html#K1" title="SEQ_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#T72" title="FUNCT_1:th.72">FUNCT_1:72</a>, <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="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="seq_1.html#K1" title="SEQ_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#T72" title="FUNCT_1:th.72">FUNCT_1:72</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="seq_1.html#K1" title="SEQ_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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c16">x0</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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"/>
<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="real_1.html#NM1" title="REAL_1: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>, <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></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" title="RCOMP_1:def.1">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#T16" title="FDIFF_1:th.16">FDIFF_1:16</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" title="FDIFF_3:th.22">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#T16" title="FDIFF_1:th.16">FDIFF_1:16</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" title="FDIFF_3:th.22">FDIFF_3:22</a></span>;<br/><a NAME="E8:7_2_1_2_3_2_1_3"/><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="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" title="FDIFF_3:th.11">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" 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="real_1.html#K9" title="REAL_1:func.9">-</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" title="FDIFF_3:th.11">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="fdiff_1.html#V1" title="FDIFF_1:attr.1">convergent_to_0</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="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   holds <font color="Olive" title="b1">h</font> <a href="seq_1.html#K1" title="SEQ_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#K6" title="NUMBERS:func.6">0</a>  ) holds <br/> <a href="seq_2.html#K2" title="SEQ_2:func.2">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#K6" title="NUMBERS:func.6">0</a> 
 
<div><a class="txt" 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="fdiff_1.html#V1" title="FDIFF_1:attr.1">convergent_to_0</a>  <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   holds <font color="Maroon" title="c16">h</font> <a href="seq_1.html#K1" title="SEQ_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#K6" title="NUMBERS:func.6">0</a>  ) holds <br/> <a href="seq_2.html#K2" title="SEQ_2:func.2">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#K6" title="NUMBERS:func.6">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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   holds <font color="Maroon" title="c16">h</font> <a href="seq_1.html#K1" title="SEQ_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#K6" title="NUMBERS:func.6">0</a>  ) implies  <a href="seq_2.html#K2" title="SEQ_2:func.2">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#K6" title="NUMBERS:func.6">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="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   holds <font color="Maroon" title="c16">h</font> <a href="seq_1.html#K1" title="SEQ_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#K6" title="NUMBERS:func.6">0</a>  ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="seq_2.html#K2" title="SEQ_2:func.2">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#K6" title="NUMBERS:func.6">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="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c16">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="fdiff_1.html#D1" title="FDIFF_1:def.1">FDIFF_1:def 1</a></span>;<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#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">b</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</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#T52" title="XREAL_1:th.52">XREAL_1:52</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c18">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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/> <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </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="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</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="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="seq_2.html#D7" title="SEQ_2:def.7">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#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st  <a href="numbers.html#K6" title="NUMBERS:func.6">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="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>for <font color="Olive" title="b3">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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/> <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</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="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="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<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="p0">(<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="p5">(<span class="default"><span class="p0">(<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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b3">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</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" 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#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="numbers.html#K6" title="NUMBERS:func.6">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="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>for <font color="Olive" title="b2">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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/> <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</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="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="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<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="p0">(<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="p5">(<span class="default"><span class="p0">(<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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</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#K6" title="NUMBERS:func.6">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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">n1</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>for <font color="Olive" title="b2">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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/> <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</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="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="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<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="p0">(<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="p5">(<span class="default"><span class="p0">(<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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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/> <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</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="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="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<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="p0">(<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="p5">(<span class="default"><span class="p0">(<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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</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="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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="seq_1.html#K1" title="SEQ_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#K6" title="NUMBERS:func.6">0</a> 
 
<div><a class="txt" 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="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="seq_1.html#K1" title="SEQ_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#K6" title="NUMBERS:func.6">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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="seq_1.html#K1" title="SEQ_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#K6" title="NUMBERS:func.6">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>: 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </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="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</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="E3:7_2_1_2_3_2_1_3_1_1_1_1"/><span class="lab"><font color="Green" title="E35">A97</font></span>: 
<font color="Maroon" title="c16">h</font> <a href="seq_1.html#K1" title="SEQ_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#K6" title="NUMBERS:func.6">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="E4:7_2_1_2_3_2_1_3_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A98</font></span>: 
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="seq_1.html#K1" title="SEQ_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="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
 
<span class="kw">by</span> <span class="lab"><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>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>;<br/>
<a NAME="E5:7_2_1_2_3_2_1_3_1_1_1_1"/><span class="lab"><font color="Green" title="E37">A99</font></span>: 
<font color="Maroon" title="c15">x0</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="seq_1.html#K1" title="SEQ_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="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">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_1_1"><span class="lab"><font color="Green" title="E35">A97</font></span></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/>
<a NAME="E6:7_2_1_2_3_2_1_3_1_1_1_1"/>
<font color="Maroon" title="c15">x0</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">x0</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</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="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="seq_1.html#K1" title="SEQ_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="#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#T15" title="XREAL_1:th.15">XREAL_1:15</a></span>;<br/>
<a NAME="E7:7_2_1_2_3_2_1_3_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A100</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c15">x0</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">x0</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</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="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">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#T15" title="XREAL_1:th.15">XREAL_1:15</a></span>;<br/>
<a NAME="E8:7_2_1_2_3_2_1_3_1_1_1_1"/><span class="lab"><font color="Green" title="E39">A101</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" title="XBOOLE_1:th.19">XBOOLE_1:19</a></span>;<br/>
<a NAME="E9:7_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="E10:7_2_1_2_3_2_1_3_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A102</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" title="VALUED_1:th.12">VALUED_1:12</a></span>;<br/>
<a NAME="E11:7_2_1_2_3_2_1_3_1_1_1_1"/><span class="lab"><font color="Green" title="E41">A103</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="#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>, <a class="ref" href="valued_1.html#T12" title="VALUED_1:th.12">VALUED_1:12</a></span>;<br/>
<a NAME="E12:7_2_1_2_3_2_1_3_1_1_1_1"/>
<font color="Maroon" title="c15">x0</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <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="real_1.html#NM1" title="REAL_1: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="#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="txt" href="#E7:7_2_1_2_3_2_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E38">A100</font></span></a></span>;<br/>
<a NAME="E13:7_2_1_2_3_2_1_3_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A104</font></span>: 
<font color="Maroon" title="c15">x0</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <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" title="RCOMP_1:def.1">RCOMP_1:def 1</a></span>;<br/>
<a NAME="E14:7_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" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E15:7_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>: 
 <a href="seq_2.html#K2" title="SEQ_2:func.2">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#T40" title="SEQ_4:th.40">SEQ_4:40</a></span>;<br/>
<span class="lab"><font color="Green" title="E44">A106</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="seq_1.html#K1" title="SEQ_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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">c1</font> <a href="seq_1.html#K1" title="SEQ_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#T11" title="SEQ_1:th.11">SEQ_1:11</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</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_3_1_1_1_1"><span class="lab"><font color="Green" title="E43">A105</font></span></a>, <a class="ref" href="seq_4.html#T41" title="SEQ_4:th.41">SEQ_4:41</a></span>
;<br/>
<a NAME="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 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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_2_1_2_3_2_1_3_1_1_1_1_2"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:7_2_1_2_3_2_1_3_1_1_1_1_2"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</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> holds <br/><font color="Olive" title="b1">x</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="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="#E10:7_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="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:7_2_1_2_3_2_1_3_1_1_1_1_2"/>
 <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="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="lab"><font color="Green" title="E46">A108</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="seq_1.html#K1" title="SEQ_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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</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="seq_1.html#K1" title="SEQ_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#T12" title="SEQ_1:th.12">SEQ_1:12</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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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="seq_1.html#K1" title="SEQ_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#T6" title="RFUNCT_2:th.6">RFUNCT_2:6</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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</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="seq_1.html#K1" title="SEQ_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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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="seq_1.html#K1" title="SEQ_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#T185" title="FUNCT_2:th.185">FUNCT_2:185</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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</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="seq_1.html#K1" title="SEQ_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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">c1</font> <a href="seq_1.html#K1" title="SEQ_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="#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="funct_2.html#T185" title="FUNCT_2:th.185">FUNCT_2:185</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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</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="seq_1.html#K1" title="SEQ_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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c15">x0</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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="seq_1.html#K1" title="SEQ_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_3_1_1_1_1"><span class="lab"><font color="Green" title="E43">A105</font></span></a>, <a class="ref" href="seq_4.html#T41" title="SEQ_4:th.41">SEQ_4:41</a>, <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></span>
;<br/>
<span class="lab"><font color="Green" title="E47">A109</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="seq_1.html#K1" title="SEQ_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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</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="seq_1.html#K1" title="SEQ_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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c15">x0</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">x2</font> <a href="seq_1.html#K1" title="SEQ_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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c15">x0</font></span>)</span></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="txt" href="#E13:7_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="ref" href="valued_1.html#T13" title="VALUED_1:th.13">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="seq_1.html#K1" title="SEQ_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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c15">x0</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">x2</font> <a href="seq_1.html#K1" title="SEQ_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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c15">x0</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:7_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="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</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="seq_1.html#K1" title="SEQ_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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c15">x0</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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="seq_1.html#K1" title="SEQ_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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c15">x0</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:7_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="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></span>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">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="seq_1.html#K1" title="SEQ_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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">x0</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">x2</font> <a href="seq_1.html#K1" title="SEQ_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="#E10:7_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#T13" title="VALUED_1:th.13">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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">x0</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">x2</font> <a href="seq_1.html#K1" title="SEQ_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#T72" title="FUNCT_1:th.72">FUNCT_1:72</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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">x0</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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="seq_1.html#K1" title="SEQ_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#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></span>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">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="E21:7_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="seq_1.html#K1" title="SEQ_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#K6" title="NUMBERS:func.6">0</a> 
 <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="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></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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/> <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</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="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="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<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="p0">(<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="p5">(<span class="default"><span class="p0">(<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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</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#T130" title="COMPLEX1:th.130">COMPLEX1:130</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_2.html#D6" title="SEQ_2:def.6">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#K2" title="SEQ_2:func.2">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#K6" title="NUMBERS:func.6">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" title="SEQ_2:def.7">SEQ_2:def 7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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#K6" title="NUMBERS:func.6">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" title="FDIFF_3:def.5">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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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" title="FDIFF_3:th.22">FDIFF_3:22</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="E8:7_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="seq_1.html#K1" title="SEQ_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="seq_1.html#K1" title="SEQ_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"><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="txt" href="#E3:7_2_1_2_3"><span class="lab"><font color="Green" title="E16">A28</font></span></a>, <a class="txt" href="#E4:7_2_1_2_3"><span class="lab"><font color="Green" title="E17">A29</font></span></a>, <a class="txt" href="#E5:7_2_1_2_3"><span class="lab"><font color="Green" title="E18">A30</font></span></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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#D17" title="FUNCT_1:def.17">FUNCT_1:def 17</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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">A110</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" 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">A111</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" 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">A112</font></span>: 
for <font color="Olive" title="b1">x0</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</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" 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">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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">A113</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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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">A113</font></span></a>, <a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</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">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> <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" title="XBOOLE_0:def.4">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#NK2" title="RELAT_1:NK.2">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="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="mesfun6c.html#K2" title="MESFUN6C:func.2">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#T21" title="FUNCT_1:th.21">FUNCT_1:21</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="seq_1.html#K1" title="SEQ_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="seq_1.html#K1" title="SEQ_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="mesfun6c.html#K2" title="MESFUN6C:func.2">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#D8" title="FUNCT_3:def.8">FUNCT_3:def 8</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="seq_1.html#K1" title="SEQ_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="seq_1.html#K1" title="SEQ_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="mesfun6c.html#K2" title="MESFUN6C:func.2">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#T72" title="FUNCT_1:th.72">FUNCT_1:72</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="seq_1.html#K1" title="SEQ_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="seq_1.html#K1" title="SEQ_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="mesfun6c.html#K2" title="MESFUN6C:func.2">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#T72" title="FUNCT_1:th.72">FUNCT_1:72</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="seq_1.html#K1" title="SEQ_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="seq_1.html#K1" title="SEQ_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="mesfun6c.html#K2" title="MESFUN6C:func.2">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#T72" title="FUNCT_1:th.72">FUNCT_1:72</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">A115</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="seq_1.html#K1" title="SEQ_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="seq_1.html#K1" title="SEQ_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="mesfun6c.html#K2" title="MESFUN6C:func.2">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#T72" title="FUNCT_1:th.72">FUNCT_1:72</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">A114</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">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#NK2" title="RELAT_1:NK.2">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#D8" title="FUNCT_3:def.8">FUNCT_3:def 8</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#NK2" title="RELAT_1:NK.2">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="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="mesfun6c.html#K2" title="MESFUN6C:func.2">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">A115</font></span></a>, <a class="ref" href="funct_3.html#D8" title="FUNCT_3:def.8">FUNCT_3:def 8</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#T21" title="FUNCT_1:th.21">FUNCT_1:21</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" title="XBOOLE_0:def.4">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#T90" title="RELAT_1:th.90">RELAT_1:90</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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">set</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" 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">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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">A116</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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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">A116</font></span></a>, <a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</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">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> <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" title="XBOOLE_0:def.4">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#NK2" title="RELAT_1:NK.2">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="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="mesfun6c.html#K2" title="MESFUN6C:func.2">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#T21" title="FUNCT_1:th.21">FUNCT_1:21</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="seq_1.html#K1" title="SEQ_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="seq_1.html#K1" title="SEQ_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="mesfun6c.html#K2" title="MESFUN6C:func.2">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#D8" title="FUNCT_3:def.8">FUNCT_3:def 8</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="seq_1.html#K1" title="SEQ_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="seq_1.html#K1" title="SEQ_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="mesfun6c.html#K2" title="MESFUN6C:func.2">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#T72" title="FUNCT_1:th.72">FUNCT_1:72</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="seq_1.html#K1" title="SEQ_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="seq_1.html#K1" title="SEQ_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="mesfun6c.html#K2" title="MESFUN6C:func.2">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#T72" title="FUNCT_1:th.72">FUNCT_1:72</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="seq_1.html#K1" title="SEQ_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="seq_1.html#K1" title="SEQ_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="mesfun6c.html#K2" title="MESFUN6C:func.2">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#T72" title="FUNCT_1:th.72">FUNCT_1:72</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">A118</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="seq_1.html#K1" title="SEQ_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="seq_1.html#K1" title="SEQ_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="mesfun6c.html#K2" title="MESFUN6C:func.2">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#T72" title="FUNCT_1:th.72">FUNCT_1:72</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">A117</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">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#NK2" title="RELAT_1:NK.2">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#D8" title="FUNCT_3:def.8">FUNCT_3:def 8</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#NK2" title="RELAT_1:NK.2">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="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="mesfun6c.html#K2" title="MESFUN6C:func.2">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">A118</font></span></a>, <a class="ref" href="funct_3.html#D8" title="FUNCT_3:def.8">FUNCT_3:def 8</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#T21" title="FUNCT_1:th.21">FUNCT_1:21</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" title="XBOOLE_0:def.4">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#T90" title="RELAT_1:th.90">RELAT_1:90</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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">A112</font></span></a>, <a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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">set</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="seq_1.html#K1" title="SEQ_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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x0</font>
 
<div><a class="txt" 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">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="seq_1.html#K1" title="SEQ_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="seq_1.html#K1" title="SEQ_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">A119</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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="seq_1.html#K1" title="SEQ_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="seq_1.html#K1" title="SEQ_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">A119</font></span></a>, <a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></span>;<br/>
<a NAME="E3:7_2_1_3_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E16">A120</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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c15">x0</font> = <font color="Maroon" title="c14">x0</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <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">A119</font></span></a></span>;<br/>
<a NAME="E5:7_2_1_3_2"/><span class="lab"><font color="Green" title="E17">A121</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" title="XBOOLE_1:th.19">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">A120</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#NK2" title="RELAT_1:NK.2">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#D8" title="FUNCT_3:def.8">FUNCT_3:def 8</a></span>;<br/>
<a NAME="E8:7_2_1_3_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E18">A122</font></span>: 
<font color="Maroon" title="c10">v01</font> <a href="seq_1.html#K1" title="SEQ_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="mesfun6c.html#K2" title="MESFUN6C:func.2">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="seq_1.html#K1" title="SEQ_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#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></span>;<br/>
<a NAME="E9:7_2_1_3_2"/><span class="lab"><font color="Green" title="E19">A123</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" title="XBOOLE_1:th.19">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">A120</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#NK2" title="RELAT_1:NK.2">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#D8" title="FUNCT_3:def.8">FUNCT_3:def 8</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="seq_1.html#K1" title="SEQ_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="mesfun6c.html#K2" title="MESFUN6C:func.2">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="seq_1.html#K1" title="SEQ_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#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></span>;<br/>
<a NAME="E13:7_2_1_3_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A124</font></span>: 
<font color="Maroon" title="c12">v02</font> <a href="seq_1.html#K1" title="SEQ_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="mesfun6c.html#K2" title="MESFUN6C:func.2">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="seq_1.html#K1" title="SEQ_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="seq_1.html#K1" title="SEQ_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="seq_1.html#K1" title="SEQ_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">A120</font></span></a>, <a class="txt" href="#E9:7_2_1_3_2"><span class="lab"><font color="Green" title="E19">A123</font></span></a>, <a class="ref" href="funct_3.html#T68" title="FUNCT_3:th.68">FUNCT_3:68</a></span>;<br/>
<span class="lab"><font color="Green" title="E21">A125</font></span>: <font color="Maroon" title="c3">x1</font> <a href="seq_1.html#K1" title="SEQ_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="seq_1.html#K1" title="SEQ_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">A120</font></span></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></span>
<br/>.= 
<font color="Maroon" title="c5">x2</font> <a href="seq_1.html#K1" title="SEQ_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">A120</font></span></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a>, <a class="txt" href="#E1:7"><span class="lab"><font color="Green" title="E6">A1</font></span></a></span>
;<br/>
<span class="lab"><font color="Green" title="E22">A126</font></span>: <font color="Maroon" title="c4">y1</font> <a href="seq_1.html#K1" title="SEQ_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="seq_1.html#K1" title="SEQ_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">A120</font></span></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></span>
<br/>.= 
<font color="Maroon" title="c6">y2</font> <a href="seq_1.html#K1" title="SEQ_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">A120</font></span></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a>, <a class="txt" href="#E1:7"><span class="lab"><font color="Green" title="E6">A1</font></span></a></span>
;<br/>
<a NAME="E16:7_2_1_3_2"/>
( <font color="Maroon" title="c10">v01</font> <a href="seq_1.html#K1" title="SEQ_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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">x0</font> &amp; <font color="Maroon" title="c12">v02</font> <a href="seq_1.html#K1" title="SEQ_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="seq_1.html#K1" title="SEQ_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">A120</font></span></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</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="seq_1.html#K1" title="SEQ_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="seq_1.html#K1" title="SEQ_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">A125</font></span></a>, <a class="txt" href="#E15:7_2_1_3_2"><span class="lab"><font color="Green" title="E22">A126</font></span></a>, <a class="txt" href="#E8:7_2_1_3_2"><span class="lab"><font color="Green" title="E18">A122</font></span></a>, <a class="txt" href="#E3:7_2_1_3_2"><span class="lab"><font color="Green" title="E16">A120</font></span></a>, <a class="txt" href="#E5:7_2_1_3_2"><span class="lab"><font color="Green" title="E17">A121</font></span></a>, <a class="ref" href="funct_3.html#T68" title="FUNCT_3:th.68">FUNCT_3:68</a>, <a class="txt" href="#E13:7_2_1_3_2"><span class="lab"><font color="Green" title="E20">A124</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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">A111</font></span></a>, <a class="ref" href="funct_1.html#D17" title="FUNCT_1:def.17">FUNCT_1:def 17</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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">A127</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" 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">A128</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#T90" title="RELAT_1:th.90">RELAT_1:90</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#D8" title="FDIFF_1:def.8">FDIFF_1:def 8</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" title="XBOOLE_1:th.28">XBOOLE_1:28</a></span>
;<br/>
<span class="lab"><font color="Green" title="E16">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="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#T90" title="RELAT_1:th.90">RELAT_1:90</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#D8" title="FDIFF_1:def.8">FDIFF_1:def 8</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" title="XBOOLE_1:th.28">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">set</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="seq_1.html#K1" title="SEQ_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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x0</font>
 
<div><a class="txt" 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">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="seq_1.html#K1" title="SEQ_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="seq_1.html#K1" title="SEQ_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">A130</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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="seq_1.html#K1" title="SEQ_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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">x0</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c15">x0</font> = <font color="Maroon" title="c14">x0</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <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">A130</font></span></a></span>;<br/>
<a NAME="E3:7_2_1_4_3"/><span class="lab"><font color="Green" title="E18">A131</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="seq_1.html#K1" title="SEQ_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="seq_1.html#K1" title="SEQ_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"><span class="lab"><font color="Green" title="E15">A128</font></span></a>, <a class="txt" href="#E1:7_2_1_4_3"><span class="lab"><font color="Green" title="E17">A130</font></span></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></span>;<br/>
<a NAME="E4:7_2_1_4_3"/><span class="lab"><font color="Green" title="E19">A132</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="seq_1.html#K1" title="SEQ_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">A130</font></span></a>, <a class="txt" href="#E1:7_2_1_4"><span class="lab"><font color="Green" title="E15">A128</font></span></a>, <a class="ref" href="fdiff_1.html#D8" title="FDIFF_1:def.8">FDIFF_1:def 8</a></span>;<br/>
<a NAME="E5:7_2_1_4_3"/><span class="lab"><font color="Green" title="E20">A133</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="seq_1.html#K1" title="SEQ_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">A130</font></span></a>, <a class="txt" href="#E1:7_2_1_4"><span class="lab"><font color="Green" title="E15">A128</font></span></a>, <a class="ref" href="fdiff_1.html#D8" title="FDIFF_1:def.8">FDIFF_1:def 8</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E21">A134</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="7_2_1_4_3_1"><span class="kw">now </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">A135</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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="real_1.html#NM1" title="REAL_1: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">A128</font></span></a>, <a class="txt" href="#E1:7_2_1_4_3"><span class="lab"><font color="Green" title="E17">A130</font></span></a>, <a class="ref" href="rcomp_1.html#D1" title="RCOMP_1:def.1">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">A136</font></span>: 
 ex <font color="Olive" title="b1">r</font> being  <a href="real_1.html#NM1" title="REAL_1: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">A137</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="7_2_1_4_3_1_1"><span class="kw">now </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">A138</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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">x0</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">a</font> or  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">a</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c15">x0</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T57" title="XREAL_1:th.57">XREAL_1:57</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="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">x0</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">a</font></span>)</span> or  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">a</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c15">x0</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</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="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</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#T31" title="XREAL_1:th.31">XREAL_1:31</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">A136</font></span></a>, <a class="txt" href="#E1:7_2_1_4_3_1_1"><span class="lab"><font color="Green" title="E25">A138</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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">A139</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="7_2_1_4_3_1_2"><span class="kw">now </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">A140</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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">x0</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">b</font> or  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">b</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c15">x0</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T57" title="XREAL_1:th.57">XREAL_1:57</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="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">x0</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">b</font></span>)</span> or  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">b</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c15">x0</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</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="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</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#T31" title="XREAL_1:th.31">XREAL_1:31</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">A136</font></span></a>, <a class="txt" href="#E1:7_2_1_4_3_1_2"><span class="lab"><font color="Green" title="E26">A140</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="real_1.html#NM1" title="REAL_1: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">A135</font></span></a>, <a class="ref" href="rcomp_1.html#D2" title="RCOMP_1:def.2">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">A135</font></span></a>, <a class="txt" href="#E4:7_2_1_4_3_1"><span class="lab"><font color="Green" title="E24">A137</font></span></a>, <a class="txt" href="#E5:7_2_1_4_3_1"><span class="lab"><font color="Green" title="E25">A139</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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"/>
 <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" 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" 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="#E6:7_2_1_4_3"><span class="lab"><font color="Green" title="E21">A134</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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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"/>
<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="real_1.html#NM1" title="REAL_1: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>, <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></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" title="RCOMP_1:def.1">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#T16" title="FDIFF_1:th.16">FDIFF_1:16</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" title="FDIFF_3:th.22">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#T16" title="FDIFF_1:th.16">FDIFF_1:16</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" title="FDIFF_3:th.22">FDIFF_3:22</a></span>;<br/><a NAME="E8:7_2_1_4_3_2_1_1"/><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="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" title="FDIFF_3:th.17">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" 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="real_1.html#K9" title="REAL_1:func.9">-</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" title="FDIFF_3:th.17">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="fdiff_1.html#V1" title="FDIFF_1:attr.1">convergent_to_0</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="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   holds <font color="Olive" title="b1">h</font> <a href="seq_1.html#K1" title="SEQ_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#K6" title="NUMBERS:func.6">0</a>  ) holds <br/> <a href="seq_2.html#K2" title="SEQ_2:func.2">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#K6" title="NUMBERS:func.6">0</a> 
 
<div><a class="txt" 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="fdiff_1.html#V1" title="FDIFF_1:attr.1">convergent_to_0</a>  <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   holds <font color="Maroon" title="c16">h</font> <a href="seq_1.html#K1" title="SEQ_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#K6" title="NUMBERS:func.6">0</a>  ) holds <br/> <a href="seq_2.html#K2" title="SEQ_2:func.2">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#K6" title="NUMBERS:func.6">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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   holds <font color="Maroon" title="c16">h</font> <a href="seq_1.html#K1" title="SEQ_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#K6" title="NUMBERS:func.6">0</a>  ) implies  <a href="seq_2.html#K2" title="SEQ_2:func.2">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#K6" title="NUMBERS:func.6">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="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   holds <font color="Maroon" title="c16">h</font> <a href="seq_1.html#K1" title="SEQ_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#K6" title="NUMBERS:func.6">0</a>  ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="seq_2.html#K2" title="SEQ_2:func.2">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#K6" title="NUMBERS:func.6">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="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c16">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="fdiff_1.html#D1" title="FDIFF_1:def.1">FDIFF_1:def 1</a></span>;<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#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">b</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</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#T52" title="XREAL_1:th.52">XREAL_1:52</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c18">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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/> <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </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="c2">b</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c15">x0</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</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="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="seq_2.html#D7" title="SEQ_2:def.7">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#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st  <a href="numbers.html#K6" title="NUMBERS:func.6">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="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>for <font color="Olive" title="b3">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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/> <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</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="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="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<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="p0">(<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="p5">(<span class="default"><span class="p0">(<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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b3">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</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" 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#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="numbers.html#K6" title="NUMBERS:func.6">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="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>for <font color="Olive" title="b2">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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/> <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</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="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="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<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="p0">(<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="p5">(<span class="default"><span class="p0">(<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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</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#K6" title="NUMBERS:func.6">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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">n1</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>for <font color="Olive" title="b2">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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/> <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</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="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="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<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="p0">(<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="p5">(<span class="default"><span class="p0">(<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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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/> <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</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="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="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<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="p0">(<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="p5">(<span class="default"><span class="p0">(<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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</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="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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="seq_1.html#K1" title="SEQ_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#K6" title="NUMBERS:func.6">0</a> 
 
<div><a class="txt" 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="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="seq_1.html#K1" title="SEQ_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#K6" title="NUMBERS:func.6">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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="seq_1.html#K1" title="SEQ_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#K6" title="NUMBERS:func.6">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>: 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </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="c2">b</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c15">x0</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</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>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">h</font> <a href="seq_1.html#K1" title="SEQ_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="E4:7_2_1_4_3_2_1_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A158</font></span>: 
<font color="Maroon" title="c16">h</font> <a href="seq_1.html#K1" title="SEQ_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="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c15">x0</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</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" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>;<br/>
<a NAME="E5:7_2_1_4_3_2_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E39">A159</font></span>: 
<font color="Maroon" title="c1">a</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">x0</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="seq_1.html#K1" title="SEQ_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="#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>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/>
<a NAME="E6:7_2_1_4_3_2_1_1_1_1_1_1"/>
<font color="Maroon" title="c15">x0</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="seq_1.html#K1" title="SEQ_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="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c15">x0</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>
 
<span class="kw">by</span> <span class="lab"><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#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/>
<a NAME="E7:7_2_1_4_3_2_1_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A160</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c15">x0</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">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="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c15">x0</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c15">x0</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</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#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/>
<a NAME="E8:7_2_1_4_3_2_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E41">A161</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" title="XBOOLE_1:th.19">XBOOLE_1:19</a></span>;<br/>
<a NAME="E9:7_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="E10:7_2_1_4_3_2_1_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A162</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" title="VALUED_1:th.12">VALUED_1:12</a></span>;<br/>
<a NAME="E11:7_2_1_4_3_2_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E43">A163</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="#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>, <a class="ref" href="valued_1.html#T12" title="VALUED_1:th.12">VALUED_1:12</a></span>;<br/>
<a NAME="E12:7_2_1_4_3_2_1_1_1_1_1_1"/>
<font color="Maroon" title="c15">x0</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <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="real_1.html#NM1" title="REAL_1: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="#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="txt" href="#E7:7_2_1_4_3_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E40">A160</font></span></a></span>;<br/>
<a NAME="E13:7_2_1_4_3_2_1_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A164</font></span>: 
<font color="Maroon" title="c15">x0</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <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" title="RCOMP_1:def.1">RCOMP_1:def 1</a></span>;<br/>
<a NAME="E14:7_2_1_4_3_2_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E45">A165</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" title="TARSKI:def.1">TARSKI: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>: 
 <a href="seq_2.html#K2" title="SEQ_2:func.2">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="seq_1.html#K1" title="SEQ_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#T41" title="SEQ_4:th.41">SEQ_4:41</a></span>;<br/>
<span class="lab"><font color="Green" title="E47">A167</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="seq_1.html#K1" title="SEQ_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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">c1</font> <a href="seq_1.html#K1" title="SEQ_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#T11" title="SEQ_1:th.11">SEQ_1:11</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c15">x0</font>
<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="#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="seq_4.html#T40" title="SEQ_4:th.40">SEQ_4:40</a></span>
;<br/>
<a NAME="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 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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_2_1_4_3_2_1_1_1_1_1_1_2"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:7_2_1_4_3_2_1_1_1_1_1_1_2"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</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> holds <br/><font color="Olive" title="b1">x</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="txt" href="#E10:7_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="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" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:7_2_1_4_3_2_1_1_1_1_1_1_2"/>
 <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="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="lab"><font color="Green" title="E49">A169</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="seq_1.html#K1" title="SEQ_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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</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="seq_1.html#K1" title="SEQ_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#T12" title="SEQ_1:th.12">SEQ_1:12</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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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="seq_1.html#K1" title="SEQ_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#T6" title="RFUNCT_2:th.6">RFUNCT_2:6</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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</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="seq_1.html#K1" title="SEQ_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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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="seq_1.html#K1" title="SEQ_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#T185" title="FUNCT_2:th.185">FUNCT_2:185</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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</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="seq_1.html#K1" title="SEQ_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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">c1</font> <a href="seq_1.html#K1" title="SEQ_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="#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="funct_2.html#T185" title="FUNCT_2:th.185">FUNCT_2:185</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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</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="seq_1.html#K1" title="SEQ_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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c15">x0</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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="seq_1.html#K1" title="SEQ_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="#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="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#T40" title="SEQ_4:th.40">SEQ_4:40</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></span>
;<br/>
<span class="lab"><font color="Green" title="E50">A170</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="seq_1.html#K1" title="SEQ_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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</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="seq_1.html#K1" title="SEQ_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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c15">x0</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">y2</font> <a href="seq_1.html#K1" title="SEQ_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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c15">x0</font></span>)</span></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="#E13:7_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="ref" href="valued_1.html#T13" title="VALUED_1:th.13">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="seq_1.html#K1" title="SEQ_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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c15">x0</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">y2</font> <a href="seq_1.html#K1" title="SEQ_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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c15">x0</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:7_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="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</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="seq_1.html#K1" title="SEQ_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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c15">x0</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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="seq_1.html#K1" title="SEQ_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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c15">x0</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:7_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="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></span>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">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="seq_1.html#K1" title="SEQ_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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">x0</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">y2</font> <a href="seq_1.html#K1" title="SEQ_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="#E10:7_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#T13" title="VALUED_1:th.13">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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">x0</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">y2</font> <a href="seq_1.html#K1" title="SEQ_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#T72" title="FUNCT_1:th.72">FUNCT_1:72</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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">x0</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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="seq_1.html#K1" title="SEQ_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#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></span>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">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="E21:7_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="seq_1.html#K1" title="SEQ_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#K6" title="NUMBERS:func.6">0</a> 
 <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="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></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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/> <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</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="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="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<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="p0">(<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="p5">(<span class="default"><span class="p0">(<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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</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#T130" title="COMPLEX1:th.130">COMPLEX1:130</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="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="seq_2.html#V4" title="SEQ_2:attr.4">convergent</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#D6" title="SEQ_2:def.6">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#K2" title="SEQ_2:func.2">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#K6" title="NUMBERS:func.6">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" title="SEQ_2:def.7">SEQ_2:def 7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="real_1.html#K9" title="REAL_1:func.9">-</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#K6" title="NUMBERS:func.6">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" title="FDIFF_3:def.6">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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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" title="FDIFF_3:th.22">FDIFF_3:22</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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">A171</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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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">A172</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#T25" title="RCOMP_1:th.25">RCOMP_1:25</a></span>;<br/><a NAME="E3:7_2_1_4_3_2_1_2"/><span class="lab"><font color="Green" title="E24">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> <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" 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">set</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" 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">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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">A174</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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="real_1.html#NM1" title="REAL_1: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">A174</font></span></a>, <a class="ref" href="rcomp_1.html#D2" title="RCOMP_1:def.2">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">A175</font></span>: 
 ex <font color="Olive" title="b1">r</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.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="real_1.html#NM1" title="REAL_1: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="real_1.html#NM1" title="REAL_1: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">A175</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" title="RCOMP_1:def.1">RCOMP_1:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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">A176</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">A172</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a>, <a class="ref" href="fdiff_1.html#T34" title="FDIFF_1:th.34">FDIFF_1:34</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">A177</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="txt" href="#E2:7_2_1_4_3_2_1_2"><span class="lab"><font color="Green" title="E23">A172</font></span></a>, <a class="ref" href="fdiff_2.html#T16" title="FDIFF_2:th.16">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">A178</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="seq_1.html#K1" title="SEQ_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">A171</font></span></a>, <a class="txt" href="#E4:7_2_1_4_3_2_1_2"><span class="lab"><font color="Green" title="E25">A176</font></span></a>, <a class="ref" href="fdiff_1.html#D8" title="FDIFF_1:def.8">FDIFF_1:def 8</a></span>;<br/><a NAME="E7:7_2_1_4_3_2_1_2"/><span class="lab"><font color="Green" title="E28">A179</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">A172</font></span></a>, <a class="txt" href="#E3:7_2_1_4_3_2_1_2"><span class="lab"><font color="Green" title="E24">A173</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a>, <a class="ref" href="fdiff_1.html#T34" title="FDIFF_1:th.34">FDIFF_1:34</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">A180</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="txt" href="#E2:7_2_1_4_3_2_1_2"><span class="lab"><font color="Green" title="E23">A172</font></span></a>, <a class="ref" href="fdiff_2.html#T16" title="FDIFF_2:th.16">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">A181</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="seq_1.html#K1" title="SEQ_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">A171</font></span></a>, <a class="txt" href="#E7:7_2_1_4_3_2_1_2"><span class="lab"><font color="Green" title="E28">A179</font></span></a>, <a class="ref" href="fdiff_1.html#D8" title="FDIFF_1:def.8">FDIFF_1:def 8</a></span>;<br/><span class="lab"><font color="Green" title="E31">A182</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#T90" title="RELAT_1:th.90">RELAT_1:90</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">A173</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">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">A173</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">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#T90" title="RELAT_1:th.90">RELAT_1:90</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">set</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="seq_1.html#K1" title="SEQ_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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x0</font>
 <div><a class="txt" 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">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="seq_1.html#K1" title="SEQ_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="seq_1.html#K1" title="SEQ_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">A183</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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="seq_1.html#K1" title="SEQ_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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c16">x0</font></span><br/>

<span class="lab"><font color="Green" title="E33">A184</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#T90" title="RELAT_1:th.90">RELAT_1:90</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">A173</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</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#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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c16">x0</font> = 
<font color="Maroon" title="c4">y1</font> <a href="seq_1.html#K1" title="SEQ_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">A183</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">A184</font></span></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</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="seq_1.html#K1" title="SEQ_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">A173</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">A183</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">A184</font></span></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></span>
<br/>.= 
<font color="Maroon" title="c6">y2</font> <a href="seq_1.html#K1" title="SEQ_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">A173</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">A183</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">A184</font></span></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a>, <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="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="seq_1.html#K1" title="SEQ_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">A183</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">A184</font></span></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</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="seq_1.html#K1" title="SEQ_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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c16">x0</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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">A177</font></span></a>, <a class="txt" href="#E6:7_2_1_4_3_2_1_2"><span class="lab"><font color="Green" title="E27">A178</font></span></a>, <a class="txt" href="#E8:7_2_1_4_3_2_1_2"><span class="lab"><font color="Green" title="E29">A180</font></span></a>, <a class="txt" href="#E9:7_2_1_4_3_2_1_2"><span class="lab"><font color="Green" title="E30">A181</font></span></a>, <a class="txt" href="#E10:7_2_1_4_3_2_1_2"><span class="lab"><font color="Green" title="E31">A182</font></span></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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">A185</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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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"/>
<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="real_1.html#NM1" title="REAL_1: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>, <a class="txt" href="#E1:7_2_1_4_3_2_1_3"><span class="lab"><font color="Green" title="E22">A185</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">A186</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" title="RCOMP_1:def.1">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">A187</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#T16" title="FDIFF_1:th.16">FDIFF_1:16</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">A188</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" title="FDIFF_3:th.22">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">A186</font></span></a>, <a class="ref" href="fdiff_1.html#T16" title="FDIFF_1:th.16">FDIFF_1:16</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">A189</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" title="FDIFF_3:th.22">FDIFF_3:22</a></span>;<br/><a NAME="E8:7_2_1_4_3_2_1_3"/><span class="lab"><font color="Green" title="E27">A190</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">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">A189</font></span></a>, <a class="ref" href="fdiff_3.html#T11" title="FDIFF_3:th.11">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" 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">A191</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="real_1.html#K9" title="REAL_1:func.9">-</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">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">A189</font></span></a>, <a class="ref" href="fdiff_3.html#T11" title="FDIFF_3:th.11">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="fdiff_1.html#V1" title="FDIFF_1:attr.1">convergent_to_0</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="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   holds <font color="Olive" title="b1">h</font> <a href="seq_1.html#K1" title="SEQ_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#K6" title="NUMBERS:func.6">0</a>  ) holds <br/> <a href="seq_2.html#K2" title="SEQ_2:func.2">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#K6" title="NUMBERS:func.6">0</a> 
 
<div><a class="txt" 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="fdiff_1.html#V1" title="FDIFF_1:attr.1">convergent_to_0</a>  <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   holds <font color="Maroon" title="c16">h</font> <a href="seq_1.html#K1" title="SEQ_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#K6" title="NUMBERS:func.6">0</a>  ) holds <br/> <a href="seq_2.html#K2" title="SEQ_2:func.2">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#K6" title="NUMBERS:func.6">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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   holds <font color="Maroon" title="c16">h</font> <a href="seq_1.html#K1" title="SEQ_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#K6" title="NUMBERS:func.6">0</a>  ) implies  <a href="seq_2.html#K2" title="SEQ_2:func.2">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#K6" title="NUMBERS:func.6">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">A192</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="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   holds <font color="Maroon" title="c16">h</font> <a href="seq_1.html#K1" title="SEQ_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#K6" title="NUMBERS:func.6">0</a>  ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="seq_2.html#K2" title="SEQ_2:func.2">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#K6" title="NUMBERS:func.6">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">A193</font></span>: 
( <font color="Maroon" title="c16">h</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c16">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="fdiff_1.html#D1" title="FDIFF_1:def.1">FDIFF_1:def 1</a></span>;<br/>
<a NAME="E3:7_2_1_4_3_2_1_3_1_1"/><span class="lab"><font color="Green" title="E31">A194</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">b</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</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#T52" title="XREAL_1:th.52">XREAL_1:52</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c18">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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">A195</font></span>: 
for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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/> <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </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="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</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">A185</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">A193</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">A194</font></span></a>, <a class="ref" href="seq_2.html#D7" title="SEQ_2:def.7">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">A196</font></span>: 
for <font color="Olive" title="b1">p1</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st  <a href="numbers.html#K6" title="NUMBERS:func.6">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="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>for <font color="Olive" title="b3">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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/> <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</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="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="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<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="p0">(<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="p5">(<span class="default"><span class="p0">(<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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b3">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</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" 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#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="numbers.html#K6" title="NUMBERS:func.6">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="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>for <font color="Olive" title="b2">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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/> <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</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="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="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<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="p0">(<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="p5">(<span class="default"><span class="p0">(<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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</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">A197</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">n1</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>for <font color="Olive" title="b2">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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/> <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</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="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="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<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="p0">(<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="p5">(<span class="default"><span class="p0">(<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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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/> <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</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="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="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<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="p0">(<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="p5">(<span class="default"><span class="p0">(<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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</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="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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="seq_1.html#K1" title="SEQ_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#K6" title="NUMBERS:func.6">0</a> 
 
<div><a class="txt" 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="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="seq_1.html#K1" title="SEQ_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#K6" title="NUMBERS:func.6">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">A198</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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="seq_1.html#K1" title="SEQ_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#K6" title="NUMBERS:func.6">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">A199</font></span>: 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </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="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</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">A195</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">A198</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">A200</font></span>: 
<font color="Maroon" title="c16">h</font> <a href="seq_1.html#K1" title="SEQ_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#K6" title="NUMBERS:func.6">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">A192</font></span></a></span>;<br/>
<a NAME="E4:7_2_1_4_3_2_1_3_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A201</font></span>: 
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="seq_1.html#K1" title="SEQ_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="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</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">A199</font></span></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>;<br/>
<a NAME="E5:7_2_1_4_3_2_1_3_1_1_1_1"/><span class="lab"><font color="Green" title="E39">A202</font></span>: 
<font color="Maroon" title="c15">x0</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="seq_1.html#K1" title="SEQ_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="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">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">A185</font></span></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">A200</font></span></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/>
<a NAME="E6:7_2_1_4_3_2_1_3_1_1_1_1"/>
<font color="Maroon" title="c15">x0</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">x0</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</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="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="seq_1.html#K1" title="SEQ_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="#E4:7_2_1_4_3_2_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E38">A201</font></span></a>, <a class="ref" href="xreal_1.html#T15" title="XREAL_1:th.15">XREAL_1:15</a></span>;<br/>
<a NAME="E7:7_2_1_4_3_2_1_3_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A203</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c15">x0</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">x0</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</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="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">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">A185</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">A194</font></span></a>, <a class="ref" href="xreal_1.html#T15" title="XREAL_1:th.15">XREAL_1:15</a></span>;<br/>
<a NAME="E8:7_2_1_4_3_2_1_3_1_1_1_1"/><span class="lab"><font color="Green" title="E41">A204</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" title="XBOOLE_1:th.19">XBOOLE_1:19</a></span>;<br/>
<a NAME="E9:7_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">A186</font></span></a></span>;<br/>
<a NAME="E10:7_2_1_4_3_2_1_3_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A205</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" title="VALUED_1:th.12">VALUED_1:12</a></span>;<br/>
<a NAME="E11:7_2_1_4_3_2_1_3_1_1_1_1"/><span class="lab"><font color="Green" title="E43">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>  <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="#E8:7_2_1_4_3_2_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E41">A204</font></span></a>, <a class="ref" href="valued_1.html#T12" title="VALUED_1:th.12">VALUED_1:12</a></span>;<br/>
<a NAME="E12:7_2_1_4_3_2_1_3_1_1_1_1"/>
<font color="Maroon" title="c15">x0</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <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="real_1.html#NM1" title="REAL_1: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="#E5:7_2_1_4_3_2_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E39">A202</font></span></a>, <a class="txt" href="#E7:7_2_1_4_3_2_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E40">A203</font></span></a></span>;<br/>
<a NAME="E13:7_2_1_4_3_2_1_3_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A207</font></span>: 
<font color="Maroon" title="c15">x0</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <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" title="RCOMP_1:def.1">RCOMP_1:def 1</a></span>;<br/>
<a NAME="E14:7_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">A192</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E15:7_2_1_4_3_2_1_3_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A208</font></span>: 
 <a href="seq_2.html#K2" title="SEQ_2:func.2">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#T40" title="SEQ_4:th.40">SEQ_4:40</a></span>;<br/>
<span class="lab"><font color="Green" title="E46">A209</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="seq_1.html#K1" title="SEQ_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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">c1</font> <a href="seq_1.html#K1" title="SEQ_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#T11" title="SEQ_1:th.11">SEQ_1:11</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c15">x0</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:7_2_1_4_3_2_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E45">A208</font></span></a>, <a class="ref" href="seq_4.html#T41" title="SEQ_4:th.41">SEQ_4:41</a></span>
;<br/>
<a NAME="E17:7_2_1_4_3_2_1_3_1_1_1_1"/><span class="lab"><font color="Green" title="E47">A210</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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_2_1_4_3_2_1_3_1_1_1_1_2"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:7_2_1_4_3_2_1_3_1_1_1_1_2"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</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> holds <br/><font color="Olive" title="b1">x</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="txt" href="#E1:7_2_1_4_3_2_1_3_1_1"><span class="lab"><font color="Green" title="E29">A192</font></span></a>, <a class="txt" href="#E10:7_2_1_4_3_2_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E42">A205</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:7_2_1_4_3_2_1_3_1_1_1_1_2"/>
 <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="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="lab"><font color="Green" title="E48">A211</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="seq_1.html#K1" title="SEQ_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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</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="seq_1.html#K1" title="SEQ_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#T12" title="SEQ_1:th.12">SEQ_1:12</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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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="seq_1.html#K1" title="SEQ_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#T6" title="RFUNCT_2:th.6">RFUNCT_2:6</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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</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="seq_1.html#K1" title="SEQ_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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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="seq_1.html#K1" title="SEQ_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">A192</font></span></a>, <a class="ref" href="funct_2.html#T185" title="FUNCT_2:th.185">FUNCT_2:185</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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</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="seq_1.html#K1" title="SEQ_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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">c1</font> <a href="seq_1.html#K1" title="SEQ_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="#E17:7_2_1_4_3_2_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E47">A210</font></span></a>, <a class="ref" href="funct_2.html#T185" title="FUNCT_2:th.185">FUNCT_2:185</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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</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="seq_1.html#K1" title="SEQ_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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c15">x0</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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="seq_1.html#K1" title="SEQ_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_3_1_1_1_1"><span class="lab"><font color="Green" title="E45">A208</font></span></a>, <a class="ref" href="seq_4.html#T41" title="SEQ_4:th.41">SEQ_4:41</a>, <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">A209</font></span></a></span>
;<br/>
<span class="lab"><font color="Green" title="E49">A212</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="seq_1.html#K1" title="SEQ_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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</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="seq_1.html#K1" title="SEQ_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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c15">x0</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">y2</font> <a href="seq_1.html#K1" title="SEQ_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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c15">x0</font></span>)</span></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">A206</font></span></a>, <a class="txt" href="#E13:7_2_1_4_3_2_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E44">A207</font></span></a>, <a class="ref" href="valued_1.html#T13" title="VALUED_1:th.13">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="seq_1.html#K1" title="SEQ_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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c15">x0</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">y2</font> <a href="seq_1.html#K1" title="SEQ_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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c15">x0</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:7_2_1_4_3_2_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E44">A207</font></span></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</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="seq_1.html#K1" title="SEQ_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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c15">x0</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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="seq_1.html#K1" title="SEQ_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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c15">x0</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:7_2_1_4_3_2_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E44">A207</font></span></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></span>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">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="seq_1.html#K1" title="SEQ_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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">x0</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">y2</font> <a href="seq_1.html#K1" title="SEQ_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="#E10:7_2_1_4_3_2_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E42">A205</font></span></a>, <a class="ref" href="valued_1.html#T13" title="VALUED_1:th.13">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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">x0</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">y2</font> <a href="seq_1.html#K1" title="SEQ_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">A186</font></span></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">x0</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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="seq_1.html#K1" title="SEQ_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">A186</font></span></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></span>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">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="E21:7_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="seq_1.html#K1" title="SEQ_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#K6" title="NUMBERS:func.6">0</a> 
 <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">A211</font></span></a>, <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">A212</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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/> <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</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="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="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<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="p0">(<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="p5">(<span class="default"><span class="p0">(<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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</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">A197</font></span></a>, <a class="ref" href="complex1.html#T130" title="COMPLEX1:th.130">COMPLEX1:130</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_2.html#D6" title="SEQ_2:def.6">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#K2" title="SEQ_2:func.2">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#K6" title="NUMBERS:func.6">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">A196</font></span></a>, <a class="ref" href="seq_2.html#D7" title="SEQ_2:def.7">SEQ_2:def 7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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#K6" title="NUMBERS:func.6">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">A190</font></span></a>, <a class="ref" href="fdiff_3.html#D5" title="FDIFF_3:def.5">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">A191</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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">A187</font></span></a>, <a class="txt" href="#E7:7_2_1_4_3_2_1_3"><span class="lab"><font color="Green" title="E26">A189</font></span></a>, <a class="ref" href="fdiff_3.html#T22" title="FDIFF_3:th.22">FDIFF_3:22</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="E8:7_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="seq_1.html#K1" title="SEQ_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="seq_1.html#K1" title="SEQ_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"><span class="lab"><font color="Green" title="E15">A128</font></span></a>, <a class="txt" href="#E1:7_2_1_4_3"><span class="lab"><font color="Green" title="E17">A130</font></span></a>, <a class="txt" href="#E3:7_2_1_4_3"><span class="lab"><font color="Green" title="E18">A131</font></span></a>, <a class="txt" href="#E4:7_2_1_4_3"><span class="lab"><font color="Green" title="E19">A132</font></span></a>, <a class="txt" href="#E5:7_2_1_4_3"><span class="lab"><font color="Green" title="E20">A133</font></span></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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">A128</font></span></a>, <a class="txt" href="#E2:7_2_1_4"><span class="lab"><font color="Green" title="E16">A129</font></span></a>, <a class="ref" href="funct_1.html#D17" title="FUNCT_1:def.17">FUNCT_1:def 17</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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">A213</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" 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">A214</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" title="NUMBERS:th.11">NUMBERS:11</a>, <a class="ref" href="relset_1.html#T17" title="RELSET_1:th.17">RELSET_1:17</a></span>;<br/>
<a NAME="E2:7_2_1_5"/><span class="lab"><font color="Green" title="E17">A215</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" title="NUMBERS:th.11">NUMBERS:11</a>, <a class="ref" href="relset_1.html#T17" title="RELSET_1:th.17">RELSET_1:17</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" title="NUMBERS:th.11">NUMBERS:11</a>, <a class="ref" href="relset_1.html#T17" title="RELSET_1:th.17">RELSET_1:17</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E18">A216</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#T67" title="CFUNCT_1:th.67">CFUNCT_1:67</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">A214</font></span></a>, <a class="txt" href="#E2:7_2_1_5"><span class="lab"><font color="Green" title="E17">A215</font></span></a>, <a class="ref" href="cfunct_1.html#T65" title="CFUNCT_1:th.65">CFUNCT_1:65</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">A214</font></span></a>, <a class="txt" href="#E2:7_2_1_5"><span class="lab"><font color="Green" title="E17">A215</font></span></a>, <a class="ref" href="cfunct_1.html#T65" title="CFUNCT_1:th.65">CFUNCT_1:65</a></span>
;<br/>
<a NAME="E5:7_2_1_5"/><span class="lab"><font color="Green" title="E19">A217</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" title="NUMBERS:th.11">NUMBERS:11</a>, <a class="ref" href="relset_1.html#T17" title="RELSET_1:th.17">RELSET_1:17</a></span>;<br/>
<a NAME="E6:7_2_1_5"/><span class="lab"><font color="Green" title="E20">A218</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" title="NUMBERS:th.11">NUMBERS:11</a>, <a class="ref" href="relset_1.html#T17" title="RELSET_1:th.17">RELSET_1:17</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" title="NUMBERS:th.11">NUMBERS:11</a>, <a class="ref" href="relset_1.html#T17" title="RELSET_1:th.17">RELSET_1:17</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#T67" title="CFUNCT_1:th.67">CFUNCT_1:67</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">A217</font></span></a>, <a class="txt" href="#E6:7_2_1_5"><span class="lab"><font color="Green" title="E20">A218</font></span></a>, <a class="ref" href="cfunct_1.html#T65" title="CFUNCT_1:th.65">CFUNCT_1:65</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">A217</font></span></a>, <a class="txt" href="#E6:7_2_1_5"><span class="lab"><font color="Green" title="E20">A218</font></span></a>, <a class="ref" href="cfunct_1.html#T65" title="CFUNCT_1:th.65">CFUNCT_1:65</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">A110</font></span></a>, <a class="txt" href="#E5:7_2_1"><span class="lab"><font color="Green" title="E14">A127</font></span></a>, <a class="txt" href="#E4:7_2_1_5"><span class="lab"><font color="Green" title="E18">A216</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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">A219</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" title="INTEGRA5:th.19">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">A213</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" title="INTEGRA5:th.19">INTEGRA5:19</a></span>
;<br/><a NAME="E8:7_2_1"/><span class="lab"><font color="Green" title="E17">A220</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" 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">A221</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" title="NUMBERS:th.11">NUMBERS:11</a>, <a class="ref" href="relset_1.html#T17" title="RELSET_1:th.17">RELSET_1:17</a></span>;<br/>
<a NAME="E2:7_2_1_7"/><span class="lab"><font color="Green" title="E19">A222</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" title="NUMBERS:th.11">NUMBERS:11</a>, <a class="ref" href="relset_1.html#T17" title="RELSET_1:th.17">RELSET_1:17</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" title="NUMBERS:th.11">NUMBERS:11</a>, <a class="ref" href="relset_1.html#T17" title="RELSET_1:th.17">RELSET_1:17</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E20">A223</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#T64" title="CFUNCT_1:th.64">CFUNCT_1:64</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">A221</font></span></a>, <a class="txt" href="#E2:7_2_1_7"><span class="lab"><font color="Green" title="E19">A222</font></span></a>, <a class="ref" href="cfunct_1.html#T65" title="CFUNCT_1:th.65">CFUNCT_1:65</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">A221</font></span></a>, <a class="txt" href="#E2:7_2_1_7"><span class="lab"><font color="Green" title="E19">A222</font></span></a>, <a class="ref" href="cfunct_1.html#T65" title="CFUNCT_1:th.65">CFUNCT_1:65</a></span>
;<br/>
<a NAME="E5:7_2_1_7"/><span class="lab"><font color="Green" title="E21">A224</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" title="NUMBERS:th.11">NUMBERS:11</a>, <a class="ref" href="relset_1.html#T17" title="RELSET_1:th.17">RELSET_1:17</a></span>;<br/>
<a NAME="E6:7_2_1_7"/><span class="lab"><font color="Green" title="E22">A225</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" title="NUMBERS:th.11">NUMBERS:11</a>, <a class="ref" href="relset_1.html#T17" title="RELSET_1:th.17">RELSET_1:17</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" title="NUMBERS:th.11">NUMBERS:11</a>, <a class="ref" href="relset_1.html#T17" title="RELSET_1:th.17">RELSET_1:17</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#T64" title="CFUNCT_1:th.64">CFUNCT_1:64</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">A224</font></span></a>, <a class="txt" href="#E6:7_2_1_7"><span class="lab"><font color="Green" title="E22">A225</font></span></a>, <a class="ref" href="cfunct_1.html#T65" title="CFUNCT_1:th.65">CFUNCT_1:65</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">A224</font></span></a>, <a class="txt" href="#E6:7_2_1_7"><span class="lab"><font color="Green" title="E22">A225</font></span></a>, <a class="ref" href="cfunct_1.html#T65" title="CFUNCT_1:th.65">CFUNCT_1:65</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">A110</font></span></a>, <a class="txt" href="#E5:7_2_1"><span class="lab"><font color="Green" title="E14">A127</font></span></a>, <a class="txt" href="#E4:7_2_1_7"><span class="lab"><font color="Green" title="E20">A223</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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" title="INTEGRA5:th.19">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">A220</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" title="INTEGRA5:th.19">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">A219</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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">A227</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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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   <a href="integra1.html#V1" title="INTEGRA1:attr.1">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="integra1.html#D1" title="INTEGRA1:def.1">INTEGRA1:def 1</a></span>;<br/><a NAME="E3:7_2_2"/><span class="lab"><font color="Green" title="E11">A228</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">A227</font></span></a>, <a class="ref" href="integra5.html#T19" title="INTEGRA5:th.19">INTEGRA5:19</a></span>;<br/><a NAME="E4:7_2_2"/><span class="lab"><font color="Green" title="E12">A229</font></span>: 
 <a href="real_1.html#K1" title="REAL_1:func.1">-</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" title="INTEGRA5:th.20">INTEGRA5:20</a></span>;<br/><a NAME="E5:7_2_2"/><span class="lab"><font color="Green" title="E13">A230</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">A227</font></span></a>, <a class="ref" href="integra5.html#T19" title="INTEGRA5:th.19">INTEGRA5:19</a></span>;<br/><a NAME="E6:7_2_2"/><span class="lab"><font color="Green" title="E14">A231</font></span>: 
 <a href="real_1.html#K1" title="REAL_1:func.1">-</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" title="INTEGRA5:th.20">INTEGRA5:20</a></span>;<br/><a NAME="E7:7_2_2"/><span class="lab"><font color="Green" title="E15">A232</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="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">A227</font></span></a>, <a class="ref" href="integra5.html#T19" title="INTEGRA5:th.19">INTEGRA5:19</a></span>;<br/><a NAME="E8:7_2_2"/><span class="lab"><font color="Green" title="E16">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="real_1.html#K1" title="REAL_1:func.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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:7_2_2"><span class="lab"><font color="Green" title="E15">A232</font></span></a>, <a class="ref" href="integra5.html#T20" title="INTEGRA5:th.20">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" title="INTEGRA5:th.19">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="real_1.html#K1" title="REAL_1:func.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="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">A227</font></span></a>, <a class="ref" href="integra5.html#T20" title="INTEGRA5:th.20">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">A228</font></span></a>, <a class="txt" href="#E4:7_2_2"><span class="lab"><font color="Green" title="E12">A229</font></span></a>, <a class="txt" href="#E5:7_2_2"><span class="lab"><font color="Green" title="E13">A230</font></span></a>, <a class="txt" href="#E6:7_2_2"><span class="lab"><font color="Green" title="E14">A231</font></span></a>, <a class="txt" href="#E8:7_2_2"><span class="lab"><font color="Green" title="E16">A233</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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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