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<span class="kw">let </span><font color="Maroon" title="c1">f</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">C</font> being   <a href="integr1c.html#NM1" title="INTEGR1C:NM.1">C1-curve</a>  st  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Olive" title="b1">C</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> <font color="Maroon" title="c1">f</font> &amp; <font color="Maroon" title="c1">f</font> <a href="integr1c.html#R1" title="INTEGR1C:pred.1">is_integrable_on</a> <font color="Olive" title="b1">C</font> &amp; <font color="Maroon" title="c1">f</font> <a href="integr1c.html#R2" title="INTEGR1C:pred.2">is_bounded_on</a> <font color="Olive" title="b1">C</font> holds <br/> for <font color="Olive" title="b2">r</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> holds   <a href="integr1c.html#K3" title="INTEGR1C:func.3">integral</a> (<span class="p1">(<span class="default"><font color="Olive" title="b2">r</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c1">f</font></span>)</span>,<font color="Olive" title="b1">C</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="integr1c.html#K3" title="INTEGR1C:func.3">integral</a> (<font color="Maroon" title="c1">f</font>,<font color="Olive" title="b1">C</font>)</span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">C</font> be   <a href="integr1c.html#NM1" title="INTEGR1C:NM.1">C1-curve</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">C</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> <font color="Maroon" title="c1">f</font> &amp; <font color="Maroon" title="c1">f</font> <a href="integr1c.html#R1" title="INTEGR1C:pred.1">is_integrable_on</a> <font color="Maroon" title="c2">C</font> &amp; <font color="Maroon" title="c1">f</font> <a href="integr1c.html#R2" title="INTEGR1C:pred.2">is_bounded_on</a> <font color="Maroon" title="c2">C</font> implies  for <font color="Olive" title="b1">r</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> holds   <a href="integr1c.html#K3" title="INTEGR1C:func.3">integral</a> (<span class="p1">(<span class="default"><font color="Olive" title="b1">r</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c1">f</font></span>)</span>,<font color="Maroon" title="c2">C</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="integr1c.html#K3" title="INTEGR1C:func.3">integral</a> (<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">C</font>)</span>)</span> )</span><br/>



<span class="kw">assume </span><a NAME="E1:12"/><span class="lab"><font color="Green" title="E7">A1</font></span>: 
(  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">C</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> <font color="Maroon" title="c1">f</font> &amp; <font color="Maroon" title="c1">f</font> <a href="integr1c.html#R1" title="INTEGR1C:pred.1">is_integrable_on</a> <font color="Maroon" title="c2">C</font> &amp; <font color="Maroon" title="c1">f</font> <a href="integr1c.html#R2" title="INTEGR1C:pred.2">is_bounded_on</a> <font color="Maroon" title="c2">C</font> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">r</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> holds   <a href="integr1c.html#K3" title="INTEGR1C:func.3">integral</a> (<span class="p1">(<span class="default"><font color="Olive" title="b1">r</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c1">f</font></span>)</span>,<font color="Maroon" title="c2">C</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="integr1c.html#K3" title="INTEGR1C:func.3">integral</a> (<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">C</font>)</span>)</span></span><br/>

<span class="kw">let </span><font color="Maroon" title="c3">r</font> be   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="integr1c.html#K3" title="INTEGR1C:func.3">integral</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c1">f</font></span>)</span>,<font color="Maroon" title="c2">C</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="integr1c.html#K3" title="INTEGR1C:func.3">integral</a> (<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">C</font>)</span>)</span></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c5">b</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>, <font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c7">y</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>, <font color="Maroon" title="c8">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><span class="kw"> such that </span><br/><a NAME="E3:12"/><span class="lab"><font color="Green" title="E8">A2</font></span>: 
( <font color="Maroon" title="c4">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">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="c4">a</font>,<font color="Maroon" title="c5">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">C</font> &amp; <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">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">x</font> &amp; <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">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="c7">y</font> &amp; <font color="Maroon" title="c8">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="c4">a</font>,<font color="Maroon" title="c5">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="c8">Z</font> &amp; <font color="Maroon" title="c6">x</font> <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a> <font color="Maroon" title="c8">Z</font> &amp; <font color="Maroon" title="c7">y</font> <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a> <font color="Maroon" title="c8">Z</font> &amp; <font color="Maroon" title="c6">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font> is  <a href="fcont_1.html#V1" title="FCONT_1:attr.1">continuous</a>  &amp; <font color="Maroon" title="c7">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font> is  <a href="fcont_1.html#V1" title="FCONT_1:attr.1">continuous</a>  &amp; <font color="Maroon" title="c2">C</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <span class="p2">(<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="c7">y</font></span>)</span></span>)</span> <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="c4">a</font>,<font color="Maroon" title="c5">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="integr1c.html#D3" target="_self" onmouseover="rs('integr1c/D3')" onmouseout="rh()">Def3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">a</font> = <font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c10">b</font> = <font color="Maroon" title="c5">b</font> as   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c11">uf0</font>, <font color="Maroon" title="c12">vf0</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="E6:12"/><span class="lab"><font color="Green" title="E9">A3</font></span>: 
( <font color="Maroon" title="c11">uf0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K5" title="COMSEQ_3:func.5">Re</a> <font color="Maroon" title="c1">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="c6">x</font>,<font color="Maroon" title="c7">y</font></span><a href="funct_3.html#K13" title="FUNCT_3:func.13">:&gt;</a></span> &amp; <font color="Maroon" title="c12">vf0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Im</a> <font color="Maroon" title="c1">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="c6">x</font>,<font color="Maroon" title="c7">y</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="c1">f</font>,<font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font>,<font color="Maroon" title="c8">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">uf0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">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">vf0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">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">vf0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p5">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">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">uf0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p5">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font>)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="integr1c.html#D2" target="_self" onmouseover="rs('integr1c/D2')" onmouseout="rh()">Def2</a></span>;<br/>
<a NAME="E7:12"/><span class="lab"><font color="Green" title="E10">A4</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">r</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c1">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#D5" onmouseover="rs('valued_1/D5')" onmouseout="rh()">VALUED_1:def 5</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c13">u0</font>, <font color="Maroon" title="c14">v0</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="E9:12"/><span class="lab"><font color="Green" title="E11">A5</font></span>: 
( <font color="Maroon" title="c13">u0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K5" title="COMSEQ_3:func.5">Re</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c1">f</font></span>)</span></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="c6">x</font>,<font color="Maroon" title="c7">y</font></span><a href="funct_3.html#K13" title="FUNCT_3:func.13">:&gt;</a></span> &amp; <font color="Maroon" title="c14">v0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Im</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c1">f</font></span>)</span></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="c6">x</font>,<font color="Maroon" title="c7">y</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> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c1">f</font></span>)</span>,<font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font>,<font color="Maroon" title="c8">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="c13">u0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">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="c14">v0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">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="c14">v0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p5">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">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="c13">u0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p5">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font>)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="integr1c.html#D2" target="_self" onmouseover="rs('integr1c/D2')" onmouseout="rh()">Def2</a></span>;<br/>
<a NAME="E10:12"/><span class="lab"><font color="Green" title="E12">A6</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c13">u0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c11">uf0</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="12_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:12_1"/><span class="lab"><font color="Green" title="E13">A7</font></span>: 
 for <font color="Olive" title="b1">x0</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c13">u0</font> 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> <font color="Maroon" title="c11">uf0</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="12_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c15">x0</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="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> <font color="Maroon" title="c13">u0</font> implies <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> <font color="Maroon" title="c11">uf0</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:12_1_1"/><span class="lab"><font color="Green" title="E14">A8</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> <font color="Maroon" title="c13">u0</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="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> <font color="Maroon" title="c11">uf0</font></span><br/>

<a NAME="E2:12_1_1"/><span class="lab"><font color="Green" title="E15">A9</font></span>: 
( <font color="Maroon" title="c15">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</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="c6">x</font>,<font color="Maroon" title="c7">y</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> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K5" title="COMSEQ_3:func.5">Re</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c1">f</font></span>)</span></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="#E9:12"><span class="lab"><font color="Green" title="E11">A5</font></span></a>, <a class="txt" href="#E1:12_1_1"><span class="lab"><font color="Green" title="E14">A8</font></span></a>, <a class="ref" href="funct_1.html#T11" onmouseover="rs('funct_1/T11')" onmouseout="rh()">FUNCT_1:11</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c16">R2</font> = <span class="p1"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</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>;<br/>
<a NAME="E3:12_1_1"/><span class="lab"><font color="Green" title="E16">A10</font></span>: 
( <span class="p1"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</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> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <a href="integr1c.html#K1" title="INTEGR1C:func.1">R2-to-C</a> &amp; <a href="integr1c.html#K1" title="INTEGR1C:func.1">R2-to-C</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</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> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K5" title="COMSEQ_3:func.5">Re</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:12_1_1"><span class="lab"><font color="Green" title="E15">A9</font></span></a>, <a class="ref" href="funct_1.html#T11" onmouseover="rs('funct_1/T11')" onmouseout="rh()">FUNCT_1:11</a></span>;<br/>
<a NAME="E4:12_1_1"/><span class="kw">then </span>
<a href="integr1c.html#K1" title="INTEGR1C:func.1">R2-to-C</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</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> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><a href="comseq_3.html#K5" title="COMSEQ_3:func.5">Re</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mesfun6c.html#T2" onmouseover="rs('mesfun6c/T2')" onmouseout="rh()">MESFUN6C:2</a></span>;<br/>
<a NAME="E5:12_1_1"/><span class="kw">then </span>
<a href="integr1c.html#K1" title="INTEGR1C:func.1">R2-to-C</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</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> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K5" title="COMSEQ_3:func.5">Re</a> <font color="Maroon" title="c1">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#D5" onmouseover="rs('valued_1/D5')" onmouseout="rh()">VALUED_1:def 5</a></span>;<br/>
<a NAME="E6:12_1_1"/><span class="kw">then </span>
<span class="p1"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</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> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K5" title="COMSEQ_3:func.5">Re</a> <font color="Maroon" title="c1">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="#E3:12_1_1"><span class="lab"><font color="Green" title="E16">A10</font></span></a>, <a class="ref" href="funct_1.html#T11" onmouseover="rs('funct_1/T11')" onmouseout="rh()">FUNCT_1:11</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:12_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#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c11">uf0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:12"><span class="lab"><font color="Green" title="E9">A3</font></span></a>, <a class="txt" href="#E2:12_1_1"><span class="lab"><font color="Green" title="E15">A9</font></span></a>, <a class="ref" href="funct_1.html#T11" onmouseover="rs('funct_1/T11')" onmouseout="rh()">FUNCT_1:11</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:12_1"/>
 for <font color="Olive" title="b1">x0</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c11">uf0</font> 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> <font color="Maroon" title="c13">u0</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="12_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c15">x0</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="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> <font color="Maroon" title="c11">uf0</font> implies <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> <font color="Maroon" title="c13">u0</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:12_1_2"/><span class="lab"><font color="Green" title="E14">A11</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> <font color="Maroon" title="c11">uf0</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="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> <font color="Maroon" title="c13">u0</font></span><br/>

<a NAME="E2:12_1_2"/><span class="lab"><font color="Green" title="E15">A12</font></span>: 
( <font color="Maroon" title="c15">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</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="c6">x</font>,<font color="Maroon" title="c7">y</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> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K5" title="COMSEQ_3:func.5">Re</a> <font color="Maroon" title="c1">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="#E6:12"><span class="lab"><font color="Green" title="E9">A3</font></span></a>, <a class="txt" href="#E1:12_1_2"><span class="lab"><font color="Green" title="E14">A11</font></span></a>, <a class="ref" href="funct_1.html#T11" onmouseover="rs('funct_1/T11')" onmouseout="rh()">FUNCT_1:11</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c16">R2</font> = <span class="p1"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</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>;<br/>
<a NAME="E3:12_1_2"/><span class="lab"><font color="Green" title="E16">A13</font></span>: 
( <span class="p1"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</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> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <a href="integr1c.html#K1" title="INTEGR1C:func.1">R2-to-C</a> &amp; <a href="integr1c.html#K1" title="INTEGR1C:func.1">R2-to-C</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</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> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K5" title="COMSEQ_3:func.5">Re</a> <font color="Maroon" title="c1">f</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:12_1_2"><span class="lab"><font color="Green" title="E15">A12</font></span></a>, <a class="ref" href="funct_1.html#T11" onmouseover="rs('funct_1/T11')" onmouseout="rh()">FUNCT_1:11</a></span>;<br/>
<a NAME="E4:12_1_2"/><span class="kw">then </span>
<a href="integr1c.html#K1" title="INTEGR1C:func.1">R2-to-C</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</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> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><a href="comseq_3.html#K5" title="COMSEQ_3:func.5">Re</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#D5" onmouseover="rs('valued_1/D5')" onmouseout="rh()">VALUED_1:def 5</a></span>;<br/>
<a NAME="E5:12_1_2"/><span class="kw">then </span>
<a href="integr1c.html#K1" title="INTEGR1C:func.1">R2-to-C</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</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> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K5" title="COMSEQ_3:func.5">Re</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mesfun6c.html#T2" onmouseover="rs('mesfun6c/T2')" onmouseout="rh()">MESFUN6C:2</a></span>;<br/>
<a NAME="E6:12_1_2"/><span class="kw">then </span>
<span class="p1"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</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> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K5" title="COMSEQ_3:func.5">Re</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c1">f</font></span>)</span></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="#E3:12_1_2"><span class="lab"><font color="Green" title="E16">A13</font></span></a>, <a class="ref" href="funct_1.html#T11" onmouseover="rs('funct_1/T11')" onmouseout="rh()">FUNCT_1:11</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:12_1_2"/>
<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> <font color="Maroon" title="c13">u0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:12"><span class="lab"><font color="Green" title="E11">A5</font></span></a>, <a class="txt" href="#E2:12_1_2"><span class="lab"><font color="Green" title="E15">A12</font></span></a>, <a class="ref" href="funct_1.html#T11" onmouseover="rs('funct_1/T11')" onmouseout="rh()">FUNCT_1:11</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:12_1"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c13">u0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c11">uf0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:12_1"><span class="lab"><font color="Green" title="E13">A7</font></span></a>, <a class="ref" href="tarski.html#T2" onmouseover="rs('tarski/T2')" onmouseout="rh()">TARSKI:2</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E11:12"/><span class="lab"><font color="Green" title="E13">A14</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c14">v0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c12">vf0</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="12_2"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:12_2"/><span class="lab"><font color="Green" title="E14">A15</font></span>: 
 for <font color="Olive" title="b1">x0</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c14">v0</font> 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> <font color="Maroon" title="c12">vf0</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="12_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c15">x0</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="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> <font color="Maroon" title="c14">v0</font> implies <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> <font color="Maroon" title="c12">vf0</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:12_2_1"/><span class="lab"><font color="Green" title="E15">A16</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> <font color="Maroon" title="c14">v0</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="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> <font color="Maroon" title="c12">vf0</font></span><br/>

<a NAME="E2:12_2_1"/><span class="lab"><font color="Green" title="E16">A17</font></span>: 
( <font color="Maroon" title="c15">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</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="c6">x</font>,<font color="Maroon" title="c7">y</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> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Im</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c1">f</font></span>)</span></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="#E1:12_2_1"><span class="lab"><font color="Green" title="E15">A16</font></span></a>, <a class="txt" href="#E9:12"><span class="lab"><font color="Green" title="E11">A5</font></span></a>, <a class="ref" href="funct_1.html#T11" onmouseover="rs('funct_1/T11')" onmouseout="rh()">FUNCT_1:11</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c16">R2</font> = <span class="p1"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</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>;<br/>
<a NAME="E3:12_2_1"/><span class="lab"><font color="Green" title="E17">A18</font></span>: 
( <span class="p1"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</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> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <a href="integr1c.html#K1" title="INTEGR1C:func.1">R2-to-C</a> &amp; <a href="integr1c.html#K1" title="INTEGR1C:func.1">R2-to-C</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</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> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Im</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:12_2_1"><span class="lab"><font color="Green" title="E16">A17</font></span></a>, <a class="ref" href="funct_1.html#T11" onmouseover="rs('funct_1/T11')" onmouseout="rh()">FUNCT_1:11</a></span>;<br/>
<a NAME="E4:12_2_1"/><span class="kw">then </span>
<a href="integr1c.html#K1" title="INTEGR1C:func.1">R2-to-C</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</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> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Im</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mesfun6c.html#T2" onmouseover="rs('mesfun6c/T2')" onmouseout="rh()">MESFUN6C:2</a></span>;<br/>
<a NAME="E5:12_2_1"/><span class="kw">then </span>
<a href="integr1c.html#K1" title="INTEGR1C:func.1">R2-to-C</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</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> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Im</a> <font color="Maroon" title="c1">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#D5" onmouseover="rs('valued_1/D5')" onmouseout="rh()">VALUED_1:def 5</a></span>;<br/>
<a NAME="E6:12_2_1"/><span class="kw">then </span>
<span class="p1"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</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> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Im</a> <font color="Maroon" title="c1">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="#E3:12_2_1"><span class="lab"><font color="Green" title="E17">A18</font></span></a>, <a class="ref" href="funct_1.html#T11" onmouseover="rs('funct_1/T11')" onmouseout="rh()">FUNCT_1:11</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:12_2_1"/>
<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> <font color="Maroon" title="c12">vf0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:12"><span class="lab"><font color="Green" title="E9">A3</font></span></a>, <a class="txt" href="#E2:12_2_1"><span class="lab"><font color="Green" title="E16">A17</font></span></a>, <a class="ref" href="funct_1.html#T11" onmouseover="rs('funct_1/T11')" onmouseout="rh()">FUNCT_1:11</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:12_2"/>
 for <font color="Olive" title="b1">x0</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c12">vf0</font> 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> <font color="Maroon" title="c14">v0</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="12_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c15">x0</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="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> <font color="Maroon" title="c12">vf0</font> implies <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> <font color="Maroon" title="c14">v0</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:12_2_2"/><span class="lab"><font color="Green" title="E15">A19</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> <font color="Maroon" title="c12">vf0</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="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> <font color="Maroon" title="c14">v0</font></span><br/>

<a NAME="E2:12_2_2"/><span class="lab"><font color="Green" title="E16">A20</font></span>: 
( <font color="Maroon" title="c15">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</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="c6">x</font>,<font color="Maroon" title="c7">y</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> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Im</a> <font color="Maroon" title="c1">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="#E1:12_2_2"><span class="lab"><font color="Green" title="E15">A19</font></span></a>, <a class="txt" href="#E6:12"><span class="lab"><font color="Green" title="E9">A3</font></span></a>, <a class="ref" href="funct_1.html#T11" onmouseover="rs('funct_1/T11')" onmouseout="rh()">FUNCT_1:11</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c16">R2</font> = <span class="p1"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</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>;<br/>
<a NAME="E3:12_2_2"/><span class="lab"><font color="Green" title="E17">A21</font></span>: 
( <span class="p1"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</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> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <a href="integr1c.html#K1" title="INTEGR1C:func.1">R2-to-C</a> &amp; <a href="integr1c.html#K1" title="INTEGR1C:func.1">R2-to-C</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</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> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Im</a> <font color="Maroon" title="c1">f</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:12_2_2"><span class="lab"><font color="Green" title="E16">A20</font></span></a>, <a class="ref" href="funct_1.html#T11" onmouseover="rs('funct_1/T11')" onmouseout="rh()">FUNCT_1:11</a></span>;<br/>
<a NAME="E4:12_2_2"/><span class="kw">then </span>
<a href="integr1c.html#K1" title="INTEGR1C:func.1">R2-to-C</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</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> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Im</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#D5" onmouseover="rs('valued_1/D5')" onmouseout="rh()">VALUED_1:def 5</a></span>;<br/>
<a NAME="E5:12_2_2"/><span class="kw">then </span>
<a href="integr1c.html#K1" title="INTEGR1C:func.1">R2-to-C</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</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> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Im</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mesfun6c.html#T2" onmouseover="rs('mesfun6c/T2')" onmouseout="rh()">MESFUN6C:2</a></span>;<br/>
<a NAME="E6:12_2_2"/><span class="kw">then </span>
<span class="p1"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</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> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Im</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c1">f</font></span>)</span></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="#E3:12_2_2"><span class="lab"><font color="Green" title="E17">A21</font></span></a>, <a class="ref" href="funct_1.html#T11" onmouseover="rs('funct_1/T11')" onmouseout="rh()">FUNCT_1:11</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:12_2_2"/>
<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> <font color="Maroon" title="c14">v0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:12"><span class="lab"><font color="Green" title="E11">A5</font></span></a>, <a class="txt" href="#E2:12_2_2"><span class="lab"><font color="Green" title="E16">A20</font></span></a>, <a class="ref" href="funct_1.html#T11" onmouseover="rs('funct_1/T11')" onmouseout="rh()">FUNCT_1:11</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:12_2"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c14">v0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c12">vf0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:12_2"><span class="lab"><font color="Green" title="E14">A15</font></span></a>, <a class="ref" href="tarski.html#T2" onmouseover="rs('tarski/T2')" onmouseout="rh()">TARSKI:2</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E12:12"/><span class="lab"><font color="Green" title="E14">A22</font></span>: 
<font color="Maroon" title="c13">u0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">uf0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="12_3"><span class="kw">proof </span></a><div class="add">

<span class="lab"><font color="Green" title="E15">A23</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">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">uf0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span></span>)</span> = 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">uf0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rfunct_1.html#T13" onmouseover="rs('rfunct_1/T13')" onmouseout="rh()">RFUNCT_1:13</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="c11">uf0</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> <span class="p2">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#D4" onmouseover="rs('valued_1/D4')" onmouseout="rh()">VALUED_1:def 4</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="c11">uf0</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> <span class="p2">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#D5" onmouseover="rs('valued_1/D5')" onmouseout="rh()">VALUED_1:def 5</a></span>
;<br/>
<a NAME="E2:12_3"/><span class="lab"><font color="Green" title="E16">A24</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="c13">u0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c13">u0</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> <span class="p2">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#D4" onmouseover="rs('valued_1/D4')" onmouseout="rh()">VALUED_1:def 4</a></span>;<br/>
<a NAME="E3:12_3"/><span class="lab"><font color="Green" title="E17">A25</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="c13">u0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">uf0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:12"><span class="lab"><font color="Green" title="E12">A6</font></span></a>, <a class="txt" href="#E1:12_3"><span class="lab"><font color="Green" title="E15">A23</font></span></a>, <a class="ref" href="valued_1.html#D4" onmouseover="rs('valued_1/D4')" onmouseout="rh()">VALUED_1:def 4</a></span>;<br/>
<a NAME="E4:12_3"/>
 for <font color="Olive" title="b1">x0</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">u0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c13">u0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">uf0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x0</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="12_3_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c15">x0</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="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="c13">u0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c13">u0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">uf0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:12_3_2"/><span class="lab"><font color="Green" title="E18">A26</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="c13">u0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c13">u0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">uf0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font></span><br/>

<a NAME="E2:12_3_2"/><span class="lab"><font color="Green" title="E19">A27</font></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="c13">u0</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> <span class="p2">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:12_3_2"><span class="lab"><font color="Green" title="E18">A26</font></span></a>, <a class="ref" href="valued_1.html#D4" onmouseover="rs('valued_1/D4')" onmouseout="rh()">VALUED_1:def 4</a></span>;<br/>
<a NAME="E3:12_3_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A28</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> <font color="Maroon" title="c13">u0</font> &amp; <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="c6">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E4:12_3_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E21">A29</font></span>: 
( <font color="Maroon" title="c15">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</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="c6">x</font>,<font color="Maroon" title="c7">y</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> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K5" title="COMSEQ_3:func.5">Re</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c1">f</font></span>)</span></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="#E9:12"><span class="lab"><font color="Green" title="E11">A5</font></span></a>, <a class="ref" href="funct_1.html#T11" onmouseover="rs('funct_1/T11')" onmouseout="rh()">FUNCT_1:11</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c16">R2</font> = <span class="p1"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</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>;<br/>
<span class="kw">set </span><font color="Maroon" title="c17">c0</font> = <a href="integr1c.html#K1" title="INTEGR1C:func.1">R2-to-C</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</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>;<br/>
<span class="lab"><font color="Green" title="E22">A30</font></span>: <font color="Maroon" title="c13">u0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K5" title="COMSEQ_3:func.5">Re</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="integr1c.html#K1" title="INTEGR1C:func.1">R2-to-C</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</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="#E9:12"><span class="lab"><font color="Green" title="E11">A5</font></span></a>, <a class="txt" href="#E3:12_3_2"><span class="lab"><font color="Green" title="E20">A28</font></span></a>, <a class="ref" href="funct_1.html#T12" onmouseover="rs('funct_1/T12')" onmouseout="rh()">FUNCT_1:12</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="comseq_3.html#K5" title="COMSEQ_3:func.5">Re</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="integr1c.html#K1" title="INTEGR1C:func.1">R2-to-C</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</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>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:12_3_2"><span class="lab"><font color="Green" title="E21">A29</font></span></a>, <a class="ref" href="funct_1.html#T12" onmouseover="rs('funct_1/T12')" onmouseout="rh()">FUNCT_1:12</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><a href="comseq_3.html#K5" title="COMSEQ_3:func.5">Re</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="integr1c.html#K1" title="INTEGR1C:func.1">R2-to-C</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</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>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="mesfun6c.html#T2" onmouseover="rs('mesfun6c/T2')" onmouseout="rh()">MESFUN6C:2</a></span>
;<br/>
<a NAME="E6:12_3_2"/>
<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> <font color="Maroon" title="c11">uf0</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:12"><span class="lab"><font color="Green" title="E12">A6</font></span></a>, <a class="txt" href="#E2:12_3_2"><span class="lab"><font color="Green" title="E19">A27</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E7:12_3_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A31</font></span>: 
( <font color="Maroon" title="c15">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</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="c6">x</font>,<font color="Maroon" title="c7">y</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> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K5" title="COMSEQ_3:func.5">Re</a> <font color="Maroon" title="c1">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="#E6:12"><span class="lab"><font color="Green" title="E9">A3</font></span></a>, <a class="ref" href="funct_1.html#T11" onmouseover="rs('funct_1/T11')" onmouseout="rh()">FUNCT_1:11</a></span>;<br/>
<span class="lab"><font color="Green" title="E24">A32</font></span>: <font color="Maroon" title="c11">uf0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K5" title="COMSEQ_3:func.5">Re</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="integr1c.html#K1" title="INTEGR1C:func.1">R2-to-C</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</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="#E6:12"><span class="lab"><font color="Green" title="E9">A3</font></span></a>, <a class="txt" href="#E10:12"><span class="lab"><font color="Green" title="E12">A6</font></span></a>, <a class="txt" href="#E3:12_3_2"><span class="lab"><font color="Green" title="E20">A28</font></span></a>, <a class="ref" href="funct_1.html#T12" onmouseover="rs('funct_1/T12')" onmouseout="rh()">FUNCT_1:12</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="comseq_3.html#K5" title="COMSEQ_3:func.5">Re</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="integr1c.html#K1" title="INTEGR1C:func.1">R2-to-C</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</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>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:12_3_2"><span class="lab"><font color="Green" title="E23">A31</font></span></a>, <a class="ref" href="funct_1.html#T12" onmouseover="rs('funct_1/T12')" onmouseout="rh()">FUNCT_1:12</a></span>
;<br/>
<a NAME="E9:12_3_2"/><span class="lab"><font color="Green" title="E25">A33</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="c11">uf0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:12_3_2"><span class="lab"><font color="Green" title="E18">A26</font></span></a>, <a class="txt" href="#E3:12_3"><span class="lab"><font color="Green" title="E17">A25</font></span></a>, <a class="ref" href="rfunct_1.html#T13" onmouseover="rs('rfunct_1/T13')" onmouseout="rh()">RFUNCT_1:13</a></span>;<br/>
<a NAME="E10:12_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="c11">uf0</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> <span class="p2">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#D4" onmouseover="rs('valued_1/D4')" onmouseout="rh()">VALUED_1:def 4</a></span>;<br/>
<a NAME="E11:12_3_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A34</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">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/>
<span class="p1">(<span class="default"><font color="Maroon" title="c13">u0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c13">u0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:12_3_2"><span class="lab"><font color="Green" title="E18">A26</font></span></a>, <a class="ref" href="valued_1.html#D4" onmouseover="rs('valued_1/D4')" onmouseout="rh()">VALUED_1:def 4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="comseq_3.html#K5" title="COMSEQ_3:func.5">Re</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="integr1c.html#K1" title="INTEGR1C:func.1">R2-to-C</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><span class="p5"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</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>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:12_3_2"><span class="lab"><font color="Green" title="E22">A30</font></span></a>, <a class="ref" href="valued_1.html#T6" onmouseover="rs('valued_1/T6')" onmouseout="rh()">VALUED_1:6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">uf0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:12_3_2"><span class="lab"><font color="Green" title="E24">A32</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">uf0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:12_3_2"><span class="lab"><font color="Green" title="E26">A34</font></span></a>, <a class="ref" href="valued_1.html#D5" onmouseover="rs('valued_1/D5')" onmouseout="rh()">VALUED_1:def 5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">uf0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:12_3_2"><span class="lab"><font color="Green" title="E25">A33</font></span></a>, <a class="ref" href="valued_1.html#D4" onmouseover="rs('valued_1/D4')" onmouseout="rh()">VALUED_1:def 4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">uf0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rfunct_1.html#T13" onmouseover="rs('rfunct_1/T13')" onmouseout="rh()">RFUNCT_1:13</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E13:12_3_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c13">u0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">uf0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E5:12_3"/>
<font color="Maroon" title="c13">u0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">uf0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:12"><span class="lab"><font color="Green" title="E12">A6</font></span></a>, <a class="txt" href="#E1:12_3"><span class="lab"><font color="Green" title="E15">A23</font></span></a>, <a class="txt" href="#E2:12_3"><span class="lab"><font color="Green" title="E16">A24</font></span></a>, <a class="ref" href="funct_1.html#T2" onmouseover="rs('funct_1/T2')" onmouseout="rh()">FUNCT_1:2</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E13:12"/><span class="lab"><font color="Green" title="E15">A35</font></span>: 
<font color="Maroon" title="c14">v0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">vf0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="12_4"><span class="kw">proof </span></a><div class="add">

<span class="lab"><font color="Green" title="E16">A36</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">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">vf0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span></span>)</span> = 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">vf0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rfunct_1.html#T13" onmouseover="rs('rfunct_1/T13')" onmouseout="rh()">RFUNCT_1:13</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="c12">vf0</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> <span class="p2">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#D4" onmouseover="rs('valued_1/D4')" onmouseout="rh()">VALUED_1:def 4</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="c12">vf0</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> <span class="p2">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#D5" onmouseover="rs('valued_1/D5')" onmouseout="rh()">VALUED_1:def 5</a></span>
;<br/>
<a NAME="E2:12_4"/><span class="lab"><font color="Green" title="E17">A37</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="c14">v0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c14">v0</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> <span class="p2">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#D4" onmouseover="rs('valued_1/D4')" onmouseout="rh()">VALUED_1:def 4</a></span>;<br/>
<a NAME="E3:12_4"/><span class="lab"><font color="Green" title="E18">A38</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="c14">v0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">vf0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:12"><span class="lab"><font color="Green" title="E13">A14</font></span></a>, <a class="txt" href="#E1:12_4"><span class="lab"><font color="Green" title="E16">A36</font></span></a>, <a class="ref" href="valued_1.html#D4" onmouseover="rs('valued_1/D4')" onmouseout="rh()">VALUED_1:def 4</a></span>;<br/>
<a NAME="E4:12_4"/>
 for <font color="Olive" title="b1">x0</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">v0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c14">v0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">vf0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x0</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="12_4_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c15">x0</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="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="c14">v0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c14">v0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">vf0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:12_4_2"/><span class="lab"><font color="Green" title="E19">A39</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="c14">v0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c14">v0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">vf0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font></span><br/>

<a NAME="E2:12_4_2"/><span class="lab"><font color="Green" title="E20">A40</font></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="c14">v0</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> <span class="p2">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:12_4_2"><span class="lab"><font color="Green" title="E19">A39</font></span></a>, <a class="ref" href="valued_1.html#D4" onmouseover="rs('valued_1/D4')" onmouseout="rh()">VALUED_1:def 4</a></span>;<br/>
<a NAME="E3:12_4_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E21">A41</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> <font color="Maroon" title="c14">v0</font> &amp; <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="c6">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E4:12_4_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">A42</font></span>: 
( <font color="Maroon" title="c15">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</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="c6">x</font>,<font color="Maroon" title="c7">y</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> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Im</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c1">f</font></span>)</span></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="#E9:12"><span class="lab"><font color="Green" title="E11">A5</font></span></a>, <a class="ref" href="funct_1.html#T11" onmouseover="rs('funct_1/T11')" onmouseout="rh()">FUNCT_1:11</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c16">R2</font> = <span class="p1"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</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>;<br/>
<span class="kw">set </span><font color="Maroon" title="c17">c0</font> = <a href="integr1c.html#K1" title="INTEGR1C:func.1">R2-to-C</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</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>;<br/>
<span class="lab"><font color="Green" title="E23">A43</font></span>: <font color="Maroon" title="c14">v0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Im</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="integr1c.html#K1" title="INTEGR1C:func.1">R2-to-C</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</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="#E9:12"><span class="lab"><font color="Green" title="E11">A5</font></span></a>, <a class="txt" href="#E3:12_4_2"><span class="lab"><font color="Green" title="E21">A41</font></span></a>, <a class="ref" href="funct_1.html#T12" onmouseover="rs('funct_1/T12')" onmouseout="rh()">FUNCT_1:12</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Im</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="integr1c.html#K1" title="INTEGR1C:func.1">R2-to-C</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</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>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:12_4_2"><span class="lab"><font color="Green" title="E22">A42</font></span></a>, <a class="ref" href="funct_1.html#T12" onmouseover="rs('funct_1/T12')" onmouseout="rh()">FUNCT_1:12</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Im</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="integr1c.html#K1" title="INTEGR1C:func.1">R2-to-C</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</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>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="mesfun6c.html#T2" onmouseover="rs('mesfun6c/T2')" onmouseout="rh()">MESFUN6C:2</a></span>
;<br/>
<a NAME="E6:12_4_2"/>
<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> <font color="Maroon" title="c12">vf0</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:12"><span class="lab"><font color="Green" title="E13">A14</font></span></a>, <a class="txt" href="#E2:12_4_2"><span class="lab"><font color="Green" title="E20">A40</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E7:12_4_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A44</font></span>: 
( <font color="Maroon" title="c15">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</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="c6">x</font>,<font color="Maroon" title="c7">y</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> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Im</a> <font color="Maroon" title="c1">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="#E6:12"><span class="lab"><font color="Green" title="E9">A3</font></span></a>, <a class="ref" href="funct_1.html#T11" onmouseover="rs('funct_1/T11')" onmouseout="rh()">FUNCT_1:11</a></span>;<br/>
<span class="lab"><font color="Green" title="E25">A45</font></span>: <font color="Maroon" title="c12">vf0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Im</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="integr1c.html#K1" title="INTEGR1C:func.1">R2-to-C</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</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="#E6:12"><span class="lab"><font color="Green" title="E9">A3</font></span></a>, <a class="txt" href="#E11:12"><span class="lab"><font color="Green" title="E13">A14</font></span></a>, <a class="txt" href="#E3:12_4_2"><span class="lab"><font color="Green" title="E21">A41</font></span></a>, <a class="ref" href="funct_1.html#T12" onmouseover="rs('funct_1/T12')" onmouseout="rh()">FUNCT_1:12</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Im</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="integr1c.html#K1" title="INTEGR1C:func.1">R2-to-C</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</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>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:12_4_2"><span class="lab"><font color="Green" title="E24">A44</font></span></a>, <a class="ref" href="funct_1.html#T12" onmouseover="rs('funct_1/T12')" onmouseout="rh()">FUNCT_1:12</a></span>
;<br/>
<a NAME="E9:12_4_2"/><span class="lab"><font color="Green" title="E26">A46</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="c12">vf0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:12_4"><span class="lab"><font color="Green" title="E18">A38</font></span></a>, <a class="txt" href="#E1:12_4_2"><span class="lab"><font color="Green" title="E19">A39</font></span></a>, <a class="ref" href="rfunct_1.html#T13" onmouseover="rs('rfunct_1/T13')" onmouseout="rh()">RFUNCT_1:13</a></span>;<br/>
<a NAME="E10:12_4_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="c12">vf0</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> <span class="p2">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#D4" onmouseover="rs('valued_1/D4')" onmouseout="rh()">VALUED_1:def 4</a></span>;<br/>
<a NAME="E11:12_4_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A47</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">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/>
<span class="p1">(<span class="default"><font color="Maroon" title="c14">v0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c14">v0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:12_4_2"><span class="lab"><font color="Green" title="E19">A39</font></span></a>, <a class="ref" href="valued_1.html#D4" onmouseover="rs('valued_1/D4')" onmouseout="rh()">VALUED_1:def 4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Im</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="integr1c.html#K1" title="INTEGR1C:func.1">R2-to-C</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><span class="p5"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</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>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:12_4_2"><span class="lab"><font color="Green" title="E23">A43</font></span></a>, <a class="ref" href="valued_1.html#T6" onmouseover="rs('valued_1/T6')" onmouseout="rh()">VALUED_1:6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">vf0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:12_4_2"><span class="lab"><font color="Green" title="E25">A45</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">vf0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:12_4_2"><span class="lab"><font color="Green" title="E27">A47</font></span></a>, <a class="ref" href="valued_1.html#D5" onmouseover="rs('valued_1/D5')" onmouseout="rh()">VALUED_1:def 5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">vf0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:12_4_2"><span class="lab"><font color="Green" title="E26">A46</font></span></a>, <a class="ref" href="valued_1.html#D4" onmouseover="rs('valued_1/D4')" onmouseout="rh()">VALUED_1:def 4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">vf0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rfunct_1.html#T13" onmouseover="rs('rfunct_1/T13')" onmouseout="rh()">RFUNCT_1:13</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E13:12_4_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c14">v0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">vf0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E5:12_4"/>
<font color="Maroon" title="c14">v0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">vf0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:12"><span class="lab"><font color="Green" title="E13">A14</font></span></a>, <a class="txt" href="#E1:12_4"><span class="lab"><font color="Green" title="E16">A36</font></span></a>, <a class="txt" href="#E2:12_4"><span class="lab"><font color="Green" title="E17">A37</font></span></a>, <a class="ref" href="funct_1.html#T2" onmouseover="rs('funct_1/T2')" onmouseout="rh()">FUNCT_1:2</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E14:12"/><span class="lab"><font color="Green" title="E16">A48</font></span>: 
<font color="Maroon" title="c13">u0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">uf0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="12_5"><span class="kw">proof </span></a><div class="add">

<span class="lab"><font color="Green" title="E17">A49</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">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">uf0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span></span>)</span> = 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">uf0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rfunct_1.html#T13" onmouseover="rs('rfunct_1/T13')" onmouseout="rh()">RFUNCT_1:13</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="c11">uf0</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> <span class="p2">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#D4" onmouseover="rs('valued_1/D4')" onmouseout="rh()">VALUED_1:def 4</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="c11">uf0</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> <span class="p2">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#D5" onmouseover="rs('valued_1/D5')" onmouseout="rh()">VALUED_1:def 5</a></span>
;<br/>
<a NAME="E2:12_5"/><span class="lab"><font color="Green" title="E18">A50</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="c13">u0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c13">u0</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> <span class="p2">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#D4" onmouseover="rs('valued_1/D4')" onmouseout="rh()">VALUED_1:def 4</a></span>;<br/>
<a NAME="E3:12_5"/><span class="lab"><font color="Green" title="E19">A51</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="c13">u0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">uf0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:12"><span class="lab"><font color="Green" title="E12">A6</font></span></a>, <a class="txt" href="#E1:12_5"><span class="lab"><font color="Green" title="E17">A49</font></span></a>, <a class="ref" href="valued_1.html#D4" onmouseover="rs('valued_1/D4')" onmouseout="rh()">VALUED_1:def 4</a></span>;<br/>
<a NAME="E4:12_5"/>
 for <font color="Olive" title="b1">x0</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">u0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c13">u0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">uf0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x0</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="12_5_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c15">x0</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="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="c13">u0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c13">u0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">uf0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:12_5_2"/><span class="lab"><font color="Green" title="E20">A52</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="c13">u0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c13">u0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">uf0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font></span><br/>

<a NAME="E2:12_5_2"/><span class="lab"><font color="Green" title="E21">A53</font></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="c13">u0</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> <span class="p2">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:12_5_2"><span class="lab"><font color="Green" title="E20">A52</font></span></a>, <a class="ref" href="valued_1.html#D4" onmouseover="rs('valued_1/D4')" onmouseout="rh()">VALUED_1:def 4</a></span>;<br/>
<a NAME="E3:12_5_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">A54</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> <font color="Maroon" title="c13">u0</font> &amp; <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="c7">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E4:12_5_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A55</font></span>: 
( <font color="Maroon" title="c15">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</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="c6">x</font>,<font color="Maroon" title="c7">y</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> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K5" title="COMSEQ_3:func.5">Re</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c1">f</font></span>)</span></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="#E9:12"><span class="lab"><font color="Green" title="E11">A5</font></span></a>, <a class="ref" href="funct_1.html#T11" onmouseover="rs('funct_1/T11')" onmouseout="rh()">FUNCT_1:11</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c16">R2</font> = <span class="p1"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</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>;<br/>
<span class="kw">set </span><font color="Maroon" title="c17">c0</font> = <a href="integr1c.html#K1" title="INTEGR1C:func.1">R2-to-C</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</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>;<br/>
<span class="lab"><font color="Green" title="E24">A56</font></span>: <font color="Maroon" title="c13">u0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K5" title="COMSEQ_3:func.5">Re</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="integr1c.html#K1" title="INTEGR1C:func.1">R2-to-C</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</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="#E9:12"><span class="lab"><font color="Green" title="E11">A5</font></span></a>, <a class="txt" href="#E3:12_5_2"><span class="lab"><font color="Green" title="E22">A54</font></span></a>, <a class="ref" href="funct_1.html#T12" onmouseover="rs('funct_1/T12')" onmouseout="rh()">FUNCT_1:12</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="comseq_3.html#K5" title="COMSEQ_3:func.5">Re</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="integr1c.html#K1" title="INTEGR1C:func.1">R2-to-C</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</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>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:12_5_2"><span class="lab"><font color="Green" title="E23">A55</font></span></a>, <a class="ref" href="funct_1.html#T12" onmouseover="rs('funct_1/T12')" onmouseout="rh()">FUNCT_1:12</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><a href="comseq_3.html#K5" title="COMSEQ_3:func.5">Re</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="integr1c.html#K1" title="INTEGR1C:func.1">R2-to-C</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</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>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="mesfun6c.html#T2" onmouseover="rs('mesfun6c/T2')" onmouseout="rh()">MESFUN6C:2</a></span>
;<br/>
<a NAME="E6:12_5_2"/>
<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> <font color="Maroon" title="c11">uf0</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:12"><span class="lab"><font color="Green" title="E12">A6</font></span></a>, <a class="txt" href="#E2:12_5_2"><span class="lab"><font color="Green" title="E21">A53</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E7:12_5_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A57</font></span>: 
( <font color="Maroon" title="c15">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</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="c6">x</font>,<font color="Maroon" title="c7">y</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> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K5" title="COMSEQ_3:func.5">Re</a> <font color="Maroon" title="c1">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="#E6:12"><span class="lab"><font color="Green" title="E9">A3</font></span></a>, <a class="ref" href="funct_1.html#T11" onmouseover="rs('funct_1/T11')" onmouseout="rh()">FUNCT_1:11</a></span>;<br/>
<span class="lab"><font color="Green" title="E26">A58</font></span>: <font color="Maroon" title="c11">uf0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K5" title="COMSEQ_3:func.5">Re</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="integr1c.html#K1" title="INTEGR1C:func.1">R2-to-C</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</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="#E6:12"><span class="lab"><font color="Green" title="E9">A3</font></span></a>, <a class="txt" href="#E10:12"><span class="lab"><font color="Green" title="E12">A6</font></span></a>, <a class="txt" href="#E3:12_5_2"><span class="lab"><font color="Green" title="E22">A54</font></span></a>, <a class="ref" href="funct_1.html#T12" onmouseover="rs('funct_1/T12')" onmouseout="rh()">FUNCT_1:12</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="comseq_3.html#K5" title="COMSEQ_3:func.5">Re</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="integr1c.html#K1" title="INTEGR1C:func.1">R2-to-C</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</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>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:12_5_2"><span class="lab"><font color="Green" title="E25">A57</font></span></a>, <a class="ref" href="funct_1.html#T12" onmouseover="rs('funct_1/T12')" onmouseout="rh()">FUNCT_1:12</a></span>
;<br/>
<a NAME="E9:12_5_2"/><span class="lab"><font color="Green" title="E27">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="c11">uf0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:12_5"><span class="lab"><font color="Green" title="E19">A51</font></span></a>, <a class="txt" href="#E1:12_5_2"><span class="lab"><font color="Green" title="E20">A52</font></span></a>, <a class="ref" href="rfunct_1.html#T13" onmouseover="rs('rfunct_1/T13')" onmouseout="rh()">RFUNCT_1:13</a></span>;<br/>
<a NAME="E10:12_5_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="c11">uf0</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> <span class="p2">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#D4" onmouseover="rs('valued_1/D4')" onmouseout="rh()">VALUED_1:def 4</a></span>;<br/>
<a NAME="E11:12_5_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A60</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">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/>
<span class="p1">(<span class="default"><font color="Maroon" title="c13">u0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c13">u0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:12_5_2"><span class="lab"><font color="Green" title="E20">A52</font></span></a>, <a class="ref" href="valued_1.html#D4" onmouseover="rs('valued_1/D4')" onmouseout="rh()">VALUED_1:def 4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="comseq_3.html#K5" title="COMSEQ_3:func.5">Re</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="integr1c.html#K1" title="INTEGR1C:func.1">R2-to-C</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><span class="p5"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</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>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:12_5_2"><span class="lab"><font color="Green" title="E24">A56</font></span></a>, <a class="ref" href="valued_1.html#T6" onmouseover="rs('valued_1/T6')" onmouseout="rh()">VALUED_1:6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">uf0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:12_5_2"><span class="lab"><font color="Green" title="E26">A58</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">uf0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:12_5_2"><span class="lab"><font color="Green" title="E28">A60</font></span></a>, <a class="ref" href="valued_1.html#D5" onmouseover="rs('valued_1/D5')" onmouseout="rh()">VALUED_1:def 5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">uf0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:12_5_2"><span class="lab"><font color="Green" title="E27">A59</font></span></a>, <a class="ref" href="valued_1.html#D4" onmouseover="rs('valued_1/D4')" onmouseout="rh()">VALUED_1:def 4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">uf0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rfunct_1.html#T13" onmouseover="rs('rfunct_1/T13')" onmouseout="rh()">RFUNCT_1:13</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E13:12_5_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c13">u0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">uf0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E5:12_5"/>
<font color="Maroon" title="c13">u0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">uf0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:12"><span class="lab"><font color="Green" title="E12">A6</font></span></a>, <a class="txt" href="#E1:12_5"><span class="lab"><font color="Green" title="E17">A49</font></span></a>, <a class="txt" href="#E2:12_5"><span class="lab"><font color="Green" title="E18">A50</font></span></a>, <a class="ref" href="funct_1.html#T2" onmouseover="rs('funct_1/T2')" onmouseout="rh()">FUNCT_1:2</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E15:12"/><span class="lab"><font color="Green" title="E17">A61</font></span>: 
<font color="Maroon" title="c14">v0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">vf0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="12_6"><span class="kw">proof </span></a><div class="add">

<span class="lab"><font color="Green" title="E18">A62</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">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">vf0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span></span>)</span> = 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">vf0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rfunct_1.html#T13" onmouseover="rs('rfunct_1/T13')" onmouseout="rh()">RFUNCT_1:13</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="c12">vf0</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> <span class="p2">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#D4" onmouseover="rs('valued_1/D4')" onmouseout="rh()">VALUED_1:def 4</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="c12">vf0</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> <span class="p2">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#D5" onmouseover="rs('valued_1/D5')" onmouseout="rh()">VALUED_1:def 5</a></span>
;<br/>
<a NAME="E2:12_6"/><span class="lab"><font color="Green" title="E19">A63</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="c14">v0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c14">v0</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> <span class="p2">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#D4" onmouseover="rs('valued_1/D4')" onmouseout="rh()">VALUED_1:def 4</a></span>;<br/>
<a NAME="E3:12_6"/><span class="lab"><font color="Green" title="E20">A64</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="c14">v0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">vf0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:12"><span class="lab"><font color="Green" title="E13">A14</font></span></a>, <a class="txt" href="#E1:12_6"><span class="lab"><font color="Green" title="E18">A62</font></span></a>, <a class="ref" href="valued_1.html#D4" onmouseover="rs('valued_1/D4')" onmouseout="rh()">VALUED_1:def 4</a></span>;<br/>
<a NAME="E4:12_6"/>
 for <font color="Olive" title="b1">x0</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">v0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c14">v0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">vf0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x0</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="12_6_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c15">x0</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="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="c14">v0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c14">v0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">vf0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:12_6_2"/><span class="lab"><font color="Green" title="E21">A65</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="c14">v0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c14">v0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">vf0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font></span><br/>

<a NAME="E2:12_6_2"/><span class="lab"><font color="Green" title="E22">A66</font></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="c14">v0</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> <span class="p2">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:12_6_2"><span class="lab"><font color="Green" title="E21">A65</font></span></a>, <a class="ref" href="valued_1.html#D4" onmouseover="rs('valued_1/D4')" onmouseout="rh()">VALUED_1:def 4</a></span>;<br/>
<a NAME="E3:12_6_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A67</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> <font color="Maroon" title="c14">v0</font> &amp; <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="c7">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E4:12_6_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A68</font></span>: 
( <font color="Maroon" title="c15">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</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="c6">x</font>,<font color="Maroon" title="c7">y</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> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Im</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c1">f</font></span>)</span></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="#E9:12"><span class="lab"><font color="Green" title="E11">A5</font></span></a>, <a class="ref" href="funct_1.html#T11" onmouseover="rs('funct_1/T11')" onmouseout="rh()">FUNCT_1:11</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c16">R2</font> = <span class="p1"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</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>;<br/>
<span class="kw">set </span><font color="Maroon" title="c17">c0</font> = <a href="integr1c.html#K1" title="INTEGR1C:func.1">R2-to-C</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</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>;<br/>
<span class="lab"><font color="Green" title="E25">A69</font></span>: <font color="Maroon" title="c14">v0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Im</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="integr1c.html#K1" title="INTEGR1C:func.1">R2-to-C</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</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="#E9:12"><span class="lab"><font color="Green" title="E11">A5</font></span></a>, <a class="txt" href="#E3:12_6_2"><span class="lab"><font color="Green" title="E23">A67</font></span></a>, <a class="ref" href="funct_1.html#T12" onmouseover="rs('funct_1/T12')" onmouseout="rh()">FUNCT_1:12</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Im</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="integr1c.html#K1" title="INTEGR1C:func.1">R2-to-C</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</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>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:12_6_2"><span class="lab"><font color="Green" title="E24">A68</font></span></a>, <a class="ref" href="funct_1.html#T12" onmouseover="rs('funct_1/T12')" onmouseout="rh()">FUNCT_1:12</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Im</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="integr1c.html#K1" title="INTEGR1C:func.1">R2-to-C</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</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>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="mesfun6c.html#T2" onmouseover="rs('mesfun6c/T2')" onmouseout="rh()">MESFUN6C:2</a></span>
;<br/>
<a NAME="E6:12_6_2"/>
<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> <font color="Maroon" title="c12">vf0</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:12"><span class="lab"><font color="Green" title="E13">A14</font></span></a>, <a class="txt" href="#E2:12_6_2"><span class="lab"><font color="Green" title="E22">A66</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E7:12_6_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A70</font></span>: 
( <font color="Maroon" title="c15">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</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="c6">x</font>,<font color="Maroon" title="c7">y</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> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Im</a> <font color="Maroon" title="c1">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="#E6:12"><span class="lab"><font color="Green" title="E9">A3</font></span></a>, <a class="ref" href="funct_1.html#T11" onmouseover="rs('funct_1/T11')" onmouseout="rh()">FUNCT_1:11</a></span>;<br/>
<span class="lab"><font color="Green" title="E27">A71</font></span>: <font color="Maroon" title="c12">vf0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Im</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <a href="integr1c.html#K1" title="INTEGR1C:func.1">R2-to-C</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</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="#E6:12"><span class="lab"><font color="Green" title="E9">A3</font></span></a>, <a class="txt" href="#E11:12"><span class="lab"><font color="Green" title="E13">A14</font></span></a>, <a class="txt" href="#E3:12_6_2"><span class="lab"><font color="Green" title="E23">A67</font></span></a>, <a class="ref" href="funct_1.html#T12" onmouseover="rs('funct_1/T12')" onmouseout="rh()">FUNCT_1:12</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Im</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="integr1c.html#K1" title="INTEGR1C:func.1">R2-to-C</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</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>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:12_6_2"><span class="lab"><font color="Green" title="E26">A70</font></span></a>, <a class="ref" href="funct_1.html#T12" onmouseover="rs('funct_1/T12')" onmouseout="rh()">FUNCT_1:12</a></span>
;<br/>
<a NAME="E9:12_6_2"/><span class="lab"><font color="Green" title="E28">A72</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="c12">vf0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:12_6"><span class="lab"><font color="Green" title="E20">A64</font></span></a>, <a class="txt" href="#E1:12_6_2"><span class="lab"><font color="Green" title="E21">A65</font></span></a>, <a class="ref" href="rfunct_1.html#T13" onmouseover="rs('rfunct_1/T13')" onmouseout="rh()">RFUNCT_1:13</a></span>;<br/>
<a NAME="E10:12_6_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="c12">vf0</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> <span class="p2">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#D4" onmouseover="rs('valued_1/D4')" onmouseout="rh()">VALUED_1:def 4</a></span>;<br/>
<a NAME="E11:12_6_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A73</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">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/>
<span class="p1">(<span class="default"><font color="Maroon" title="c14">v0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c14">v0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:12_6_2"><span class="lab"><font color="Green" title="E21">A65</font></span></a>, <a class="ref" href="valued_1.html#D4" onmouseover="rs('valued_1/D4')" onmouseout="rh()">VALUED_1:def 4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Im</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="integr1c.html#K1" title="INTEGR1C:func.1">R2-to-C</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><span class="p5"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</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>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:12_6_2"><span class="lab"><font color="Green" title="E25">A69</font></span></a>, <a class="ref" href="valued_1.html#T6" onmouseover="rs('valued_1/T6')" onmouseout="rh()">VALUED_1:6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">vf0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:12_6_2"><span class="lab"><font color="Green" title="E27">A71</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">vf0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:12_6_2"><span class="lab"><font color="Green" title="E29">A73</font></span></a>, <a class="ref" href="valued_1.html#D5" onmouseover="rs('valued_1/D5')" onmouseout="rh()">VALUED_1:def 5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">vf0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:12_6_2"><span class="lab"><font color="Green" title="E28">A72</font></span></a>, <a class="ref" href="valued_1.html#D4" onmouseover="rs('valued_1/D4')" onmouseout="rh()">VALUED_1:def 4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">vf0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rfunct_1.html#T13" onmouseover="rs('rfunct_1/T13')" onmouseout="rh()">RFUNCT_1:13</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E13:12_6_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c14">v0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">vf0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E5:12_6"/>
<font color="Maroon" title="c14">v0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">vf0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:12"><span class="lab"><font color="Green" title="E13">A14</font></span></a>, <a class="txt" href="#E1:12_6"><span class="lab"><font color="Green" title="E18">A62</font></span></a>, <a class="txt" href="#E2:12_6"><span class="lab"><font color="Green" title="E19">A63</font></span></a>, <a class="ref" href="funct_1.html#T2" onmouseover="rs('funct_1/T2')" onmouseout="rh()">FUNCT_1:2</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E16:12"/><span class="lab"><font color="Green" title="E18">A74</font></span>: 
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">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="c11">uf0</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="12_7"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c15">x0</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  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#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> or <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> <font color="Maroon" title="c11">uf0</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:12_7"/><span class="lab"><font color="Green" title="E19">A75</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="c9">a</font>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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> <font color="Maroon" title="c11">uf0</font></span><br/>

<a NAME="E2:12_7"/><span class="lab"><font color="Green" title="E20">A76</font></span>: 
<font color="Maroon" title="c2">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:12"><span class="lab"><font color="Green" title="E8">A2</font></span></a>, <a class="txt" href="#E1:12_7"><span class="lab"><font color="Green" title="E19">A75</font></span></a>, <a class="ref" href="funct_1.html#T3" onmouseover="rs('funct_1/T3')" onmouseout="rh()">FUNCT_1:3</a></span>;<br/>
<a NAME="E3:12_7"/><span class="lab"><font color="Green" title="E21">A77</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> <font color="Maroon" title="c6">x</font> &amp; <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> <font color="Maroon" title="c7">y</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:12"><span class="lab"><font color="Green" title="E8">A2</font></span></a>, <a class="txt" href="#E1:12_7"><span class="lab"><font color="Green" title="E19">A75</font></span></a></span>;<br/>
<a NAME="E4:12_7"/><span class="lab"><font color="Green" title="E22">A78</font></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="c6">x</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="c7">y</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:12"><span class="lab"><font color="Green" title="E8">A2</font></span></a>, <a class="txt" href="#E1:12_7"><span class="lab"><font color="Green" title="E19">A75</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E5:12_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A79</font></span>: 
<font color="Maroon" title="c15">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</font></span><a href="funct_3.html#K13" title="FUNCT_3:func.13">:&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_3.html#D7" onmouseover="rs('funct_3/D7')" onmouseout="rh()">FUNCT_3:def 7</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c16">R2</font> = <span class="p1"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</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>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c17">xx0</font> = <font color="Maroon" title="c6">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font>, <font color="Maroon" title="c18">yx0</font> = <font color="Maroon" title="c7">y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font> as    <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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" onmouseover="rs('xreal_0/D1')" onmouseout="rh()">XREAL_0:def 1</a></span>;<br/>
<a NAME="E7:12_7"/>
<span class="p1"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c17">xx0</font>,<font color="Maroon" title="c18">yx0</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:12_7"><span class="lab"><font color="Green" title="E22">A78</font></span></a>, <a class="ref" href="funct_3.html#T48" onmouseover="rs('funct_3/T48')" onmouseout="rh()">FUNCT_3:48</a></span>;<br/>
<a NAME="E8:12_7"/><span class="kw">then </span>
<span class="p1"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</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> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#D2" onmouseover="rs('zfmisc_1/D2')" onmouseout="rh()">ZFMISC_1:def 2</a></span>;<br/>
<a NAME="E9:12_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A80</font></span>: 
<span class="p1"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</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> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <a href="integr1c.html#K1" title="INTEGR1C:func.1">R2-to-C</a>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E10:12_7"/>
<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"><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="c7">y</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:12_7"><span class="lab"><font color="Green" title="E21">A77</font></span></a>, <a class="ref" href="valued_1.html#D5" onmouseover="rs('valued_1/D5')" onmouseout="rh()">VALUED_1:def 5</a></span>;<br/>
<a NAME="E11:12_7"/><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="c6">x</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> <span class="p2">(<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="c7">y</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:12"><span class="lab"><font color="Green" title="E8">A2</font></span></a>, <a class="txt" href="#E1:12_7"><span class="lab"><font color="Green" title="E19">A75</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E12:12_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A81</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="c6">x</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <span class="p2">(<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="c7">y</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#D1" onmouseover="rs('valued_1/D1')" onmouseout="rh()">VALUED_1:def 1</a></span>;<br/>
<a NAME="E13:12_7"/><span class="lab"><font color="Green" title="E26">A82</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c17">xx0</font>,<font color="Maroon" title="c18">yx0</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#D2" onmouseover="rs('zfmisc_1/D2')" onmouseout="rh()">ZFMISC_1:def 2</a></span>;<br/>
<a NAME="E14:12_7"/><span class="lab"><font color="Green" title="E27">A83</font></span>: 
( <font color="Maroon" title="c6">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="xtuple_0.html#K1" title="XTUPLE_0:func.1">`1</a>  &amp; <font color="Maroon" title="c7">y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a>  )
 
;<br/>
<font color="Maroon" title="c2">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <span class="p2">(<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="c7">y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:12_7"><span class="lab"><font color="Green" title="E19">A75</font></span></a>, <a class="txt" href="#E3:12"><span class="lab"><font color="Green" title="E8">A2</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<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="c7">y</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:12_7"><span class="lab"><font color="Green" title="E25">A81</font></span></a>, <a class="ref" href="valued_1.html#D1" onmouseover="rs('valued_1/D1')" onmouseout="rh()">VALUED_1:def 1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#T6" onmouseover="rs('valued_1/T6')" onmouseout="rh()">VALUED_1:6</a></span>
<br/>.= 
<a href="integr1c.html#K1" title="INTEGR1C:func.1">R2-to-C</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c17">xx0</font>,<font color="Maroon" title="c18">yx0</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:12_7"><span class="lab"><font color="Green" title="E26">A82</font></span></a>, <a class="txt" href="#E14:12_7"><span class="lab"><font color="Green" title="E27">A83</font></span></a>, <a class="ref" href="integr1c.html#D1" target="_self" onmouseover="rs('integr1c/D1')" onmouseout="rh()">Def1</a></span>
<br/>.= 
<a href="integr1c.html#K1" title="INTEGR1C:func.1">R2-to-C</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</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="#E4:12_7"><span class="lab"><font color="Green" title="E22">A78</font></span></a>, <a class="ref" href="funct_3.html#T48" onmouseover="rs('funct_3/T48')" onmouseout="rh()">FUNCT_3:48</a></span>
;<br/>
<a NAME="E16:12_7"/><span class="kw">then </span>
<a href="integr1c.html#K1" title="INTEGR1C:func.1">R2-to-C</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</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> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c1">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:12"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="txt" href="#E2:12_7"><span class="lab"><font color="Green" title="E20">A76</font></span></a></span>;<br/>
<a NAME="E17:12_7"/><span class="kw">then </span>
<a href="integr1c.html#K1" title="INTEGR1C:func.1">R2-to-C</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</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> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K5" title="COMSEQ_3:func.5">Re</a> <font color="Maroon" title="c1">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="comseq_3.html#D3" onmouseover="rs('comseq_3/D3')" onmouseout="rh()">COMSEQ_3:def 3</a></span>;<br/>
<a NAME="E18:12_7"/><span class="kw">then </span>
<span class="p1"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</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> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K5" title="COMSEQ_3:func.5">Re</a> <font color="Maroon" title="c1">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="#E9:12_7"><span class="lab"><font color="Green" title="E24">A80</font></span></a>, <a class="ref" href="funct_1.html#T11" onmouseover="rs('funct_1/T11')" onmouseout="rh()">FUNCT_1:11</a></span>;<br/>
<span class="kw">hence </span><a NAME="E19:12_7"/>
<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> <font color="Maroon" title="c11">uf0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:12"><span class="lab"><font color="Green" title="E9">A3</font></span></a>, <a class="txt" href="#E5:12_7"><span class="lab"><font color="Green" title="E23">A79</font></span></a>, <a class="ref" href="funct_1.html#T11" onmouseover="rs('funct_1/T11')" onmouseout="rh()">FUNCT_1:11</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E17:12"/><span class="lab"><font color="Green" title="E19">A84</font></span>: 
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">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="c12">vf0</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="12_8"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c15">x0</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  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#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> or <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> <font color="Maroon" title="c12">vf0</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:12_8"/><span class="lab"><font color="Green" title="E20">A85</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="c9">a</font>,<font color="Maroon" title="c10">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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> <font color="Maroon" title="c12">vf0</font></span><br/>

<a NAME="E2:12_8"/><span class="lab"><font color="Green" title="E21">A86</font></span>: 
<font color="Maroon" title="c2">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:12"><span class="lab"><font color="Green" title="E8">A2</font></span></a>, <a class="txt" href="#E1:12_8"><span class="lab"><font color="Green" title="E20">A85</font></span></a>, <a class="ref" href="funct_1.html#T3" onmouseover="rs('funct_1/T3')" onmouseout="rh()">FUNCT_1:3</a></span>;<br/>
<a NAME="E3:12_8"/><span class="lab"><font color="Green" title="E22">A87</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> <font color="Maroon" title="c6">x</font> &amp; <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> <font color="Maroon" title="c7">y</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:12"><span class="lab"><font color="Green" title="E8">A2</font></span></a>, <a class="txt" href="#E1:12_8"><span class="lab"><font color="Green" title="E20">A85</font></span></a></span>;<br/>
<a NAME="E4:12_8"/><span class="lab"><font color="Green" title="E23">A88</font></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="c6">x</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="c7">y</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:12"><span class="lab"><font color="Green" title="E8">A2</font></span></a>, <a class="txt" href="#E1:12_8"><span class="lab"><font color="Green" title="E20">A85</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E5:12_8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A89</font></span>: 
<font color="Maroon" title="c15">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</font></span><a href="funct_3.html#K13" title="FUNCT_3:func.13">:&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_3.html#D7" onmouseover="rs('funct_3/D7')" onmouseout="rh()">FUNCT_3:def 7</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c16">R2</font> = <span class="p1"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</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>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c17">xx0</font> = <font color="Maroon" title="c6">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font>, <font color="Maroon" title="c18">yx0</font> = <font color="Maroon" title="c7">y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font> as    <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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" onmouseover="rs('xreal_0/D1')" onmouseout="rh()">XREAL_0:def 1</a></span>;<br/>
<a NAME="E7:12_8"/>
<span class="p1"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c17">xx0</font>,<font color="Maroon" title="c18">yx0</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:12_8"><span class="lab"><font color="Green" title="E23">A88</font></span></a>, <a class="ref" href="funct_3.html#T48" onmouseover="rs('funct_3/T48')" onmouseout="rh()">FUNCT_3:48</a></span>;<br/>
<a NAME="E8:12_8"/><span class="kw">then </span>
<span class="p1"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</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> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#D2" onmouseover="rs('zfmisc_1/D2')" onmouseout="rh()">ZFMISC_1:def 2</a></span>;<br/>
<a NAME="E9:12_8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A90</font></span>: 
<span class="p1"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</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> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <a href="integr1c.html#K1" title="INTEGR1C:func.1">R2-to-C</a>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E10:12_8"/>
<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"><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="c7">y</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:12_8"><span class="lab"><font color="Green" title="E22">A87</font></span></a>, <a class="ref" href="valued_1.html#D5" onmouseover="rs('valued_1/D5')" onmouseout="rh()">VALUED_1:def 5</a></span>;<br/>
<a NAME="E11:12_8"/><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="c6">x</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> <span class="p2">(<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="c7">y</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:12"><span class="lab"><font color="Green" title="E8">A2</font></span></a>, <a class="txt" href="#E1:12_8"><span class="lab"><font color="Green" title="E20">A85</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E12:12_8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A91</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="c6">x</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <span class="p2">(<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="c7">y</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#D1" onmouseover="rs('valued_1/D1')" onmouseout="rh()">VALUED_1:def 1</a></span>;<br/>
<a NAME="E13:12_8"/><span class="lab"><font color="Green" title="E27">A92</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c17">xx0</font>,<font color="Maroon" title="c18">yx0</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#D2" onmouseover="rs('zfmisc_1/D2')" onmouseout="rh()">ZFMISC_1:def 2</a></span>;<br/>
<a NAME="E14:12_8"/><span class="lab"><font color="Green" title="E28">A93</font></span>: 
( <font color="Maroon" title="c6">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="xtuple_0.html#K1" title="XTUPLE_0:func.1">`1</a>  &amp; <font color="Maroon" title="c7">y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a>  )
 
;<br/>
<font color="Maroon" title="c2">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <span class="p2">(<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="c7">y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:12_8"><span class="lab"><font color="Green" title="E20">A85</font></span></a>, <a class="txt" href="#E3:12"><span class="lab"><font color="Green" title="E8">A2</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<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="c7">y</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:12_8"><span class="lab"><font color="Green" title="E26">A91</font></span></a>, <a class="ref" href="valued_1.html#D1" onmouseover="rs('valued_1/D1')" onmouseout="rh()">VALUED_1:def 1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x0</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#T6" onmouseover="rs('valued_1/T6')" onmouseout="rh()">VALUED_1:6</a></span>
<br/>.= 
<a href="integr1c.html#K1" title="INTEGR1C:func.1">R2-to-C</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c17">xx0</font>,<font color="Maroon" title="c18">yx0</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:12_8"><span class="lab"><font color="Green" title="E27">A92</font></span></a>, <a class="txt" href="#E14:12_8"><span class="lab"><font color="Green" title="E28">A93</font></span></a>, <a class="ref" href="integr1c.html#D1" target="_self" onmouseover="rs('integr1c/D1')" onmouseout="rh()">Def1</a></span>
<br/>.= 
<a href="integr1c.html#K1" title="INTEGR1C:func.1">R2-to-C</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</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="#E4:12_8"><span class="lab"><font color="Green" title="E23">A88</font></span></a>, <a class="ref" href="funct_3.html#T48" onmouseover="rs('funct_3/T48')" onmouseout="rh()">FUNCT_3:48</a></span>
;<br/>
<a NAME="E16:12_8"/><span class="kw">then </span>
<a href="integr1c.html#K1" title="INTEGR1C:func.1">R2-to-C</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</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> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c1">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:12"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="txt" href="#E2:12_8"><span class="lab"><font color="Green" title="E21">A86</font></span></a></span>;<br/>
<a NAME="E17:12_8"/><span class="kw">then </span>
<a href="integr1c.html#K1" title="INTEGR1C:func.1">R2-to-C</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</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> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Im</a> <font color="Maroon" title="c1">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="comseq_3.html#D4" onmouseover="rs('comseq_3/D4')" onmouseout="rh()">COMSEQ_3:def 4</a></span>;<br/>
<a NAME="E18:12_8"/><span class="kw">then </span>
<span class="p1"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</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> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Im</a> <font color="Maroon" title="c1">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="#E9:12_8"><span class="lab"><font color="Green" title="E25">A90</font></span></a>, <a class="ref" href="funct_1.html#T11" onmouseover="rs('funct_1/T11')" onmouseout="rh()">FUNCT_1:11</a></span>;<br/>
<span class="kw">hence </span><a NAME="E19:12_8"/>
<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> <font color="Maroon" title="c12">vf0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:12"><span class="lab"><font color="Green" title="E9">A3</font></span></a>, <a class="txt" href="#E5:12_8"><span class="lab"><font color="Green" title="E24">A89</font></span></a>, <a class="ref" href="funct_1.html#T11" onmouseover="rs('funct_1/T11')" onmouseout="rh()">FUNCT_1:11</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E18:12"/><span class="lab"><font color="Green" title="E20">A94</font></span>: 
<span class="p1"><a href="integra5.html#K3" title="INTEGRA5:func.3">['</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font></span><a href="integra5.html#K3" title="INTEGRA5:func.3">']</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"><span class="p2">(<span class="default"><font color="Maroon" title="c11">uf0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">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">vf0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span></span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="12_9"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:12_9"/><span class="lab"><font color="Green" title="E21">A95</font></span>: 
<span class="p1"><a href="integra5.html#K3" title="INTEGRA5:func.3">['</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font></span><a href="integra5.html#K3" title="INTEGRA5:func.3">']</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">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="txt" href="#E3:12"><span class="lab"><font color="Green" title="E8">A2</font></span></a>, <a class="ref" href="integra5.html#D3" onmouseover="rs('integra5/D3')" onmouseout="rh()">INTEGRA5:def 3</a></span>;<br/>
<span class="lab"><font color="Green" title="E22">A96</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="c11">uf0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">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">vf0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span></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="c11">uf0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span></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> <span class="p2">(<span class="default"><font color="Maroon" title="c12">vf0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#T12" onmouseover="rs('valued_1/T12')" onmouseout="rh()">VALUED_1:12</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c11">uf0</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span></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> <span class="p2">(<span class="default"><font color="Maroon" title="c12">vf0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#D4" onmouseover="rs('valued_1/D4')" onmouseout="rh()">VALUED_1:def 4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c11">uf0</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c12">vf0</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#D4" onmouseover="rs('valued_1/D4')" onmouseout="rh()">VALUED_1:def 4</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="c11">uf0</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c12">vf0</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T16" onmouseover="rs('xboole_1/T16')" onmouseout="rh()">XBOOLE_1:16</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="c11">uf0</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">Z</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c12">vf0</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:12"><span class="lab"><font color="Green" title="E8">A2</font></span></a>, <a class="ref" href="fdiff_1.html#D7" onmouseover="rs('fdiff_1/D7')" onmouseout="rh()">FDIFF_1:def 7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c11">uf0</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">Z</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">Z</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c12">vf0</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:12"><span class="lab"><font color="Green" title="E8">A2</font></span></a>, <a class="ref" href="fdiff_1.html#D7" onmouseover="rs('fdiff_1/D7')" onmouseout="rh()">FDIFF_1:def 7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c11">uf0</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">Z</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c8">Z</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c12">vf0</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T16" onmouseover="rs('xboole_1/T16')" onmouseout="rh()">XBOOLE_1:16</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c11">uf0</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c12">vf0</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c8">Z</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T16" onmouseover="rs('xboole_1/T16')" onmouseout="rh()">XBOOLE_1:16</a></span>
;<br/>
<a NAME="E3:12_9"/>
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">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="c11">uf0</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="c12">vf0</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:12"><span class="lab"><font color="Green" title="E18">A74</font></span></a>, <a class="txt" href="#E17:12"><span class="lab"><font color="Green" title="E19">A84</font></span></a>, <a class="ref" href="xboole_1.html#T19" onmouseover="rs('xboole_1/T19')" onmouseout="rh()">XBOOLE_1:19</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:12_9"/>
<span class="p1"><a href="integra5.html#K3" title="INTEGRA5:func.3">['</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font></span><a href="integra5.html#K3" title="INTEGRA5:func.3">']</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"><span class="p2">(<span class="default"><font color="Maroon" title="c11">uf0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">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">vf0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:12"><span class="lab"><font color="Green" title="E8">A2</font></span></a>, <a class="txt" href="#E1:12_9"><span class="lab"><font color="Green" title="E21">A95</font></span></a>, <a class="txt" href="#E2:12_9"><span class="lab"><font color="Green" title="E22">A96</font></span></a>, <a class="ref" href="xboole_1.html#T19" onmouseover="rs('xboole_1/T19')" onmouseout="rh()">XBOOLE_1:19</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E19:12"/><span class="lab"><font color="Green" title="E21">A97</font></span>: 
<span class="p1"><a href="integra5.html#K3" title="INTEGRA5:func.3">['</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font></span><a href="integra5.html#K3" title="INTEGRA5:func.3">']</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"><span class="p2">(<span class="default"><font color="Maroon" title="c12">vf0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">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">uf0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span></span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="12_10"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:12_10"/><span class="lab"><font color="Green" title="E22">A98</font></span>: 
<span class="p1"><a href="integra5.html#K3" title="INTEGRA5:func.3">['</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font></span><a href="integra5.html#K3" title="INTEGRA5:func.3">']</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">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="txt" href="#E3:12"><span class="lab"><font color="Green" title="E8">A2</font></span></a>, <a class="ref" href="integra5.html#D3" onmouseover="rs('integra5/D3')" onmouseout="rh()">INTEGRA5:def 3</a></span>;<br/>
<span class="lab"><font color="Green" title="E23">A99</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="c12">vf0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">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">uf0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span></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="c12">vf0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span></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> <span class="p2">(<span class="default"><font color="Maroon" title="c11">uf0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#D1" onmouseover="rs('valued_1/D1')" onmouseout="rh()">VALUED_1:def 1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c12">vf0</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span></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> <span class="p2">(<span class="default"><font color="Maroon" title="c11">uf0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#D4" onmouseover="rs('valued_1/D4')" onmouseout="rh()">VALUED_1:def 4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c12">vf0</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c11">uf0</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#D4" onmouseover="rs('valued_1/D4')" onmouseout="rh()">VALUED_1:def 4</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="c12">vf0</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c11">uf0</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T16" onmouseover="rs('xboole_1/T16')" onmouseout="rh()">XBOOLE_1:16</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="c12">vf0</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">Z</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c11">uf0</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:12"><span class="lab"><font color="Green" title="E8">A2</font></span></a>, <a class="ref" href="fdiff_1.html#D7" onmouseover="rs('fdiff_1/D7')" onmouseout="rh()">FDIFF_1:def 7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c12">vf0</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">Z</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">Z</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c11">uf0</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:12"><span class="lab"><font color="Green" title="E8">A2</font></span></a>, <a class="ref" href="fdiff_1.html#D7" onmouseover="rs('fdiff_1/D7')" onmouseout="rh()">FDIFF_1:def 7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c12">vf0</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">Z</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c8">Z</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c11">uf0</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T16" onmouseover="rs('xboole_1/T16')" onmouseout="rh()">XBOOLE_1:16</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c12">vf0</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c11">uf0</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c8">Z</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T16" onmouseover="rs('xboole_1/T16')" onmouseout="rh()">XBOOLE_1:16</a></span>
;<br/>
<a NAME="E3:12_10"/>
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">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="c12">vf0</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="c11">uf0</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:12"><span class="lab"><font color="Green" title="E18">A74</font></span></a>, <a class="txt" href="#E17:12"><span class="lab"><font color="Green" title="E19">A84</font></span></a>, <a class="ref" href="xboole_1.html#T19" onmouseover="rs('xboole_1/T19')" onmouseout="rh()">XBOOLE_1:19</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:12_10"/>
<span class="p1"><a href="integra5.html#K3" title="INTEGRA5:func.3">['</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font></span><a href="integra5.html#K3" title="INTEGRA5:func.3">']</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"><span class="p2">(<span class="default"><font color="Maroon" title="c12">vf0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">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">uf0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:12"><span class="lab"><font color="Green" title="E8">A2</font></span></a>, <a class="txt" href="#E1:12_10"><span class="lab"><font color="Green" title="E22">A98</font></span></a>, <a class="txt" href="#E2:12_10"><span class="lab"><font color="Green" title="E23">A99</font></span></a>, <a class="ref" href="xboole_1.html#T19" onmouseover="rs('xboole_1/T19')" onmouseout="rh()">XBOOLE_1:19</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">reconsider </span><font color="Maroon" title="c15">a</font> = <font color="Maroon" title="c9">a</font>, <font color="Maroon" title="c16">b</font> = <font color="Maroon" title="c10">b</font> as   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> ;<br/>
<a NAME="E21:12"/><span class="lab"><font color="Green" title="E22">A100</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">uf0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">vf0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span> <a href="integra5.html#R1" title="INTEGRA5:pred.1">is_integrable_on</a> <span class="p1"><a href="integra5.html#K3" title="INTEGRA5:func.3">['</a><span class="default"><font color="Maroon" title="c15">a</font>,<font color="Maroon" title="c16">b</font></span><a href="integra5.html#K3" title="INTEGRA5:func.3">']</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:12"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="txt" href="#E3:12"><span class="lab"><font color="Green" title="E8">A2</font></span></a></span>;<br/>
<a NAME="E22:12"/><span class="lab"><font color="Green" title="E23">A101</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">vf0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">uf0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span> <a href="integra5.html#R1" title="INTEGRA5:pred.1">is_integrable_on</a> <span class="p1"><a href="integra5.html#K3" title="INTEGRA5:func.3">['</a><span class="default"><font color="Maroon" title="c15">a</font>,<font color="Maroon" title="c16">b</font></span><a href="integra5.html#K3" title="INTEGRA5:func.3">']</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:12"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="txt" href="#E3:12"><span class="lab"><font color="Green" title="E8">A2</font></span></a></span>;<br/>
<a NAME="E23:12"/><span class="lab"><font color="Green" title="E24">A102</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">uf0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">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">vf0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="integra5.html#K3" title="INTEGRA5:func.3">['</a><span class="default"><font color="Maroon" title="c15">a</font>,<font color="Maroon" title="c16">b</font></span><a href="integra5.html#K3" title="INTEGRA5:func.3">']</a></span> is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">bounded</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:12"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="txt" href="#E3:12"><span class="lab"><font color="Green" title="E8">A2</font></span></a></span>;<br/>
<a NAME="E24:12"/><span class="lab"><font color="Green" title="E25">A103</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">vf0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">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">uf0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="integra5.html#K3" title="INTEGRA5:func.3">['</a><span class="default"><font color="Maroon" title="c15">a</font>,<font color="Maroon" title="c16">b</font></span><a href="integra5.html#K3" title="INTEGRA5:func.3">']</a></span> is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">bounded</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:12"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="txt" href="#E3:12"><span class="lab"><font color="Green" title="E8">A2</font></span></a></span>;<br/>
 <a href="integr1c.html#K3" title="INTEGR1C:func.3">integral</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c1">f</font></span>)</span>,<font color="Maroon" title="c2">C</font>) = 
<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="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c11">uf0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p5">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></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="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c12">vf0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p5">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c15">a</font>,<font color="Maroon" title="c16">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="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p5">(<span class="default"><font color="Maroon" title="c12">vf0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p0">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></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="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p5">(<span class="default"><font color="Maroon" title="c11">uf0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p0">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c15">a</font>,<font color="Maroon" title="c16">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="txt" href="#E13:12"><span class="lab"><font color="Green" title="E15">A35</font></span></a>, <a class="txt" href="#E15:12"><span class="lab"><font color="Green" title="E17">A61</font></span></a>, <a class="txt" href="#E14:12"><span class="lab"><font color="Green" title="E16">A48</font></span></a>, <a class="txt" href="#E12:12"><span class="lab"><font color="Green" title="E14">A22</font></span></a>, <a class="txt" href="#E9:12"><span class="lab"><font color="Green" title="E11">A5</font></span></a>, <a class="txt" href="#E1:12"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="txt" href="#E3:12"><span class="lab"><font color="Green" title="E8">A2</font></span></a>, <a class="txt" href="#E7:12"><span class="lab"><font color="Green" title="E10">A4</font></span></a>, <a class="ref" href="integr1c.html#D4" target="_self" onmouseover="rs('integr1c/D4')" onmouseout="rh()">Def4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="integra5.html#K4" title="INTEGRA5:func.4">integral</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c11">uf0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p5">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c12">vf0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p5">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c15">a</font>,<font color="Maroon" title="c16">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="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p5">(<span class="default"><font color="Maroon" title="c12">vf0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p0">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></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="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p5">(<span class="default"><font color="Maroon" title="c11">uf0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p0">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c15">a</font>,<font color="Maroon" title="c16">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="rfunct_1.html#T18" onmouseover="rs('rfunct_1/T18')" onmouseout="rh()">RFUNCT_1:18</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="integra5.html#K4" title="INTEGRA5:func.4">integral</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c11">uf0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p5">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c12">vf0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p5">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c15">a</font>,<font color="Maroon" title="c16">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"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c12">vf0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p0">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p5">(<span class="default"><font color="Maroon" title="c11">uf0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p0">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c15">a</font>,<font color="Maroon" title="c16">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="rfunct_1.html#T16" onmouseover="rs('rfunct_1/T16')" onmouseout="rh()">RFUNCT_1:16</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <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="c11">uf0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p5">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c12">vf0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p5">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c15">a</font>,<font color="Maroon" title="c16">b</font>)</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra5.html#K4" title="INTEGRA5:func.4">integral</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c12">vf0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p0">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p5">(<span class="default"><font color="Maroon" title="c11">uf0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p0">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c15">a</font>,<font color="Maroon" title="c16">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="txt" href="#E3:12"><span class="lab"><font color="Green" title="E8">A2</font></span></a>, <a class="txt" href="#E18:12"><span class="lab"><font color="Green" title="E20">A94</font></span></a>, <a class="txt" href="#E21:12"><span class="lab"><font color="Green" title="E22">A100</font></span></a>, <a class="txt" href="#E23:12"><span class="lab"><font color="Green" title="E24">A102</font></span></a>, <a class="ref" href="integra6.html#T10" onmouseover="rs('integra6/T10')" onmouseout="rh()">INTEGRA6:10</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <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="c11">uf0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p5">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c12">vf0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p5">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c15">a</font>,<font color="Maroon" title="c16">b</font>)</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><a href="integra5.html#K4" title="INTEGRA5:func.4">integral</a> (<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c12">vf0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p0">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p5">(<span class="default"><font color="Maroon" title="c11">uf0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p0">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c15">a</font>,<font color="Maroon" title="c16">b</font>)</span>)</span></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="txt" href="#E3:12"><span class="lab"><font color="Green" title="E8">A2</font></span></a>, <a class="txt" href="#E19:12"><span class="lab"><font color="Green" title="E21">A97</font></span></a>, <a class="txt" href="#E22:12"><span class="lab"><font color="Green" title="E23">A101</font></span></a>, <a class="txt" href="#E24:12"><span class="lab"><font color="Green" title="E25">A103</font></span></a>, <a class="ref" href="integra6.html#T10" onmouseover="rs('integra6/T10')" onmouseout="rh()">INTEGRA6:10</a></span>
<br/>.= 
<font color="Maroon" title="c3">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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="c11">uf0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p5">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c12">vf0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p5">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c15">a</font>,<font color="Maroon" title="c16">b</font>)</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra5.html#K4" title="INTEGRA5:func.4">integral</a> (<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c12">vf0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p0">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p5">(<span class="default"><font color="Maroon" title="c11">uf0</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p0">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c8">Z</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c15">a</font>,<font color="Maroon" title="c16">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>)</span>

<br/>.= 
<font color="Maroon" title="c3">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="integr1c.html#K3" title="INTEGR1C:func.3">integral</a> (<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">C</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:12"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="txt" href="#E3:12"><span class="lab"><font color="Green" title="E8">A2</font></span></a>, <a class="txt" href="#E6:12"><span class="lab"><font color="Green" title="E9">A3</font></span></a>, <a class="ref" href="integr1c.html#D4" target="_self" onmouseover="rs('integr1c/D4')" onmouseout="rh()">Def4</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E26:12"/>
 <a href="integr1c.html#K3" title="INTEGR1C:func.3">integral</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c1">f</font></span>)</span>,<font color="Maroon" title="c2">C</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="integr1c.html#K3" title="INTEGR1C:func.3">integral</a> (<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">C</font>)</span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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