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<span class="kw">reconsider </span><font color="Maroon" title="c1">ey0</font> = <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1</span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> 2</span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="cfdiff_2.html#E11"><span class="lab"><font color="Green" title="E11">Lm9</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c2">ex0</font> = <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default">1,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> 2</span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="cfdiff_2.html#E11"><span class="lab"><font color="Green" title="E11">Lm9</font></span></a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c3">u</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1">(<span class="default"><a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> 2</span>)</span>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x0</font>, <font color="Olive" title="b2">y0</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><br/>  for <font color="Olive" title="b3">xy0</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> 2  st <font color="Olive" title="b3">xy0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Olive" title="b1">x0</font>,<font color="Olive" title="b2">y0</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> &amp;  <a href="pdiff_1.html#K3" title="PDIFF_1:func.3">&lt;&gt;*</a> <font color="Maroon" title="c3">u</font> <a href="pdiff_1.html#R1" title="PDIFF_1:pred.1">is_differentiable_in</a> <font color="Olive" title="b3">xy0</font> holds <br/>( <font color="Maroon" title="c3">u</font> <a href="pdiff_1.html#R3" title="PDIFF_1:pred.3">is_partial_differentiable_in</a> <font color="Olive" title="b3">xy0</font>,1 &amp; <font color="Maroon" title="c3">u</font> <a href="pdiff_1.html#R3" title="PDIFF_1:pred.3">is_partial_differentiable_in</a> <font color="Olive" title="b3">xy0</font>,2 &amp; <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K10" title="PDIFF_1:func.10">partdiff</a> (<font color="Maroon" title="c3">u</font>,<font color="Olive" title="b3">xy0</font>,1)</span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K8" title="PDIFF_1:func.8">diff</a> (<span class="p2">(<span class="default"><a href="pdiff_1.html#K3" title="PDIFF_1:func.3">&lt;&gt;*</a> <font color="Maroon" title="c3">u</font></span>)</span>,<font color="Olive" title="b3">xy0</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default">1,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> &amp; <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K10" title="PDIFF_1:func.10">partdiff</a> (<font color="Maroon" title="c3">u</font>,<font color="Olive" title="b3">xy0</font>,2)</span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K8" title="PDIFF_1:func.8">diff</a> (<span class="p2">(<span class="default"><a href="pdiff_1.html#K3" title="PDIFF_1:func.3">&lt;&gt;*</a> <font color="Maroon" title="c3">u</font></span>)</span>,<font color="Olive" title="b3">xy0</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1</span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">x00</font>, <font color="Maroon" title="c5">y00</font> be   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">xy0</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> 2  st <font color="Olive" title="b1">xy0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c4">x00</font>,<font color="Maroon" title="c5">y00</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> &amp;  <a href="pdiff_1.html#K3" title="PDIFF_1:func.3">&lt;&gt;*</a> <font color="Maroon" title="c3">u</font> <a href="pdiff_1.html#R1" title="PDIFF_1:pred.1">is_differentiable_in</a> <font color="Olive" title="b1">xy0</font> holds <br/>( <font color="Maroon" title="c3">u</font> <a href="pdiff_1.html#R3" title="PDIFF_1:pred.3">is_partial_differentiable_in</a> <font color="Olive" title="b1">xy0</font>,1 &amp; <font color="Maroon" title="c3">u</font> <a href="pdiff_1.html#R3" title="PDIFF_1:pred.3">is_partial_differentiable_in</a> <font color="Olive" title="b1">xy0</font>,2 &amp; <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K10" title="PDIFF_1:func.10">partdiff</a> (<font color="Maroon" title="c3">u</font>,<font color="Olive" title="b1">xy0</font>,1)</span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K8" title="PDIFF_1:func.8">diff</a> (<span class="p2">(<span class="default"><a href="pdiff_1.html#K3" title="PDIFF_1:func.3">&lt;&gt;*</a> <font color="Maroon" title="c3">u</font></span>)</span>,<font color="Olive" title="b1">xy0</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default">1,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> &amp; <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K10" title="PDIFF_1:func.10">partdiff</a> (<font color="Maroon" title="c3">u</font>,<font color="Olive" title="b1">xy0</font>,2)</span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K8" title="PDIFF_1:func.8">diff</a> (<span class="p2">(<span class="default"><a href="pdiff_1.html#K3" title="PDIFF_1:func.3">&lt;&gt;*</a> <font color="Maroon" title="c3">u</font></span>)</span>,<font color="Olive" title="b1">xy0</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1</span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c6">xy0</font> be    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> 2; <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="c6">xy0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c4">x00</font>,<font color="Maroon" title="c5">y00</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> &amp;  <a href="pdiff_1.html#K3" title="PDIFF_1:func.3">&lt;&gt;*</a> <font color="Maroon" title="c3">u</font> <a href="pdiff_1.html#R1" title="PDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c6">xy0</font> implies ( <font color="Maroon" title="c3">u</font> <a href="pdiff_1.html#R3" title="PDIFF_1:pred.3">is_partial_differentiable_in</a> <font color="Maroon" title="c6">xy0</font>,1 &amp; <font color="Maroon" title="c3">u</font> <a href="pdiff_1.html#R3" title="PDIFF_1:pred.3">is_partial_differentiable_in</a> <font color="Maroon" title="c6">xy0</font>,2 &amp; <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K10" title="PDIFF_1:func.10">partdiff</a> (<font color="Maroon" title="c3">u</font>,<font color="Maroon" title="c6">xy0</font>,1)</span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K8" title="PDIFF_1:func.8">diff</a> (<span class="p2">(<span class="default"><a href="pdiff_1.html#K3" title="PDIFF_1:func.3">&lt;&gt;*</a> <font color="Maroon" title="c3">u</font></span>)</span>,<font color="Maroon" title="c6">xy0</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default">1,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> &amp; <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K10" title="PDIFF_1:func.10">partdiff</a> (<font color="Maroon" title="c3">u</font>,<font color="Maroon" title="c6">xy0</font>,2)</span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K8" title="PDIFF_1:func.8">diff</a> (<span class="p2">(<span class="default"><a href="pdiff_1.html#K3" title="PDIFF_1:func.3">&lt;&gt;*</a> <font color="Maroon" title="c3">u</font></span>)</span>,<font color="Maroon" title="c6">xy0</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1</span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> ) )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E3:26"/><span class="lab"><font color="Green" title="E27">A1</font></span>: 
<font color="Maroon" title="c6">xy0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c4">x00</font>,<font color="Maroon" title="c5">y00</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 <span class="kw">and </span><br/><a NAME="E4:26"/><span class="lab"><font color="Green" title="E28">A2</font></span>: 
 <a href="pdiff_1.html#K3" title="PDIFF_1:func.3">&lt;&gt;*</a> <font color="Maroon" title="c3">u</font> <a href="pdiff_1.html#R1" title="PDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c6">xy0</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="c3">u</font> <a href="pdiff_1.html#R3" title="PDIFF_1:pred.3">is_partial_differentiable_in</a> <font color="Maroon" title="c6">xy0</font>,1 &amp; <font color="Maroon" title="c3">u</font> <a href="pdiff_1.html#R3" title="PDIFF_1:pred.3">is_partial_differentiable_in</a> <font color="Maroon" title="c6">xy0</font>,2 &amp; <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K10" title="PDIFF_1:func.10">partdiff</a> (<font color="Maroon" title="c3">u</font>,<font color="Maroon" title="c6">xy0</font>,1)</span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K8" title="PDIFF_1:func.8">diff</a> (<span class="p2">(<span class="default"><a href="pdiff_1.html#K3" title="PDIFF_1:func.3">&lt;&gt;*</a> <font color="Maroon" title="c3">u</font></span>)</span>,<font color="Maroon" title="c6">xy0</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default">1,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> &amp; <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K10" title="PDIFF_1:func.10">partdiff</a> (<font color="Maroon" title="c3">u</font>,<font color="Maroon" title="c6">xy0</font>,2)</span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K8" title="PDIFF_1:func.8">diff</a> (<span class="p2">(<span class="default"><a href="pdiff_1.html#K3" title="PDIFF_1:func.3">&lt;&gt;*</a> <font color="Maroon" title="c3">u</font></span>)</span>,<font color="Maroon" title="c6">xy0</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1</span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c7">x0</font> = <font color="Maroon" title="c4">x00</font>, <font color="Maroon" title="c8">y0</font> = <font color="Maroon" title="c5">y00</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="E6:26"/><span class="lab"><font color="Green" title="E29">A3</font></span>: 
<font color="Maroon" title="c6">xy0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c7">x0</font>,<font color="Maroon" title="c8">y0</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:26"><span class="lab"><font color="Green" title="E27">A1</font></span></a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c9">W</font> = <span class="p1">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> (1,1)</span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c10">g</font> being   <a href="struct_0.html#NM12" title="STRUCT_0:NM.12">PartFunc</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> 1</span>)</span>, <font color="Maroon" title="c11">gxy0</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> 2</span>)</span><span class="kw"> such that </span><br/><a NAME="E8:26"/><span class="lab"><font color="Green" title="E30">A4</font></span>: 
 <a href="pdiff_1.html#K3" title="PDIFF_1:func.3">&lt;&gt;*</a> <font color="Maroon" title="c3">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">g</font>
 <span class="kw">and </span><br/><a NAME="E9:26"/><span class="lab"><font color="Green" title="E31">A5</font></span>: 
<font color="Maroon" title="c6">xy0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">gxy0</font>
 <span class="kw">and </span><br/><a NAME="E10:26"/><span class="lab"><font color="Green" title="E32">A6</font></span>: 
<font color="Maroon" title="c10">g</font> <a href="ndiff_1.html#R1" title="NDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c11">gxy0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:26"><span class="lab"><font color="Green" title="E28">A2</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c12">N</font> being    <a href="nfcont_1.html#M1" title="NFCONT_1:mode.1">Neighbourhood</a> of <font color="Maroon" title="c11">gxy0</font><span class="kw"> such that </span><br/><a NAME="E12:26"/><span class="lab"><font color="Green" title="E33">A7</font></span>: 
<font color="Maroon" title="c12">N</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="c10">g</font>
 <span class="kw">and </span><br/><a NAME="E13:26"/><span class="lab"><font color="Green" title="E34">A8</font></span>: 
 ex <font color="Olive" title="b1">R</font> being   <a href="ndiff_1.html#NM1" title="NDIFF_1:NM.1">RestFunc</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> 1</span>)</span> st <br/> for <font color="Olive" title="b2">xy</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> 2</span>)</span>  st <font color="Olive" title="b2">xy</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c12">N</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">xy</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">gxy0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> (<font color="Maroon" title="c10">g</font>,<font color="Maroon" title="c11">gxy0</font>)</span>)</span> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">xy</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c11">gxy0</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">R</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">xy</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c11">gxy0</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:26"><span class="lab"><font color="Green" title="E32">A6</font></span></a>, <a class="ref" href="ndiff_1.html#D7" onmouseover="rs('ndiff_1/D7')" onmouseout="rh()">NDIFF_1:def 7</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c13">R</font> being   <a href="ndiff_1.html#NM1" title="NDIFF_1:NM.1">RestFunc</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> 1</span>)</span><span class="kw"> such that </span><br/><a NAME="E15:26"/><span class="lab"><font color="Green" title="E35">A9</font></span>: 
 for <font color="Olive" title="b1">xy</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> 2</span>)</span>  st <font color="Olive" title="b1">xy</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c12">N</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">xy</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">gxy0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> (<font color="Maroon" title="c10">g</font>,<font color="Maroon" title="c11">gxy0</font>)</span>)</span> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">xy</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c11">gxy0</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">R</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">xy</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c11">gxy0</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:26"><span class="lab"><font color="Green" title="E34">A8</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c14">r0</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E17:26"/><span class="lab"><font color="Green" title="E36">A10</font></span>: 
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">r0</font>
 <span class="kw">and </span><br/><a NAME="E18:26"/><span class="lab"><font color="Green" title="E37">A11</font></span>: 
<span class="p1"> { <span class="default"> <font color="Olive" title="b1">xy</font> where <font color="Olive" title="b1">xy</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> 2</span>)</span> : <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">xy</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c11">gxy0</font></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">r0</font> </span> } </span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">N</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nfcont_1.html#D1" onmouseover="rs('nfcont_1/D1')" onmouseout="rh()">NFCONT_1:def 1</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c15">f</font> being   <a href="struct_0.html#NM12" title="STRUCT_0:NM.12">PartFunc</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> 1</span>)</span>, <font color="Maroon" title="c16">fxy0</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> 2</span>)</span><span class="kw"> such that </span><br/><a NAME="E20:26"/><span class="lab"><font color="Green" title="E38">A12</font></span>: 
(  <a href="pdiff_1.html#K3" title="PDIFF_1:func.3">&lt;&gt;*</a> <font color="Maroon" title="c3">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">f</font> &amp; <font color="Maroon" title="c6">xy0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">fxy0</font> )
 <span class="kw">and </span><br/><a NAME="E21:26"/><span class="lab"><font color="Green" title="E39">A13</font></span>: 
 <a href="pdiff_1.html#K8" title="PDIFF_1:func.8">diff</a> (<span class="p1">(<span class="default"><a href="pdiff_1.html#K3" title="PDIFF_1:func.3">&lt;&gt;*</a> <font color="Maroon" title="c3">u</font></span>)</span>,<font color="Maroon" title="c6">xy0</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> (<font color="Maroon" title="c15">f</font>,<font color="Maroon" title="c16">fxy0</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:26"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="ref" href="pdiff_1.html#D8" onmouseover="rs('pdiff_1/D8')" onmouseout="rh()">PDIFF_1:def 8</a></span>;<br/>
<a NAME="E22:26"/>
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c3">u</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> (1,1)</span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pdiff_1.html#T2" onmouseover="rs('pdiff_1/T2')" onmouseout="rh()">PDIFF_1:2</a></span>;<br/>
<a NAME="E23:26"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A14</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K3" title="PDIFF_1:func.3">&lt;&gt;*</a> <font color="Maroon" title="c3">u</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">u</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T27" onmouseover="rs('relat_1/T27')" onmouseout="rh()">RELAT_1:27</a></span>;<br/>
<a NAME="E24:26"/><span class="lab"><font color="Green" title="E41">A15</font></span>: 
 for <font color="Olive" title="b1">xy</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> 2<br/>  for <font color="Olive" title="b2">L1</font>, <font color="Olive" title="b3">R1</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><br/>  for <font color="Olive" title="b4">z</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> 2  st <font color="Olive" title="b1">xy</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c12">N</font> &amp; <font color="Olive" title="b4">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">xy</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">xy0</font> &amp; <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Olive" title="b2">L1</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K8" title="PDIFF_1:func.8">diff</a> (<span class="p2">(<span class="default"><a href="pdiff_1.html#K3" title="PDIFF_1:func.3">&lt;&gt;*</a> <font color="Maroon" title="c3">u</font></span>)</span>,<font color="Maroon" title="c6">xy0</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">z</font> &amp; <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Olive" title="b3">R1</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">z</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c3">u</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">xy</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">u</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">xy0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">L1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b3">R1</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="26_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c17">xy</font> be    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> 2; <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">L1</font>, <font color="Olive" title="b2">R1</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><br/>  for <font color="Olive" title="b3">z</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> 2  st <font color="Maroon" title="c17">xy</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c12">N</font> &amp; <font color="Olive" title="b3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">xy</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">xy0</font> &amp; <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Olive" title="b1">L1</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K8" title="PDIFF_1:func.8">diff</a> (<span class="p2">(<span class="default"><a href="pdiff_1.html#K3" title="PDIFF_1:func.3">&lt;&gt;*</a> <font color="Maroon" title="c3">u</font></span>)</span>,<font color="Maroon" title="c6">xy0</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">z</font> &amp; <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Olive" title="b2">R1</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">z</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c3">u</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">xy</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">u</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">xy0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">L1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b2">R1</font></span><br/><span class="kw">let </span><font color="Maroon" title="c18">L1</font>, <font color="Maroon" title="c19">R1</font> be   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">z</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> 2  st <font color="Maroon" title="c17">xy</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c12">N</font> &amp; <font color="Olive" title="b1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">xy</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">xy0</font> &amp; <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c18">L1</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K8" title="PDIFF_1:func.8">diff</a> (<span class="p2">(<span class="default"><a href="pdiff_1.html#K3" title="PDIFF_1:func.3">&lt;&gt;*</a> <font color="Maroon" title="c3">u</font></span>)</span>,<font color="Maroon" title="c6">xy0</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">z</font> &amp; <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c19">R1</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">z</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c3">u</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">xy</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">u</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">xy0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">L1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c19">R1</font></span><br/><span class="kw">let </span><font color="Maroon" title="c20">z</font> be    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> 2; <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="c17">xy</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c12">N</font> &amp; <font color="Maroon" title="c20">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">xy</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">xy0</font> &amp; <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c18">L1</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K8" title="PDIFF_1:func.8">diff</a> (<span class="p2">(<span class="default"><a href="pdiff_1.html#K3" title="PDIFF_1:func.3">&lt;&gt;*</a> <font color="Maroon" title="c3">u</font></span>)</span>,<font color="Maroon" title="c6">xy0</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">z</font> &amp; <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c19">R1</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">z</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c3">u</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">xy</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">u</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">xy0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">L1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c19">R1</font> )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:26_1"/><span class="lab"><font color="Green" title="E42">A16</font></span>: 
<font color="Maroon" title="c17">xy</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c12">N</font>
 <span class="kw">and </span><br/><a NAME="E2:26_1"/><span class="lab"><font color="Green" title="E43">A17</font></span>: 
<font color="Maroon" title="c20">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">xy</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">xy0</font>
 <span class="kw">and </span><br/><a NAME="E3:26_1"/><span class="lab"><font color="Green" title="E44">A18</font></span>: 
<span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c18">L1</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K8" title="PDIFF_1:func.8">diff</a> (<span class="p2">(<span class="default"><a href="pdiff_1.html#K3" title="PDIFF_1:func.3">&lt;&gt;*</a> <font color="Maroon" title="c3">u</font></span>)</span>,<font color="Maroon" title="c6">xy0</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">z</font>
 <span class="kw">and </span><br/><a NAME="E4:26_1"/><span class="lab"><font color="Green" title="E45">A19</font></span>: 
<span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c19">R1</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">z</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c3">u</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">xy</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">u</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">xy0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">L1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c19">R1</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c21">zxy</font> = <font color="Maroon" title="c17">xy</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> 2</span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="real_ns1.html#D4" onmouseover="rs('real_ns1/D4')" onmouseout="rh()">REAL_NS1:def 4</a></span>;<br/>
<a NAME="E6:26_1"/><span class="lab"><font color="Green" title="E46">A20</font></span>: 
<font color="Maroon" title="c21">zxy</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c11">gxy0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">z</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:26"><span class="lab"><font color="Green" title="E31">A5</font></span></a>, <a class="txt" href="#E2:26_1"><span class="lab"><font color="Green" title="E43">A17</font></span></a>, <a class="ref" href="real_ns1.html#T5" onmouseover="rs('real_ns1/T5')" onmouseout="rh()">REAL_NS1:5</a></span>;<br/>
<a NAME="E7:26_1"/>
<font color="Maroon" title="c13">R</font> is  <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ndiff_1.html#D5" onmouseover="rs('ndiff_1/D5')" onmouseout="rh()">NDIFF_1:def 5</a></span>;<br/>
<a NAME="E8:26_1"/><span class="kw">then </span>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c13">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> 2</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D2" onmouseover="rs('partfun1/D2')" onmouseout="rh()">PARTFUN1:def 2</a></span>;<br/>
<a NAME="E9:26_1"/><span class="kw">then </span>
<font color="Maroon" title="c13">R</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">zxy</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c11">gxy0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c19">R1</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:26_1"><span class="lab"><font color="Green" title="E45">A19</font></span></a>, <a class="txt" href="#E6:26_1"><span class="lab"><font color="Green" title="E46">A20</font></span></a>, <a class="ref" href="partfun1.html#D6" onmouseover="rs('partfun1/D6')" onmouseout="rh()">PARTFUN1:def 6</a></span>;<br/>
<a NAME="E10:26_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A21</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> (<font color="Maroon" title="c10">g</font>,<font color="Maroon" title="c11">gxy0</font>)</span>)</span> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">zxy</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c11">gxy0</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">R</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">zxy</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c11">gxy0</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">L1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c19">R1</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:26"><span class="lab"><font color="Green" title="E30">A4</font></span></a>, <a class="txt" href="#E9:26"><span class="lab"><font color="Green" title="E31">A5</font></span></a>, <a class="txt" href="#E20:26"><span class="lab"><font color="Green" title="E38">A12</font></span></a>, <a class="txt" href="#E21:26"><span class="lab"><font color="Green" title="E39">A13</font></span></a>, <a class="txt" href="#E3:26_1"><span class="lab"><font color="Green" title="E44">A18</font></span></a>, <a class="txt" href="#E6:26_1"><span class="lab"><font color="Green" title="E46">A20</font></span></a>, <a class="txt" href="cfdiff_2.html#E8"><span class="lab"><font color="Green" title="E8">Lm6</font></span></a></span>;<br/>
<a NAME="E11:26_1"/><span class="lab"><font color="Green" title="E48">A22</font></span>: 
<font color="Maroon" title="c6">xy0</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c12">N</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:26"><span class="lab"><font color="Green" title="E31">A5</font></span></a>, <a class="ref" href="nfcont_1.html#T4" onmouseover="rs('nfcont_1/T4')" onmouseout="rh()">NFCONT_1:4</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E49">A23</font></span>: <font color="Maroon" title="c10">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">gxy0</font> = 
<font color="Maroon" title="c10">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">xy0</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:26"><span class="lab"><font color="Green" title="E31">A5</font></span></a>, <a class="txt" href="#E12:26"><span class="lab"><font color="Green" title="E33">A7</font></span></a>, <a class="ref" href="partfun1.html#D6" onmouseover="rs('partfun1/D6')" onmouseout="rh()">PARTFUN1:def 6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="pdiff_1.html#K3" title="PDIFF_1:func.3">&lt;&gt;*</a> <font color="Maroon" title="c3">u</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">xy0</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:26"><span class="lab"><font color="Green" title="E30">A4</font></span></a>, <a class="txt" href="#E12:26"><span class="lab"><font color="Green" title="E33">A7</font></span></a>, <a class="txt" href="#E11:26_1"><span class="lab"><font color="Green" title="E48">A22</font></span></a>, <a class="ref" href="partfun1.html#D6" onmouseover="rs('partfun1/D6')" onmouseout="rh()">PARTFUN1:def 6</a></span>
;<br/>
<span class="lab"><font color="Green" title="E50">A24</font></span>: <font color="Maroon" title="c10">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c21">zxy</font> = 
<font color="Maroon" title="c10">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">xy</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:26"><span class="lab"><font color="Green" title="E33">A7</font></span></a>, <a class="txt" href="#E1:26_1"><span class="lab"><font color="Green" title="E42">A16</font></span></a>, <a class="ref" href="partfun1.html#D6" onmouseover="rs('partfun1/D6')" onmouseout="rh()">PARTFUN1:def 6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="pdiff_1.html#K3" title="PDIFF_1:func.3">&lt;&gt;*</a> <font color="Maroon" title="c3">u</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">xy</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:26"><span class="lab"><font color="Green" title="E30">A4</font></span></a>, <a class="txt" href="#E12:26"><span class="lab"><font color="Green" title="E33">A7</font></span></a>, <a class="txt" href="#E1:26_1"><span class="lab"><font color="Green" title="E42">A16</font></span></a>, <a class="ref" href="partfun1.html#D6" onmouseover="rs('partfun1/D6')" onmouseout="rh()">PARTFUN1:def 6</a></span>
;<br/>
<a NAME="E14:26_1"/><span class="lab"><font color="Green" title="E51">A25</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c21">zxy</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">gxy0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> (<font color="Maroon" title="c10">g</font>,<font color="Maroon" title="c11">gxy0</font>)</span>)</span> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">zxy</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c11">gxy0</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">R</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">zxy</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c11">gxy0</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:26"><span class="lab"><font color="Green" title="E35">A9</font></span></a>, <a class="txt" href="#E1:26_1"><span class="lab"><font color="Green" title="E42">A16</font></span></a></span>;<br/>
<span class="lab"><font color="Green" title="E52">A26</font></span>: <span class="p1">(<span class="default"><a href="pdiff_1.html#K3" title="PDIFF_1:func.3">&lt;&gt;*</a> <font color="Maroon" title="c3">u</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">xy0</font> = 
<span class="p1">(<span class="default"><a href="pdiff_1.html#K3" title="PDIFF_1:func.3">&lt;&gt;*</a> <font color="Maroon" title="c3">u</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">xy0</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:26"><span class="lab"><font color="Green" title="E30">A4</font></span></a>, <a class="txt" href="#E12:26"><span class="lab"><font color="Green" title="E33">A7</font></span></a>, <a class="txt" href="#E11:26_1"><span class="lab"><font color="Green" title="E48">A22</font></span></a>, <a class="ref" href="partfun1.html#D6" onmouseover="rs('partfun1/D6')" onmouseout="rh()">PARTFUN1:def 6</a></span>
<br/>.= 
<span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">u</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">xy0</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:26"><span class="lab"><font color="Green" title="E30">A4</font></span></a>, <a class="txt" href="#E12:26"><span class="lab"><font color="Green" title="E33">A7</font></span></a>, <a class="txt" href="#E23:26"><span class="lab"><font color="Green" title="E40">A14</font></span></a>, <a class="txt" href="#E11:26_1"><span class="lab"><font color="Green" title="E48">A22</font></span></a>, <a class="txt" href="cfdiff_2.html#E14"><span class="lab"><font color="Green" title="E14">Lm12</font></span></a></span>
;<br/>
<span class="p1">(<span class="default"><a href="pdiff_1.html#K3" title="PDIFF_1:func.3">&lt;&gt;*</a> <font color="Maroon" title="c3">u</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">xy</font> = 
<span class="p1">(<span class="default"><a href="pdiff_1.html#K3" title="PDIFF_1:func.3">&lt;&gt;*</a> <font color="Maroon" title="c3">u</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">xy</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:26"><span class="lab"><font color="Green" title="E30">A4</font></span></a>, <a class="txt" href="#E12:26"><span class="lab"><font color="Green" title="E33">A7</font></span></a>, <a class="txt" href="#E1:26_1"><span class="lab"><font color="Green" title="E42">A16</font></span></a>, <a class="ref" href="partfun1.html#D6" onmouseover="rs('partfun1/D6')" onmouseout="rh()">PARTFUN1:def 6</a></span>
<br/>.= 
<span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">u</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">xy</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:26"><span class="lab"><font color="Green" title="E30">A4</font></span></a>, <a class="txt" href="#E12:26"><span class="lab"><font color="Green" title="E33">A7</font></span></a>, <a class="txt" href="#E23:26"><span class="lab"><font color="Green" title="E40">A14</font></span></a>, <a class="txt" href="#E1:26_1"><span class="lab"><font color="Green" title="E42">A16</font></span></a>, <a class="txt" href="cfdiff_2.html#E14"><span class="lab"><font color="Green" title="E14">Lm12</font></span></a></span>
;<br/>
<span class="kw">then </span><span class="p1">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c21">zxy</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">gxy0</font></span>)</span> = 
<span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">u</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">xy</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="valued_1.html#K45" title="VALUED_1:func.45">-</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">u</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">xy0</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:26_1"><span class="lab"><font color="Green" title="E50">A24</font></span></a>, <a class="txt" href="#E12:26_1"><span class="lab"><font color="Green" title="E49">A23</font></span></a>, <a class="txt" href="#E15:26_1"><span class="lab"><font color="Green" title="E52">A26</font></span></a>, <a class="ref" href="real_ns1.html#T5" onmouseover="rs('real_ns1/T5')" onmouseout="rh()">REAL_NS1:5</a></span>
<br/>.= 
<span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">u</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">xy</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="valued_1.html#K1" title="VALUED_1:func.1">+</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">u</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">xy0</font></span>)</span></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T20" onmouseover="rs('rvsum_1/T20')" onmouseout="rh()">RVSUM_1:20</a></span>
<br/>.= 
<span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">u</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">xy</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">u</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">xy0</font></span>)</span></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T13" onmouseover="rs('rvsum_1/T13')" onmouseout="rh()">RVSUM_1:13</a></span>
;<br/>
<span class="kw">hence </span><span class="p1">(<span class="default"><font color="Maroon" title="c3">u</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">xy</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">u</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">xy0</font></span>)</span> = 
<span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">L1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c19">R1</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:26_1"><span class="lab"><font color="Green" title="E51">A25</font></span></a>, <a class="txt" href="#E10:26_1"><span class="lab"><font color="Green" title="E47">A21</font></span></a>, <a class="ref" href="finseq_1.html#D8" onmouseover="rs('finseq_1/D8')" onmouseout="rh()">FINSEQ_1:def 8</a></span>
<br/>.= 
<font color="Maroon" title="c18">L1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c19">R1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D8" onmouseover="rs('finseq_1/D8')" onmouseout="rh()">FINSEQ_1:def 8</a></span>
;<br/> <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">set </span><font color="Maroon" title="c17">Nx0</font> = <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c14">r0</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c14">r0</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c18">Nx0</font> = <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c14">r0</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c14">r0</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> as    <a href="rcomp_1.html#M1" title="RCOMP_1:mode.1">Neighbourhood</a> of <font color="Maroon" title="c7">x0</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:26"><span class="lab"><font color="Green" title="E36">A10</font></span></a>, <a class="ref" href="rcomp_1.html#D6" onmouseover="rs('rcomp_1/D6')" onmouseout="rh()">RCOMP_1:def 6</a></span>;<br/>
<a NAME="E26:26"/><span class="lab"><font color="Green" title="E42">A27</font></span>: 
 for <font color="Olive" title="b1">x</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">Nx0</font> holds <br/><span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Maroon" title="c8">y0</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c12">N</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="26_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c19">x</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"> ( <font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">Nx0</font> implies <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c19">x</font>,<font color="Maroon" title="c8">y0</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c12">N</font> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c20">xy</font> = <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c19">x</font>,<font color="Maroon" title="c8">y0</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> 2</span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="cfdiff_2.html#E11"><span class="lab"><font color="Green" title="E11">Lm9</font></span></a></span>;<br/>
<a NAME="E2:26_2"/>
( <font color="Maroon" title="c19">x</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c7">x0</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  &amp;  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <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/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c21">xx</font> = <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">x</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c7">x0</font></span>)</span>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> 2 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T101" onmouseover="rs('finseq_2/T101')" onmouseout="rh()">FINSEQ_2:101</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c22">xx1</font> = <font color="Maroon" title="c21">xx</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T22" onmouseover="rs('euclid/T22')" onmouseout="rh()">EUCLID:22</a></span>;<br/>
<span class="kw">assume </span><a NAME="E5:26_2"/>
<font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">Nx0</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c19">x</font>,<font color="Maroon" title="c8">y0</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c12">N</font></span><br/>

<a NAME="E6:26_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A28</font></span>: 
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">x</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c7">x0</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">r0</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rcomp_1.html#T1" onmouseover="rs('rcomp_1/T1')" onmouseout="rh()">RCOMP_1:1</a></span>;<br/>
<a NAME="E7:26_2"/>
<font color="Maroon" title="c22">xx1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T44" onmouseover="rs('finseq_1/T44')" onmouseout="rh()">FINSEQ_1:44</a></span>;<br/>
<a NAME="E8:26_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A29</font></span>: 
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c22">xx1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="absvalue.html#T2" onmouseover="rs('absvalue/T2')" onmouseout="rh()">ABSVALUE:2</a></span>;<br/>
<font color="Maroon" title="c20">xy</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c11">gxy0</font> = 
<span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">x</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c7">x0</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">y0</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c8">y0</font></span>)</span></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:26"><span class="lab"><font color="Green" title="E29">A3</font></span></a>, <a class="txt" href="#E9:26"><span class="lab"><font color="Green" title="E31">A5</font></span></a>, <a class="txt" href="cfdiff_2.html#E12"><span class="lab"><font color="Green" title="E12">Lm10</font></span></a></span>
<br/>.= 
<span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">x</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c7">x0</font></span>)</span>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>

;<br/>
<a NAME="E10:26_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A30</font></span>: 
<span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">xy</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c11">gxy0</font></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c21">xx</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="real_ns1.html#T1" onmouseover="rs('real_ns1/T1')" onmouseout="rh()">REAL_NS1:1</a></span>;<br/>
<a NAME="E11:26_2"/>
( <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c22">xx1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">x</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c7">x0</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> &amp; <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c21">xx</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c22">xx1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c22">xx1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T44" onmouseover="rs('finseq_1/T44')" onmouseout="rh()">FINSEQ_1:44</a>, <a class="ref" href="jgraph_1.html#T31" onmouseover="rs('jgraph_1/T31')" onmouseout="rh()">JGRAPH_1:31</a></span>;<br/>
<a NAME="E12:26_2"/><span class="kw">then </span>
<span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">xy</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c11">gxy0</font></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">r0</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:26_2"><span class="lab"><font color="Green" title="E43">A28</font></span></a>, <a class="txt" href="#E8:26_2"><span class="lab"><font color="Green" title="E44">A29</font></span></a>, <a class="txt" href="#E10:26_2"><span class="lab"><font color="Green" title="E45">A30</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/>
<a NAME="E13:26_2"/><span class="kw">then </span>
<font color="Maroon" title="c20">xy</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">z</font> where <font color="Olive" title="b1">z</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> 2</span>)</span> : <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">z</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c11">gxy0</font></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">r0</font> </span> } </span> 
 
;<br/>
<span class="kw">hence </span><a NAME="E14:26_2"/>
<span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c19">x</font>,<font color="Maroon" title="c8">y0</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c12">N</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:26"><span class="lab"><font color="Green" title="E37">A11</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="26_3"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">s</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K6" title="PDIFF_1:func.6">reproj</a> (1,<font color="Maroon" title="c6">xy0</font>)</span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c18">Nx0</font> holds <br/><font color="Olive" title="b1">s</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="c3">u</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c19">s</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="c19">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K6" title="PDIFF_1:func.6">reproj</a> (1,<font color="Maroon" title="c6">xy0</font>)</span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c18">Nx0</font> implies <font color="Maroon" title="c19">s</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="c3">u</font> )</span><br/><span class="kw">assume </span><a NAME="E1:26_3"/>
<font color="Maroon" title="c19">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K6" title="PDIFF_1:func.6">reproj</a> (1,<font color="Maroon" title="c6">xy0</font>)</span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c18">Nx0</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="c19">s</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="c3">u</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c20">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <span class="kw"> such that </span><br/><a NAME="E3:26_3"/><span class="lab"><font color="Green" title="E43">A31</font></span>: 
<font color="Maroon" title="c20">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c18">Nx0</font>
 <span class="kw">and </span><br/><a NAME="E4:26_3"/><span class="lab"><font color="Green" title="E44">A32</font></span>: 
<font color="Maroon" title="c19">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K6" title="PDIFF_1:func.6">reproj</a> (1,<font color="Maroon" title="c6">xy0</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T65" onmouseover="rs('funct_2/T65')" onmouseout="rh()">FUNCT_2:65</a></span>;<br/><a NAME="E5:26_3"/><span class="lab"><font color="Green" title="E45">A33</font></span>: 
<span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c20">x</font>,<font color="Maroon" title="c8">y0</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c12">N</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:26"><span class="lab"><font color="Green" title="E42">A27</font></span></a>, <a class="txt" href="#E3:26_3"><span class="lab"><font color="Green" title="E43">A31</font></span></a></span>;<br/><font color="Maroon" title="c19">s</font> = 
 <a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> (<font color="Maroon" title="c6">xy0</font>,1,<font color="Maroon" title="c20">x</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:26_3"><span class="lab"><font color="Green" title="E44">A32</font></span></a>, <a class="ref" href="pdiff_1.html#D5" onmouseover="rs('pdiff_1/D5')" onmouseout="rh()">PDIFF_1:def 5</a></span>
<br/>.= 
<span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c20">x</font>,<font color="Maroon" title="c8">y0</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:26"><span class="lab"><font color="Green" title="E29">A3</font></span></a>, <a class="ref" href="finseq_7.html#T13" onmouseover="rs('finseq_7/T13')" onmouseout="rh()">FINSEQ_7:13</a></span>
;<br/><span class="kw">hence </span><a NAME="E7:26_3"/>
<font color="Maroon" title="c19">s</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="c3">u</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:26"><span class="lab"><font color="Green" title="E30">A4</font></span></a>, <a class="txt" href="#E12:26"><span class="lab"><font color="Green" title="E33">A7</font></span></a>, <a class="txt" href="#E23:26"><span class="lab"><font color="Green" title="E40">A14</font></span></a>, <a class="txt" href="#E5:26_3"><span class="lab"><font color="Green" title="E45">A33</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E28:26"/><span class="kw">then </span>
(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K6" title="PDIFF_1:func.6">reproj</a> (1,<font color="Maroon" title="c6">xy0</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  &amp; <span class="p1">(<span class="default"><a href="pdiff_1.html#K6" title="PDIFF_1:func.6">reproj</a> (1,<font color="Maroon" title="c6">xy0</font>)</span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c18">Nx0</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="c3">u</font> )
 
<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>, <a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a></span>;<br/>
<a NAME="E29:26"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A34</font></span>: 
<font color="Maroon" title="c18">Nx0</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">u</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="pdiff_1.html#K6" title="PDIFF_1:func.6">reproj</a> (1,<font color="Maroon" title="c6">xy0</font>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_3.html#T3" onmouseover="rs('funct_3/T3')" onmouseout="rh()">FUNCT_3:3</a></span>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <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> ,   <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">means </span> ex <font color="Olive" title="b1">vx</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> 2 st <br/>( <font color="Olive" title="b1">vx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default">$1,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> &amp; <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">$2</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">vx</font> );<br/>
<div><span class="lab"><font color="Green" title="E44">A35</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="26_4"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>   ex <font color="Olive" title="b2">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>]</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c19">x</font> be    <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> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c19">x</font>,<font color="Olive" title="b1">y</font>]</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c20">vx</font> = <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c19">x</font>,<span class="p2">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>)</span>)</span></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> 2 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T101" onmouseover="rs('finseq_2/T101')" onmouseout="rh()">FINSEQ_2:101</a></span>;<br/><a NAME="E2:26_4"/>
<font color="Maroon" title="c13">R</font> is  <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ndiff_1.html#D5" onmouseover="rs('ndiff_1/D5')" onmouseout="rh()">NDIFF_1:def 5</a></span>;<br/><a NAME="E3:26_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A36</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c13">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> 2</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D2" onmouseover="rs('partfun1/D2')" onmouseout="rh()">PARTFUN1:def 2</a></span>;<br/><a NAME="E4:26_4"/>
<font color="Maroon" title="c20">vx</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> 2</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="real_ns1.html#D4" onmouseover="rs('real_ns1/D4')" onmouseout="rh()">REAL_NS1:def 4</a></span>;<br/><a NAME="E5:26_4"/><span class="kw">then </span>
<font color="Maroon" title="c13">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">vx</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:26_4"><span class="lab"><font color="Green" title="E45">A36</font></span></a>, <a class="ref" href="partfun1.html#T4" onmouseover="rs('partfun1/T4')" onmouseout="rh()">PARTFUN1:4</a></span>;<br/><a NAME="E6:26_4"/><span class="kw">then </span>
<font color="Maroon" title="c13">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">vx</font> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="real_ns1.html#D4" onmouseover="rs('real_ns1/D4')" onmouseout="rh()">REAL_NS1:def 4</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c21">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <span class="kw"> such that </span><br/><a NAME="E8:26_4"/><span class="lab"><font color="Green" title="E46">A37</font></span>: 
<span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c21">y</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">vx</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T97" onmouseover="rs('finseq_2/T97')" onmouseout="rh()">FINSEQ_2:97</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c22">y</font> = <font color="Maroon" title="c21">y</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">S<sub>1</sub></font>[<font color="Maroon" title="c19">x</font>,<font color="Maroon" title="c22">y</font>]</span><br/><span class="kw">thus </span><a NAME="E9:26_4"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c19">x</font>,<font color="Maroon" title="c22">y</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:26_4"><span class="lab"><font color="Green" title="E46">A37</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c19">R1</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</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="E32:26"/><span class="lab"><font color="Green" title="E45">A38</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c19">R1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S3" onmouseover="rs('funct_2/S3')" onmouseout="rh()">FUNCT_2:sch 3</a>(<a class="txt" href="#E30:26"><span class="lab"><font color="Green" title="E44">A35</font></span></a>);<br/></span>
<div><span class="lab"><font color="Green" title="E46">A39</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="26_5"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><br/>  for <font color="Olive" title="b2">vx</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> 2  st <font color="Olive" title="b2">vx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Olive" title="b1">x</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> holds <br/><span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">R1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">vx</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c20">x</font> be   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">vx</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> 2  st <font color="Olive" title="b1">vx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c20">x</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> holds <br/><span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">R1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">x</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">vx</font></span><br/><span class="kw">let </span><font color="Maroon" title="c21">vx</font> be    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> 2; <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="c21">vx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c20">x</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> implies <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">R1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">x</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">vx</font> )</span><br/><span class="kw">assume </span><a NAME="E1:26_5"/><span class="lab"><font color="Green" title="E47">A40</font></span>: 
<font color="Maroon" title="c21">vx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c20">x</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</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"> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">R1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">x</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">vx</font></span><br/><a NAME="E2:26_5"/>
<font color="Maroon" title="c20">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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="E3:26_5"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">vx1</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> 2 st <br/>( <font color="Olive" title="b1">vx1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c20">x</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> &amp; <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">R1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">x</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">vx1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:26"><span class="lab"><font color="Green" title="E45">A38</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:26_5"/>
<span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">R1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">x</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">vx</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26_5"><span class="lab"><font color="Green" title="E47">A40</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E34:26"/>
 for <font color="Olive" title="b1">h</font> being   <a href="ordinal1.html#V9" title="ORDINAL1:attr.9">non-zero</a>   <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="fdiff_1.html#V1" title="FDIFF_1:attr.1">-convergent</a>  <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a> holds <br/> ( <span class="p1">(<span class="default"><font color="Olive" title="b1">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">R1</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Olive" title="b1">h</font></span>)</span> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a>  &amp;  <a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">R1</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Olive" title="b1">h</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="26_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c20">h</font> be   <a href="ordinal1.html#V9" title="ORDINAL1:attr.9">non-zero</a>   <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="fdiff_1.html#V1" title="FDIFF_1:attr.1">-convergent</a>  <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><font color="Maroon" title="c20">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">R1</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c20">h</font></span>)</span> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a>  &amp;  <a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">R1</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c20">h</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )</span><br/>

<span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>,  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> 2</span>)</span>] <span class="kw">means </span>$2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E47">A41</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="26_6_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   ex <font color="Olive" title="b2">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> 2</span>)</span> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Olive" title="b2">y</font>]</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c21">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> 2</span>)</span> st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c21">n</font>,<font color="Olive" title="b1">y</font>]</span><br/><a NAME="E1:26_6_1"/>
<span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>)</span>)</span></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T101" onmouseover="rs('finseq_2/T101')" onmouseout="rh()">FINSEQ_2:101</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c22">y</font> = <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>)</span>)</span></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> 2</span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="real_ns1.html#D4" onmouseover="rs('real_ns1/D4')" onmouseout="rh()">REAL_NS1:def 4</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c23">y</font> = <font color="Maroon" title="c22">y</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">S<sub>2</sub></font>[<font color="Maroon" title="c21">n</font>,<font color="Maroon" title="c23">y</font>]</span><br/><span class="kw">thus </span><a NAME="E3:26_6_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c21">n</font>,<font color="Maroon" title="c23">y</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">consider </span><font color="Maroon" title="c21">h1</font> being   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> 2</span>)</span><span class="kw"> such that </span><br/><a NAME="E3:26_6"/><span class="lab"><font color="Green" title="E48">A42</font></span>: 
 for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Maroon" title="c21">h1</font> <a href="normsp_0.html#K1" title="NORMSP_0:func.1">.</a> <font color="Olive" title="b1">n</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S3" onmouseover="rs('funct_2/S3')" onmouseout="rh()">FUNCT_2:sch 3</a>(<a class="txt" href="#E1:26_6"><span class="lab"><font color="Green" title="E47">A41</font></span></a>);<br/></span>
<a NAME="E4:26_6"/><span class="lab"><font color="Green" title="E49">A43</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Maroon" title="c21">h1</font> <a href="normsp_0.html#K1" title="NORMSP_0:func.1">.</a> <font color="Olive" title="b1">n</font>]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="26_6_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c22">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c22">n</font>,<font color="Maroon" title="c21">h1</font> <a href="normsp_0.html#K1" title="NORMSP_0:func.1">.</a> <font color="Maroon" title="c22">n</font>]</span><br/>

<a NAME="E1:26_6_2"/>
<font color="Maroon" title="c22">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" onmouseover="rs('ordinal1/D12')" onmouseout="rh()">ORDINAL1:def 12</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:26_6_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c22">n</font>,<font color="Maroon" title="c21">h1</font> <a href="normsp_0.html#K1" title="NORMSP_0:func.1">.</a> <font color="Maroon" title="c22">n</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:26_6"><span class="lab"><font color="Green" title="E48">A42</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">reconsider </span><font color="Maroon" title="c22">h1</font> = <font color="Maroon" title="c21">h1</font> as   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> 2</span>)</span> ;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="26_6_3"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  holds  <span class="p1"><a href="normsp_0.html#K5" title="NORMSP_0:func.5">||.</a><span class="default"><font color="Maroon" title="c22">h1</font></span><a href="normsp_0.html#K5" title="NORMSP_0:func.5">.||</a></span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c20">h</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c23">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="normsp_0.html#K5" title="NORMSP_0:func.5">||.</a><span class="default"><font color="Maroon" title="c22">h1</font></span><a href="normsp_0.html#K5" title="NORMSP_0:func.5">.||</a></span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c20">h</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c23">n</font></span><br/><a NAME="E1:26_6_3"/><span class="lab"><font color="Green" title="E50">A44</font></span>: 
<font color="Maroon" title="c22">h1</font> <a href="normsp_0.html#K1" title="NORMSP_0:func.1">.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c23">n</font></span>)</span>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:26_6"><span class="lab"><font color="Green" title="E49">A43</font></span></a></span>;<br/><span class="kw">thus </span><span class="p1"><a href="normsp_0.html#K5" title="NORMSP_0:func.5">||.</a><span class="default"><font color="Maroon" title="c22">h1</font></span><a href="normsp_0.html#K5" title="NORMSP_0:func.5">.||</a></span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c23">n</font> = 
<span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c22">h1</font> <a href="normsp_0.html#K1" title="NORMSP_0:func.1">.</a> <font color="Maroon" title="c23">n</font></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="normsp_0.html#D4" onmouseover="rs('normsp_0/D4')" onmouseout="rh()">NORMSP_0:def 4</a></span>
<br/>.= 
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c23">n</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26_6_3"><span class="lab"><font color="Green" title="E50">A44</font></span></a>, <a class="txt" href="cfdiff_2.html#E22"><span class="lab"><font color="Green" title="E22">Lm18</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c20">h</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c23">n</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#T18" onmouseover="rs('valued_1/T18')" onmouseout="rh()">VALUED_1:18</a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E7:26_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A45</font></span>: 
<span class="p1"><a href="normsp_0.html#K5" title="NORMSP_0:func.5">||.</a><span class="default"><font color="Maroon" title="c22">h1</font></span><a href="normsp_0.html#K5" title="NORMSP_0:func.5">.||</a></span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c20">h</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T63" onmouseover="rs('funct_2/T63')" onmouseout="rh()">FUNCT_2:63</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c23">g</font> = <span class="p1">(<span class="default"><span class="p2"><a href="normsp_0.html#K5" title="NORMSP_0:func.5">||.</a><span class="default"><font color="Maroon" title="c22">h1</font></span><a href="normsp_0.html#K5" title="NORMSP_0:func.5">.||</a></span> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="ndiff_1.html#K1" title="NDIFF_1:func.1">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">R</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c22">h1</font></span>)</span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="26_6_4"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  holds  <span class="p1"><a href="normsp_0.html#K5" title="NORMSP_0:func.5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="normsp_0.html#K5" title="NORMSP_0:func.5">||.</a><span class="default"><font color="Maroon" title="c22">h1</font></span><a href="normsp_0.html#K5" title="NORMSP_0:func.5">.||</a></span> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="ndiff_1.html#K1" title="NDIFF_1:func.1">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">R</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c22">h1</font></span>)</span></span>)</span></span><a href="normsp_0.html#K5" title="NORMSP_0:func.5">.||</a></span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c20">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c19">R1</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c20">h</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c24">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="normsp_0.html#K5" title="NORMSP_0:func.5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="normsp_0.html#K5" title="NORMSP_0:func.5">||.</a><span class="default"><font color="Maroon" title="c22">h1</font></span><a href="normsp_0.html#K5" title="NORMSP_0:func.5">.||</a></span> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="ndiff_1.html#K1" title="NDIFF_1:func.1">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">R</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c22">h1</font></span>)</span></span>)</span></span><a href="normsp_0.html#K5" title="NORMSP_0:func.5">.||</a></span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c20">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c19">R1</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c20">h</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">n</font></span><br/><span class="kw">reconsider </span><font color="Maroon" title="c25">v</font> = <font color="Maroon" title="c22">h1</font> <a href="normsp_0.html#K1" title="NORMSP_0:func.1">.</a> <font color="Maroon" title="c24">n</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> 2 <span class="kw">by</span> <span class="lab"><a class="ref" href="real_ns1.html#D4" onmouseover="rs('real_ns1/D4')" onmouseout="rh()">REAL_NS1:def 4</a></span>;<br/><a NAME="E2:26_6_4"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c19">R1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D2" onmouseover="rs('partfun1/D2')" onmouseout="rh()">PARTFUN1:def 2</a></span>;<br/><a NAME="E3:26_6_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A46</font></span>: 
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c20">h</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="c19">R1</font>
 ;<br/><a NAME="E4:26_6_4"/><span class="lab"><font color="Green" title="E52">A47</font></span>: 
<font color="Maroon" title="c22">h1</font> <a href="normsp_0.html#K1" title="NORMSP_0:func.1">.</a> <font color="Maroon" title="c24">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">n</font></span>)</span>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:26_6"><span class="lab"><font color="Green" title="E49">A43</font></span></a></span>;<br/><a NAME="E5:26_6_4"/>
<font color="Maroon" title="c13">R</font> is  <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ndiff_1.html#D5" onmouseover="rs('ndiff_1/D5')" onmouseout="rh()">NDIFF_1:def 5</a></span>;<br/><a NAME="E6:26_6_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A48</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c13">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> 2</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D2" onmouseover="rs('partfun1/D2')" onmouseout="rh()">PARTFUN1:def 2</a></span>;<br/><a NAME="E7:26_6_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A49</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c22">h1</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="c13">R</font>
 ;<br/><a NAME="E8:26_6_4"/><span class="lab"><font color="Green" title="E55">A50</font></span>: 
<font color="Maroon" title="c22">h1</font> <a href="normsp_0.html#K1" title="NORMSP_0:func.1">.</a> <font color="Maroon" title="c24">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">n</font></span>)</span>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:26_6"><span class="lab"><font color="Green" title="E49">A43</font></span></a></span>;<br/><span class="lab"><font color="Green" title="E56">A51</font></span>: <font color="Maroon" title="c13">R</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">h1</font> <a href="normsp_0.html#K1" title="NORMSP_0:func.1">.</a> <font color="Maroon" title="c24">n</font></span>)</span> = 
<font color="Maroon" title="c13">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">v</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:26_6_4"><span class="lab"><font color="Green" title="E53">A48</font></span></a>, <a class="ref" href="partfun1.html#D6" onmouseover="rs('partfun1/D6')" onmouseout="rh()">PARTFUN1:def 6</a></span>
<br/>.= 
<span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">R1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c20">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">n</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:26"><span class="lab"><font color="Green" title="E46">A39</font></span></a>, <a class="txt" href="#E4:26_6_4"><span class="lab"><font color="Green" title="E52">A47</font></span></a></span>
;<br/><span class="kw">thus </span><span class="p1"><a href="normsp_0.html#K5" title="NORMSP_0:func.5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="normsp_0.html#K5" title="NORMSP_0:func.5">||.</a><span class="default"><font color="Maroon" title="c22">h1</font></span><a href="normsp_0.html#K5" title="NORMSP_0:func.5">.||</a></span> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="ndiff_1.html#K1" title="NDIFF_1:func.1">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">R</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c22">h1</font></span>)</span></span>)</span></span><a href="normsp_0.html#K5" title="NORMSP_0:func.5">.||</a></span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">n</font> = 
<span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5"><a href="normsp_0.html#K5" title="NORMSP_0:func.5">||.</a><span class="default"><font color="Maroon" title="c22">h1</font></span><a href="normsp_0.html#K5" title="NORMSP_0:func.5">.||</a></span> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="ndiff_1.html#K1" title="NDIFF_1:func.1">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c13">R</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c22">h1</font></span>)</span></span>)</span> <a href="normsp_0.html#K1" title="NORMSP_0:func.1">.</a> <font color="Maroon" title="c24">n</font></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="normsp_0.html#D4" onmouseover="rs('normsp_0/D4')" onmouseout="rh()">NORMSP_0:def 4</a></span>
<br/>.= 
<span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5"><a href="normsp_0.html#K5" title="NORMSP_0:func.5">||.</a><span class="default"><font color="Maroon" title="c22">h1</font></span><a href="normsp_0.html#K5" title="NORMSP_0:func.5">.||</a></span> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">n</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c13">R</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c22">h1</font></span>)</span> <a href="normsp_0.html#K1" title="NORMSP_0:func.1">.</a> <font color="Maroon" title="c24">n</font></span>)</span></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ndiff_1.html#D2" onmouseover="rs('ndiff_1/D2')" onmouseout="rh()">NDIFF_1:def 2</a></span>
<br/>.= 
<span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5"><a href="normsp_0.html#K5" title="NORMSP_0:func.5">||.</a><span class="default"><font color="Maroon" title="c22">h1</font></span><a href="normsp_0.html#K5" title="NORMSP_0:func.5">.||</a></span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">n</font></span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c13">R</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c22">h1</font></span>)</span> <a href="normsp_0.html#K1" title="NORMSP_0:func.1">.</a> <font color="Maroon" title="c24">n</font></span>)</span></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#T10" onmouseover="rs('valued_1/T10')" onmouseout="rh()">VALUED_1:10</a></span>
<br/>.= 
<span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c22">h1</font> <a href="normsp_0.html#K1" title="NORMSP_0:func.1">.</a> <font color="Maroon" title="c24">n</font></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c13">R</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c22">h1</font></span>)</span> <a href="normsp_0.html#K1" title="NORMSP_0:func.1">.</a> <font color="Maroon" title="c24">n</font></span>)</span></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="normsp_0.html#D4" onmouseover="rs('normsp_0/D4')" onmouseout="rh()">NORMSP_0:def 4</a></span>
<br/>.= 
<span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c20">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">n</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c13">R</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c22">h1</font></span>)</span> <a href="normsp_0.html#K1" title="NORMSP_0:func.1">.</a> <font color="Maroon" title="c24">n</font></span>)</span></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:26_6_4"><span class="lab"><font color="Green" title="E55">A50</font></span></a>, <a class="txt" href="cfdiff_2.html#E22"><span class="lab"><font color="Green" title="E22">Lm18</font></span></a></span>
<br/>.= 
<span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c20">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">n</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">R</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c22">h1</font> <a href="normsp_0.html#K1" title="NORMSP_0:func.1">.</a> <font color="Maroon" title="c24">n</font></span>)</span></span>)</span></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:26_6_4"><span class="lab"><font color="Green" title="E54">A49</font></span></a>, <a class="ref" href="funct_2.html#T109" onmouseover="rs('funct_2/T109')" onmouseout="rh()">FUNCT_2:109</a></span>
<br/>.= 
<span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c20">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">n</font></span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">R</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c22">h1</font> <a href="normsp_0.html#K1" title="NORMSP_0:func.1">.</a> <font color="Maroon" title="c24">n</font></span>)</span></span>)</span></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T66" onmouseover="rs('complex1/T66')" onmouseout="rh()">COMPLEX1:66</a></span>
<br/>.= 
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c20">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">n</font></span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">R</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c22">h1</font> <a href="normsp_0.html#K1" title="NORMSP_0:func.1">.</a> <font color="Maroon" title="c24">n</font></span>)</span></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="normsp_1.html#D1" onmouseover="rs('normsp_1/D1')" onmouseout="rh()">NORMSP_1:def 1</a></span>
<br/>.= 
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c20">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">n</font></span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">R1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c20">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">n</font></span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:26_6_4"><span class="lab"><font color="Green" title="E56">A51</font></span></a>, <a class="txt" href="cfdiff_2.html#E24"><span class="lab"><font color="Green" title="E24">Lm20</font></span></a></span>
<br/>.= 
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c20">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">n</font></span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c19">R1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c20">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">n</font></span>)</span></span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T65" onmouseover="rs('complex1/T65')" onmouseout="rh()">COMPLEX1:65</a></span>
<br/>.= 
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c20">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">n</font></span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c19">R1</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c20">h</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">n</font></span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:26_6_4"><span class="lab"><font color="Green" title="E51">A46</font></span></a>, <a class="ref" href="funct_2.html#T108" onmouseover="rs('funct_2/T108')" onmouseout="rh()">FUNCT_2:108</a></span>
<br/>.= 
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c20">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">n</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c19">R1</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c20">h</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">n</font></span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#T10" onmouseover="rs('valued_1/T10')" onmouseout="rh()">VALUED_1:10</a></span>
<br/>.= 
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c20">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c19">R1</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c20">h</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">n</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#T5" onmouseover="rs('valued_1/T5')" onmouseout="rh()">VALUED_1:5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c20">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c19">R1</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c20">h</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">n</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#T18" onmouseover="rs('valued_1/T18')" onmouseout="rh()">VALUED_1:18</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="E9:26_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A52</font></span>: 
 <a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">R1</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c20">h</font></span>)</span></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1"><a href="normsp_0.html#K5" title="NORMSP_0:func.5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="normsp_0.html#K5" title="NORMSP_0:func.5">||.</a><span class="default"><font color="Maroon" title="c22">h1</font></span><a href="normsp_0.html#K5" title="NORMSP_0:func.5">.||</a></span> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="ndiff_1.html#K1" title="NDIFF_1:func.1">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">R</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c22">h1</font></span>)</span></span>)</span></span><a href="normsp_0.html#K5" title="NORMSP_0:func.5">.||</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T63" onmouseover="rs('funct_2/T63')" onmouseout="rh()">FUNCT_2:63</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="26_6_5"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  holds <font color="Maroon" title="c22">h1</font> <a href="normsp_0.html#K1" title="NORMSP_0:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> 2</span>)</span></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c24">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c22">h1</font> <a href="normsp_0.html#K1" title="NORMSP_0:func.1">.</a> <font color="Maroon" title="c24">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> 2</span>)</span></span><br/><a NAME="E1:26_6_5"/><span class="lab"><font color="Green" title="E52">A53</font></span>: 
<font color="Maroon" title="c20">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="seq_1.html#T5" onmouseover="rs('seq_1/T5')" onmouseout="rh()">SEQ_1:5</a></span>;<br/><a NAME="E2:26_6_5"/><span class="lab"><font color="Green" title="E53">A54</font></span>: 
<font color="Maroon" title="c22">h1</font> <a href="normsp_0.html#K1" title="NORMSP_0:func.1">.</a> <font color="Maroon" title="c24">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">n</font></span>)</span>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:26_6"><span class="lab"><font color="Green" title="E49">A43</font></span></a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="26_6_5_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <font color="Maroon" title="c22">h1</font> <a href="normsp_0.html#K1" title="NORMSP_0:func.1">.</a> <font color="Maroon" title="c24">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> 2</span>)</span></span></a><div class="add"><span class="kw">assume </span><a NAME="E1:26_6_5_1"/>
<font color="Maroon" title="c22">h1</font> <a href="normsp_0.html#K1" title="NORMSP_0:func.1">.</a> <font color="Maroon" title="c24">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> 2</span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">then </span><font color="Maroon" title="c22">h1</font> <a href="normsp_0.html#K1" title="NORMSP_0:func.1">.</a> <font color="Maroon" title="c24">n</font> = 
 <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> 2
<span class="kw">by</span> <span class="lab"><a class="ref" href="real_ns1.html#D4" onmouseover="rs('real_ns1/D4')" onmouseout="rh()">REAL_NS1:def 4</a></span>
<br/>.= 
<span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T54" onmouseover="rs('euclid/T54')" onmouseout="rh()">EUCLID:54</a>, <a class="ref" href="euclid.html#T70" onmouseover="rs('euclid/T70')" onmouseout="rh()">EUCLID:70</a></span>
;<br/><span class="kw">hence </span><a NAME="E3:26_6_5_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:26_6_5"><span class="lab"><font color="Green" title="E53">A54</font></span></a>, <a class="txt" href="#E1:26_6_5"><span class="lab"><font color="Green" title="E52">A53</font></span></a>, <a class="ref" href="finseq_1.html#T77" onmouseover="rs('finseq_1/T77')" onmouseout="rh()">FINSEQ_1:77</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E4:26_6_5"/>
<font color="Maroon" title="c22">h1</font> <a href="normsp_0.html#K1" title="NORMSP_0:func.1">.</a> <font color="Maroon" title="c24">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> 2</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/></div><span class="kw">end;</span></div>
<a NAME="E11:26_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A55</font></span>: 
<font color="Maroon" title="c22">h1</font> is  <a href="struct_0.html#V12" title="STRUCT_0:attr.12">non-zero</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ndiff_1.html#T7" onmouseover="rs('ndiff_1/T7')" onmouseout="rh()">NDIFF_1:7</a></span>;<br/>
<a NAME="E12:26_6"/>
( <font color="Maroon" title="c20">h</font> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a>  &amp;  <a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c20">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 
;<br/>
<a NAME="E13:26_6"/><span class="kw">then </span>
(  <a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c20">h</font> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a>  &amp;  <a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p1">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c20">h</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cfdiff_2.html#T3" target="_self" onmouseover="rs('cfdiff_2/T3')" onmouseout="rh()">Th3</a></span>;<br/>
<a NAME="E14:26_6"/><span class="kw">then </span>
( <font color="Maroon" title="c22">h1</font> is  <a href="normsp_1.html#V3" title="NORMSP_1:attr.3">convergent</a>  &amp;  <a href="normsp_1.html#K5" title="NORMSP_1:func.5">lim</a> <font color="Maroon" title="c22">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> 2</span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:26_6"><span class="lab"><font color="Green" title="E50">A45</font></span></a>, <a class="ref" href="cfdiff_2.html#T4" target="_self" onmouseover="rs('cfdiff_2/T4')" onmouseout="rh()">Th4</a></span>;<br/>
<a NAME="E15:26_6"/><span class="kw">then </span>
( <font color="Maroon" title="c22">h1</font> is  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> 2</span>)</span> <a href="ndiff_1.html#V1" title="NDIFF_1:attr.1">-convergent</a>  &amp; <font color="Maroon" title="c22">h1</font> is  <a href="struct_0.html#V12" title="STRUCT_0:attr.12">non-zero</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:26_6"><span class="lab"><font color="Green" title="E52">A55</font></span></a>, <a class="ref" href="ndiff_1.html#D4" onmouseover="rs('ndiff_1/D4')" onmouseout="rh()">NDIFF_1:def 4</a></span>;<br/>
<a NAME="E16:26_6"/><span class="kw">then </span>
( <span class="p1">(<span class="default"><span class="p2"><a href="normsp_0.html#K5" title="NORMSP_0:func.5">||.</a><span class="default"><font color="Maroon" title="c22">h1</font></span><a href="normsp_0.html#K5" title="NORMSP_0:func.5">.||</a></span> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="ndiff_1.html#K1" title="NDIFF_1:func.1">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">R</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c22">h1</font></span>)</span> is  <a href="normsp_1.html#V3" title="NORMSP_1:attr.3">convergent</a>  &amp;  <a href="normsp_1.html#K5" title="NORMSP_1:func.5">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="normsp_0.html#K5" title="NORMSP_0:func.5">||.</a><span class="default"><font color="Maroon" title="c22">h1</font></span><a href="normsp_0.html#K5" title="NORMSP_0:func.5">.||</a></span> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="ndiff_1.html#K1" title="NDIFF_1:func.1">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">R</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c22">h1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> 1</span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ndiff_1.html#D5" onmouseover="rs('ndiff_1/D5')" onmouseout="rh()">NDIFF_1:def 5</a></span>;<br/>
<a NAME="E17:26_6"/><span class="kw">then </span>
( <span class="p1"><a href="normsp_0.html#K5" title="NORMSP_0:func.5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="normsp_0.html#K5" title="NORMSP_0:func.5">||.</a><span class="default"><font color="Maroon" title="c22">h1</font></span><a href="normsp_0.html#K5" title="NORMSP_0:func.5">.||</a></span> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="ndiff_1.html#K1" title="NDIFF_1:func.1">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">R</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c22">h1</font></span>)</span></span>)</span></span><a href="normsp_0.html#K5" title="NORMSP_0:func.5">.||</a></span> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a>  &amp;  <a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p1"><a href="normsp_0.html#K5" title="NORMSP_0:func.5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="normsp_0.html#K5" title="NORMSP_0:func.5">||.</a><span class="default"><font color="Maroon" title="c22">h1</font></span><a href="normsp_0.html#K5" title="NORMSP_0:func.5">.||</a></span> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="ndiff_1.html#K1" title="NDIFF_1:func.1">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">R</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c22">h1</font></span>)</span></span>)</span></span><a href="normsp_0.html#K5" title="NORMSP_0:func.5">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cfdiff_2.html#T4" target="_self" onmouseover="rs('cfdiff_2/T4')" onmouseout="rh()">Th4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E18:26_6"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c20">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">R1</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c20">h</font></span>)</span> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a>  &amp;  <a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">R1</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c20">h</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:26_6"><span class="lab"><font color="Green" title="E51">A52</font></span></a>, <a class="ref" href="cfdiff_2.html#T3" target="_self" onmouseover="rs('cfdiff_2/T3')" onmouseout="rh()">Th3</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">then </span><span class="kw">reconsider </span><font color="Maroon" title="c20">R1</font> = <font color="Maroon" title="c19">R1</font> as   <a href="fdiff_1.html#NM1" title="FDIFF_1:NM.1">RestFunc</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="fdiff_1.html#D2" onmouseover="rs('fdiff_1/D2')" onmouseout="rh()">FDIFF_1:def 2</a></span>;<br/>
<a NAME="E36:26"/>
<font color="Maroon" title="c2">ex0</font> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="real_ns1.html#D4" onmouseover="rs('real_ns1/D4')" onmouseout="rh()">REAL_NS1:def 4</a></span>;<br/>
<a NAME="E37:26"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="pdiff_1.html#K8" title="PDIFF_1:func.8">diff</a> (<span class="p2">(<span class="default"><a href="pdiff_1.html#K3" title="PDIFF_1:func.3">&lt;&gt;*</a> <font color="Maroon" title="c3">u</font></span>)</span>,<font color="Maroon" title="c6">xy0</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">ex0</font> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T5" onmouseover="rs('funct_2/T5')" onmouseout="rh()">FUNCT_2:5</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c21">Dux</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <span class="kw"> such that </span><br/><a NAME="E39:26"/><span class="lab"><font color="Green" title="E47">A56</font></span>: 
<span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c21">Dux</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K8" title="PDIFF_1:func.8">diff</a> (<span class="p2">(<span class="default"><a href="pdiff_1.html#K3" title="PDIFF_1:func.3">&lt;&gt;*</a> <font color="Maroon" title="c3">u</font></span>)</span>,<font color="Maroon" title="c6">xy0</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">ex0</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T97" onmouseover="rs('finseq_2/T97')" onmouseout="rh()">FINSEQ_2:97</a></span>;<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(  <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>) <span class="kw">-&gt; </span>   <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>  =  <a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c21">Dux</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> $1</span>)</span>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>);<br/>
<span class="kw">consider </span><font color="Maroon" title="c22">LD1</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</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="E41:26"/><span class="lab"><font color="Green" title="E48">A57</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  holds  <font color="Maroon" title="c22">LD1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">x</font>)
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S4" onmouseover="rs('funct_2/S4')" onmouseout="rh()">FUNCT_2:sch 4</a>();<br/></span>
<a NAME="E42:26"/><span class="lab"><font color="Green" title="E49">A58</font></span>: 
 for <font color="Olive" title="b1">x</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> holds  <font color="Maroon" title="c22">LD1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">Dux</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="26_7"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c23">x</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"> <font color="Maroon" title="c22">LD1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">Dux</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c23">x</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c24">x</font> = <font color="Maroon" title="c23">x</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="E2:26_7"/>
<font color="Maroon" title="c22">LD1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c24">x</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:26"><span class="lab"><font color="Green" title="E48">A57</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:26_7"/>
<font color="Maroon" title="c22">LD1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">Dux</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c23">x</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">set </span><font color="Maroon" title="c23">Ny0</font> = <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">y0</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c14">r0</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">y0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c14">r0</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c24">Ny0</font> = <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">y0</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c14">r0</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">y0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c14">r0</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> as    <a href="rcomp_1.html#M1" title="RCOMP_1:mode.1">Neighbourhood</a> of <font color="Maroon" title="c8">y0</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:26"><span class="lab"><font color="Green" title="E36">A10</font></span></a>, <a class="ref" href="rcomp_1.html#D6" onmouseover="rs('rcomp_1/D6')" onmouseout="rh()">RCOMP_1:def 6</a></span>;<br/>
<a NAME="E44:26"/><span class="lab"><font color="Green" title="E50">A59</font></span>: 
 for <font color="Olive" title="b1">y</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c24">Ny0</font> holds <br/><span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c7">x0</font>,<font color="Olive" title="b1">y</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c12">N</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="26_8"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c25">y</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"> ( <font color="Maroon" title="c25">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c24">Ny0</font> implies <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c7">x0</font>,<font color="Maroon" title="c25">y</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c12">N</font> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c26">xy</font> = <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c7">x0</font>,<font color="Maroon" title="c25">y</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> 2</span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="cfdiff_2.html#E11"><span class="lab"><font color="Green" title="E11">Lm9</font></span></a></span>;<br/>
<a NAME="E2:26_8"/>
(  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  &amp; <font color="Maroon" title="c25">y</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c8">y0</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <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/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c27">xx</font> = <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<span class="p2">(<span class="default"><font color="Maroon" title="c25">y</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c8">y0</font></span>)</span></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> 2 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T101" onmouseover="rs('finseq_2/T101')" onmouseout="rh()">FINSEQ_2:101</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c28">xx1</font> = <font color="Maroon" title="c27">xx</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T22" onmouseover="rs('euclid/T22')" onmouseout="rh()">EUCLID:22</a></span>;<br/>
<span class="kw">assume </span><a NAME="E5:26_8"/>
<font color="Maroon" title="c25">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c24">Ny0</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c7">x0</font>,<font color="Maroon" title="c25">y</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c12">N</font></span><br/>

<a NAME="E6:26_8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A60</font></span>: 
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c25">y</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c8">y0</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">r0</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rcomp_1.html#T1" onmouseover="rs('rcomp_1/T1')" onmouseout="rh()">RCOMP_1:1</a></span>;<br/>
<a NAME="E7:26_8"/>
<font color="Maroon" title="c28">xx1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T44" onmouseover="rs('finseq_1/T44')" onmouseout="rh()">FINSEQ_1:44</a></span>;<br/>
<a NAME="E8:26_8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A61</font></span>: 
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c28">xx1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="absvalue.html#T2" onmouseover="rs('absvalue/T2')" onmouseout="rh()">ABSVALUE:2</a></span>;<br/>
<a NAME="E9:26_8"/>
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c27">xx</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c28">xx1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c28">xx1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jgraph_1.html#T31" onmouseover="rs('jgraph_1/T31')" onmouseout="rh()">JGRAPH_1:31</a></span>;<br/>
<a NAME="E10:26_8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A62</font></span>: 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c27">xx</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c25">y</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c8">y0</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:26_8"><span class="lab"><font color="Green" title="E52">A61</font></span></a>, <a class="ref" href="finseq_1.html#T44" onmouseover="rs('finseq_1/T44')" onmouseout="rh()">FINSEQ_1:44</a></span>;<br/>
<font color="Maroon" title="c26">xy</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c11">gxy0</font> = 
<span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c7">x0</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c25">y</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c8">y0</font></span>)</span></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:26"><span class="lab"><font color="Green" title="E29">A3</font></span></a>, <a class="txt" href="#E9:26"><span class="lab"><font color="Green" title="E31">A5</font></span></a>, <a class="txt" href="cfdiff_2.html#E12"><span class="lab"><font color="Green" title="E12">Lm10</font></span></a></span>
<br/>.= 
<span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<span class="p2">(<span class="default"><font color="Maroon" title="c25">y</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c8">y0</font></span>)</span></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>

;<br/>
<a NAME="E12:26_8"/><span class="kw">then </span>
<span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c26">xy</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c11">gxy0</font></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c27">xx</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="real_ns1.html#T1" onmouseover="rs('real_ns1/T1')" onmouseout="rh()">REAL_NS1:1</a></span>;<br/>
<a NAME="E13:26_8"/><span class="kw">then </span>
<span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c26">xy</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c11">gxy0</font></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">r0</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:26_8"><span class="lab"><font color="Green" title="E51">A60</font></span></a>, <a class="txt" href="#E10:26_8"><span class="lab"><font color="Green" title="E53">A62</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/>
<a NAME="E14:26_8"/><span class="kw">then </span>
<font color="Maroon" title="c26">xy</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">z</font> where <font color="Olive" title="b1">z</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> 2</span>)</span> : <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">z</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c11">gxy0</font></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">r0</font> </span> } </span> 
 
;<br/>
<span class="kw">hence </span><a NAME="E15:26_8"/>
<span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c7">x0</font>,<font color="Maroon" title="c25">y</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c12">N</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:26"><span class="lab"><font color="Green" title="E37">A11</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="26_9"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">s</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K6" title="PDIFF_1:func.6">reproj</a> (2,<font color="Maroon" title="c6">xy0</font>)</span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c24">Ny0</font> holds <br/><font color="Olive" title="b1">s</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="c3">u</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c25">s</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="c25">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K6" title="PDIFF_1:func.6">reproj</a> (2,<font color="Maroon" title="c6">xy0</font>)</span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c24">Ny0</font> implies <font color="Maroon" title="c25">s</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="c3">u</font> )</span><br/><span class="kw">assume </span><a NAME="E1:26_9"/>
<font color="Maroon" title="c25">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K6" title="PDIFF_1:func.6">reproj</a> (2,<font color="Maroon" title="c6">xy0</font>)</span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c24">Ny0</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="c25">s</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="c3">u</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c26">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <span class="kw"> such that </span><br/><a NAME="E3:26_9"/><span class="lab"><font color="Green" title="E51">A63</font></span>: 
<font color="Maroon" title="c26">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c24">Ny0</font>
 <span class="kw">and </span><br/><a NAME="E4:26_9"/><span class="lab"><font color="Green" title="E52">A64</font></span>: 
<font color="Maroon" title="c25">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K6" title="PDIFF_1:func.6">reproj</a> (2,<font color="Maroon" title="c6">xy0</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c26">y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T65" onmouseover="rs('funct_2/T65')" onmouseout="rh()">FUNCT_2:65</a></span>;<br/><a NAME="E5:26_9"/><span class="lab"><font color="Green" title="E53">A65</font></span>: 
<span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c7">x0</font>,<font color="Maroon" title="c26">y</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c12">N</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E44:26"><span class="lab"><font color="Green" title="E50">A59</font></span></a>, <a class="txt" href="#E3:26_9"><span class="lab"><font color="Green" title="E51">A63</font></span></a></span>;<br/><font color="Maroon" title="c25">s</font> = 
 <a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> (<font color="Maroon" title="c6">xy0</font>,2,<font color="Maroon" title="c26">y</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:26_9"><span class="lab"><font color="Green" title="E52">A64</font></span></a>, <a class="ref" href="pdiff_1.html#D5" onmouseover="rs('pdiff_1/D5')" onmouseout="rh()">PDIFF_1:def 5</a></span>
<br/>.= 
<span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c7">x0</font>,<font color="Maroon" title="c26">y</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:26"><span class="lab"><font color="Green" title="E29">A3</font></span></a>, <a class="ref" href="finseq_7.html#T14" onmouseover="rs('finseq_7/T14')" onmouseout="rh()">FINSEQ_7:14</a></span>
;<br/><span class="kw">hence </span><a NAME="E7:26_9"/>
<font color="Maroon" title="c25">s</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="c3">u</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:26"><span class="lab"><font color="Green" title="E30">A4</font></span></a>, <a class="txt" href="#E12:26"><span class="lab"><font color="Green" title="E33">A7</font></span></a>, <a class="txt" href="#E23:26"><span class="lab"><font color="Green" title="E40">A14</font></span></a>, <a class="txt" href="#E5:26_9"><span class="lab"><font color="Green" title="E53">A65</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E46:26"/><span class="kw">then </span>
(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K6" title="PDIFF_1:func.6">reproj</a> (2,<font color="Maroon" title="c6">xy0</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  &amp; <span class="p1">(<span class="default"><a href="pdiff_1.html#K6" title="PDIFF_1:func.6">reproj</a> (2,<font color="Maroon" title="c6">xy0</font>)</span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c24">Ny0</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="c3">u</font> )
 
<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>, <a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a></span>;<br/>
<a NAME="E47:26"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A66</font></span>: 
<font color="Maroon" title="c24">Ny0</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">u</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="pdiff_1.html#K6" title="PDIFF_1:func.6">reproj</a> (2,<font color="Maroon" title="c6">xy0</font>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_3.html#T3" onmouseover="rs('funct_3/T3')" onmouseout="rh()">FUNCT_3:3</a></span>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[   <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> ,   <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">means </span> ex <font color="Olive" title="b1">vy</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> 2 st <br/>( <font color="Olive" title="b1">vy</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,$1</span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> &amp; <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">$2</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">vy</font> );<br/>
<div><span class="lab"><font color="Green" title="E52">A67</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="26_10"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>   ex <font color="Olive" title="b2">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>]</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c25">x</font> be    <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> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c25">x</font>,<font color="Olive" title="b1">y</font>]</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c26">vx</font> = <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>)</span>)</span>,<font color="Maroon" title="c25">x</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> 2 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T101" onmouseover="rs('finseq_2/T101')" onmouseout="rh()">FINSEQ_2:101</a></span>;<br/><a NAME="E2:26_10"/>
<font color="Maroon" title="c13">R</font> is  <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ndiff_1.html#D5" onmouseover="rs('ndiff_1/D5')" onmouseout="rh()">NDIFF_1:def 5</a></span>;<br/><a NAME="E3:26_10"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A68</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c13">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> 2</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D2" onmouseover="rs('partfun1/D2')" onmouseout="rh()">PARTFUN1:def 2</a></span>;<br/><a NAME="E4:26_10"/>
<font color="Maroon" title="c26">vx</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> 2</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="real_ns1.html#D4" onmouseover="rs('real_ns1/D4')" onmouseout="rh()">REAL_NS1:def 4</a></span>;<br/><a NAME="E5:26_10"/><span class="kw">then </span>
<font color="Maroon" title="c13">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">vx</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:26_10"><span class="lab"><font color="Green" title="E53">A68</font></span></a>, <a class="ref" href="partfun1.html#T4" onmouseover="rs('partfun1/T4')" onmouseout="rh()">PARTFUN1:4</a></span>;<br/><a NAME="E6:26_10"/><span class="kw">then </span>
<font color="Maroon" title="c13">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">vx</font> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="real_ns1.html#D4" onmouseover="rs('real_ns1/D4')" onmouseout="rh()">REAL_NS1:def 4</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c27">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <span class="kw"> such that </span><br/><a NAME="E8:26_10"/><span class="lab"><font color="Green" title="E54">A69</font></span>: 
<span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c27">y</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">vx</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T97" onmouseover="rs('finseq_2/T97')" onmouseout="rh()">FINSEQ_2:97</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c28">y</font> = <font color="Maroon" title="c27">y</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">S<sub>2</sub></font>[<font color="Maroon" title="c25">x</font>,<font color="Maroon" title="c28">y</font>]</span><br/><span class="kw">thus </span><a NAME="E9:26_10"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c25">x</font>,<font color="Maroon" title="c28">y</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:26_10"><span class="lab"><font color="Green" title="E54">A69</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c25">R3</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</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="E50:26"/><span class="lab"><font color="Green" title="E53">A70</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c25">R3</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S3" onmouseover="rs('funct_2/S3')" onmouseout="rh()">FUNCT_2:sch 3</a>(<a class="txt" href="#E48:26"><span class="lab"><font color="Green" title="E52">A67</font></span></a>);<br/></span>
<div><span class="lab"><font color="Green" title="E54">A71</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="26_11"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">y</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><br/>  for <font color="Olive" title="b2">vy</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> 2  st <font color="Olive" title="b2">vy</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<font color="Olive" title="b1">y</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> holds <br/><span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c25">R3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">vy</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c26">y</font> be   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">vy</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> 2  st <font color="Olive" title="b1">vy</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<font color="Maroon" title="c26">y</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> holds <br/><span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c25">R3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">y</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">vy</font></span><br/><span class="kw">let </span><font color="Maroon" title="c27">vy</font> be    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> 2; <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="c27">vy</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<font color="Maroon" title="c26">y</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> implies <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c25">R3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">y</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">vy</font> )</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c28">yy</font> = <font color="Maroon" title="c26">y</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/><span class="kw">assume </span><a NAME="E2:26_11"/><span class="lab"><font color="Green" title="E55">A72</font></span>: 
<font color="Maroon" title="c27">vy</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<font color="Maroon" title="c26">y</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</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"> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c25">R3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">y</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">vy</font></span><br/><a NAME="E3:26_11"/>
 ex <font color="Olive" title="b1">vy1</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> 2 st <br/>( <font color="Olive" title="b1">vy1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<font color="Maroon" title="c26">y</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> &amp; <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c25">R3</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c28">yy</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">vy1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E50:26"><span class="lab"><font color="Green" title="E53">A70</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:26_11"/>
<span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c25">R3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">y</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">vy</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:26_11"><span class="lab"><font color="Green" title="E55">A72</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E52:26"/>
 for <font color="Olive" title="b1">h</font> being   <a href="ordinal1.html#V9" title="ORDINAL1:attr.9">non-zero</a>   <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="fdiff_1.html#V1" title="FDIFF_1:attr.1">-convergent</a>  <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a> holds <br/> ( <span class="p1">(<span class="default"><font color="Olive" title="b1">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">R3</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Olive" title="b1">h</font></span>)</span> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a>  &amp;  <a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c25">R3</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Olive" title="b1">h</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="26_12"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c26">h</font> be   <a href="ordinal1.html#V9" title="ORDINAL1:attr.9">non-zero</a>   <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="fdiff_1.html#V1" title="FDIFF_1:attr.1">-convergent</a>  <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><font color="Maroon" title="c26">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">R3</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c26">h</font></span>)</span> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a>  &amp;  <a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c26">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c25">R3</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c26">h</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )</span><br/>

<span class="kw">defpred </span><font color="Maroon">S<sub>3</sub></font>[  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>,  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> 2</span>)</span>] <span class="kw">means </span>$2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<span class="p2">(<span class="default"><font color="Maroon" title="c26">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E55">A73</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="26_12_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   ex <font color="Olive" title="b2">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> 2</span>)</span> st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Olive" title="b2">y</font>]</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c27">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> 2</span>)</span> st <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c27">n</font>,<font color="Olive" title="b1">y</font>]</span><br/><a NAME="E1:26_12_1"/>
<span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>)</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c26">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c27">n</font></span>)</span></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T101" onmouseover="rs('finseq_2/T101')" onmouseout="rh()">FINSEQ_2:101</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c28">y</font> = <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<span class="p2">(<span class="default"><font color="Maroon" title="c26">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c27">n</font></span>)</span></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> 2</span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="real_ns1.html#D4" onmouseover="rs('real_ns1/D4')" onmouseout="rh()">REAL_NS1:def 4</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c29">y</font> = <font color="Maroon" title="c28">y</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">S<sub>3</sub></font>[<font color="Maroon" title="c27">n</font>,<font color="Maroon" title="c29">y</font>]</span><br/><span class="kw">thus </span><a NAME="E3:26_12_1"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c27">n</font>,<font color="Maroon" title="c29">y</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">consider </span><font color="Maroon" title="c27">h1</font> being   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> 2</span>)</span><span class="kw"> such that </span><br/><a NAME="E3:26_12"/><span class="lab"><font color="Green" title="E56">A74</font></span>: 
 for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  holds  <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Maroon" title="c27">h1</font> <a href="normsp_0.html#K1" title="NORMSP_0:func.1">.</a> <font color="Olive" title="b1">n</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S3" onmouseover="rs('funct_2/S3')" onmouseout="rh()">FUNCT_2:sch 3</a>(<a class="txt" href="#E1:26_12"><span class="lab"><font color="Green" title="E55">A73</font></span></a>);<br/></span>
<a NAME="E4:26_12"/><span class="lab"><font color="Green" title="E57">A75</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Maroon" title="c27">h1</font> <a href="normsp_0.html#K1" title="NORMSP_0:func.1">.</a> <font color="Olive" title="b1">n</font>]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="26_12_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c28">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c28">n</font>,<font color="Maroon" title="c27">h1</font> <a href="normsp_0.html#K1" title="NORMSP_0:func.1">.</a> <font color="Maroon" title="c28">n</font>]</span><br/>

<a NAME="E1:26_12_2"/>
<font color="Maroon" title="c28">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" onmouseover="rs('ordinal1/D12')" onmouseout="rh()">ORDINAL1:def 12</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:26_12_2"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c28">n</font>,<font color="Maroon" title="c27">h1</font> <a href="normsp_0.html#K1" title="NORMSP_0:func.1">.</a> <font color="Maroon" title="c28">n</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:26_12"><span class="lab"><font color="Green" title="E56">A74</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">reconsider </span><font color="Maroon" title="c28">h1</font> = <font color="Maroon" title="c27">h1</font> as   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> 2</span>)</span> ;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="26_12_3"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  holds  <span class="p1"><a href="normsp_0.html#K5" title="NORMSP_0:func.5">||.</a><span class="default"><font color="Maroon" title="c28">h1</font></span><a href="normsp_0.html#K5" title="NORMSP_0:func.5">.||</a></span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c26">h</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c29">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="normsp_0.html#K5" title="NORMSP_0:func.5">||.</a><span class="default"><font color="Maroon" title="c28">h1</font></span><a href="normsp_0.html#K5" title="NORMSP_0:func.5">.||</a></span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c29">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c26">h</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c29">n</font></span><br/><a NAME="E1:26_12_3"/><span class="lab"><font color="Green" title="E58">A76</font></span>: 
<font color="Maroon" title="c28">h1</font> <a href="normsp_0.html#K1" title="NORMSP_0:func.1">.</a> <font color="Maroon" title="c29">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<span class="p2">(<span class="default"><font color="Maroon" title="c26">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c29">n</font></span>)</span></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:26_12"><span class="lab"><font color="Green" title="E57">A75</font></span></a></span>;<br/><span class="kw">thus </span><span class="p1"><a href="normsp_0.html#K5" title="NORMSP_0:func.5">||.</a><span class="default"><font color="Maroon" title="c28">h1</font></span><a href="normsp_0.html#K5" title="NORMSP_0:func.5">.||</a></span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c29">n</font> = 
<span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c28">h1</font> <a href="normsp_0.html#K1" title="NORMSP_0:func.1">.</a> <font color="Maroon" title="c29">n</font></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="normsp_0.html#D4" onmouseover="rs('normsp_0/D4')" onmouseout="rh()">NORMSP_0:def 4</a></span>
<br/>.= 
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c26">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c29">n</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26_12_3"><span class="lab"><font color="Green" title="E58">A76</font></span></a>, <a class="txt" href="cfdiff_2.html#E23"><span class="lab"><font color="Green" title="E23">Lm19</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c26">h</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c29">n</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#T18" onmouseover="rs('valued_1/T18')" onmouseout="rh()">VALUED_1:18</a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E7:26_12"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A77</font></span>: 
<span class="p1"><a href="normsp_0.html#K5" title="NORMSP_0:func.5">||.</a><span class="default"><font color="Maroon" title="c28">h1</font></span><a href="normsp_0.html#K5" title="NORMSP_0:func.5">.||</a></span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c26">h</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T63" onmouseover="rs('funct_2/T63')" onmouseout="rh()">FUNCT_2:63</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c29">g</font> = <span class="p1">(<span class="default"><span class="p2"><a href="normsp_0.html#K5" title="NORMSP_0:func.5">||.</a><span class="default"><font color="Maroon" title="c28">h1</font></span><a href="normsp_0.html#K5" title="NORMSP_0:func.5">.||</a></span> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="ndiff_1.html#K1" title="NDIFF_1:func.1">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">R</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c28">h1</font></span>)</span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="26_12_4"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  holds  <span class="p1"><a href="normsp_0.html#K5" title="NORMSP_0:func.5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="normsp_0.html#K5" title="NORMSP_0:func.5">||.</a><span class="default"><font color="Maroon" title="c28">h1</font></span><a href="normsp_0.html#K5" title="NORMSP_0:func.5">.||</a></span> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="ndiff_1.html#K1" title="NDIFF_1:func.1">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">R</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c28">h1</font></span>)</span></span>)</span></span><a href="normsp_0.html#K5" title="NORMSP_0:func.5">.||</a></span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c26">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c25">R3</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c26">h</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c30">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="normsp_0.html#K5" title="NORMSP_0:func.5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="normsp_0.html#K5" title="NORMSP_0:func.5">||.</a><span class="default"><font color="Maroon" title="c28">h1</font></span><a href="normsp_0.html#K5" title="NORMSP_0:func.5">.||</a></span> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="ndiff_1.html#K1" title="NDIFF_1:func.1">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">R</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c28">h1</font></span>)</span></span>)</span></span><a href="normsp_0.html#K5" title="NORMSP_0:func.5">.||</a></span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c26">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c25">R3</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c26">h</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">n</font></span><br/><span class="kw">reconsider </span><font color="Maroon" title="c31">v</font> = <font color="Maroon" title="c28">h1</font> <a href="normsp_0.html#K1" title="NORMSP_0:func.1">.</a> <font color="Maroon" title="c30">n</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> 2 <span class="kw">by</span> <span class="lab"><a class="ref" href="real_ns1.html#D4" onmouseover="rs('real_ns1/D4')" onmouseout="rh()">REAL_NS1:def 4</a></span>;<br/><a NAME="E2:26_12_4"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c25">R3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D2" onmouseover="rs('partfun1/D2')" onmouseout="rh()">PARTFUN1:def 2</a></span>;<br/><a NAME="E3:26_12_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A78</font></span>: 
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c26">h</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="c25">R3</font>
 ;<br/><a NAME="E4:26_12_4"/><span class="lab"><font color="Green" title="E60">A79</font></span>: 
<font color="Maroon" title="c28">h1</font> <a href="normsp_0.html#K1" title="NORMSP_0:func.1">.</a> <font color="Maroon" title="c30">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<span class="p2">(<span class="default"><font color="Maroon" title="c26">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">n</font></span>)</span></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:26_12"><span class="lab"><font color="Green" title="E57">A75</font></span></a></span>;<br/><a NAME="E5:26_12_4"/>
<font color="Maroon" title="c13">R</font> is  <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ndiff_1.html#D5" onmouseover="rs('ndiff_1/D5')" onmouseout="rh()">NDIFF_1:def 5</a></span>;<br/><a NAME="E6:26_12_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A80</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c13">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> 2</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D2" onmouseover="rs('partfun1/D2')" onmouseout="rh()">PARTFUN1:def 2</a></span>;<br/><a NAME="E7:26_12_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A81</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c28">h1</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="c13">R</font>
 ;<br/><a NAME="E8:26_12_4"/><span class="lab"><font color="Green" title="E63">A82</font></span>: 
<font color="Maroon" title="c28">h1</font> <a href="normsp_0.html#K1" title="NORMSP_0:func.1">.</a> <font color="Maroon" title="c30">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<span class="p2">(<span class="default"><font color="Maroon" title="c26">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">n</font></span>)</span></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:26_12"><span class="lab"><font color="Green" title="E57">A75</font></span></a></span>;<br/><span class="lab"><font color="Green" title="E64">A83</font></span>: <font color="Maroon" title="c13">R</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">h1</font> <a href="normsp_0.html#K1" title="NORMSP_0:func.1">.</a> <font color="Maroon" title="c30">n</font></span>)</span> = 
<font color="Maroon" title="c13">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">v</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:26_12_4"><span class="lab"><font color="Green" title="E61">A80</font></span></a>, <a class="ref" href="partfun1.html#D6" onmouseover="rs('partfun1/D6')" onmouseout="rh()">PARTFUN1:def 6</a></span>
<br/>.= 
<span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c25">R3</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c26">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">n</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E51:26"><span class="lab"><font color="Green" title="E54">A71</font></span></a>, <a class="txt" href="#E4:26_12_4"><span class="lab"><font color="Green" title="E60">A79</font></span></a></span>
;<br/><span class="kw">thus </span><span class="p1"><a href="normsp_0.html#K5" title="NORMSP_0:func.5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="normsp_0.html#K5" title="NORMSP_0:func.5">||.</a><span class="default"><font color="Maroon" title="c28">h1</font></span><a href="normsp_0.html#K5" title="NORMSP_0:func.5">.||</a></span> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="ndiff_1.html#K1" title="NDIFF_1:func.1">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">R</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c28">h1</font></span>)</span></span>)</span></span><a href="normsp_0.html#K5" title="NORMSP_0:func.5">.||</a></span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">n</font> = 
<span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5"><a href="normsp_0.html#K5" title="NORMSP_0:func.5">||.</a><span class="default"><font color="Maroon" title="c28">h1</font></span><a href="normsp_0.html#K5" title="NORMSP_0:func.5">.||</a></span> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="ndiff_1.html#K1" title="NDIFF_1:func.1">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c13">R</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c28">h1</font></span>)</span></span>)</span> <a href="normsp_0.html#K1" title="NORMSP_0:func.1">.</a> <font color="Maroon" title="c30">n</font></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="normsp_0.html#D4" onmouseover="rs('normsp_0/D4')" onmouseout="rh()">NORMSP_0:def 4</a></span>
<br/>.= 
<span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5"><a href="normsp_0.html#K5" title="NORMSP_0:func.5">||.</a><span class="default"><font color="Maroon" title="c28">h1</font></span><a href="normsp_0.html#K5" title="NORMSP_0:func.5">.||</a></span> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">n</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c13">R</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c28">h1</font></span>)</span> <a href="normsp_0.html#K1" title="NORMSP_0:func.1">.</a> <font color="Maroon" title="c30">n</font></span>)</span></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ndiff_1.html#D2" onmouseover="rs('ndiff_1/D2')" onmouseout="rh()">NDIFF_1:def 2</a></span>
<br/>.= 
<span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5"><a href="normsp_0.html#K5" title="NORMSP_0:func.5">||.</a><span class="default"><font color="Maroon" title="c28">h1</font></span><a href="normsp_0.html#K5" title="NORMSP_0:func.5">.||</a></span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">n</font></span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c13">R</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c28">h1</font></span>)</span> <a href="normsp_0.html#K1" title="NORMSP_0:func.1">.</a> <font color="Maroon" title="c30">n</font></span>)</span></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#T10" onmouseover="rs('valued_1/T10')" onmouseout="rh()">VALUED_1:10</a></span>
<br/>.= 
<span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c28">h1</font> <a href="normsp_0.html#K1" title="NORMSP_0:func.1">.</a> <font color="Maroon" title="c30">n</font></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c13">R</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c28">h1</font></span>)</span> <a href="normsp_0.html#K1" title="NORMSP_0:func.1">.</a> <font color="Maroon" title="c30">n</font></span>)</span></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="normsp_0.html#D4" onmouseover="rs('normsp_0/D4')" onmouseout="rh()">NORMSP_0:def 4</a></span>
<br/>.= 
<span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c26">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">n</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c13">R</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c28">h1</font></span>)</span> <a href="normsp_0.html#K1" title="NORMSP_0:func.1">.</a> <font color="Maroon" title="c30">n</font></span>)</span></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:26_12_4"><span class="lab"><font color="Green" title="E63">A82</font></span></a>, <a class="txt" href="cfdiff_2.html#E23"><span class="lab"><font color="Green" title="E23">Lm19</font></span></a></span>
<br/>.= 
<span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c26">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">n</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">R</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c28">h1</font> <a href="normsp_0.html#K1" title="NORMSP_0:func.1">.</a> <font color="Maroon" title="c30">n</font></span>)</span></span>)</span></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:26_12_4"><span class="lab"><font color="Green" title="E62">A81</font></span></a>, <a class="ref" href="funct_2.html#T109" onmouseover="rs('funct_2/T109')" onmouseout="rh()">FUNCT_2:109</a></span>
<br/>.= 
<span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c26">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">n</font></span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">R</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c28">h1</font> <a href="normsp_0.html#K1" title="NORMSP_0:func.1">.</a> <font color="Maroon" title="c30">n</font></span>)</span></span>)</span></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T66" onmouseover="rs('complex1/T66')" onmouseout="rh()">COMPLEX1:66</a></span>
<br/>.= 
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c26">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">n</font></span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">R</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c28">h1</font> <a href="normsp_0.html#K1" title="NORMSP_0:func.1">.</a> <font color="Maroon" title="c30">n</font></span>)</span></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="normsp_1.html#D1" onmouseover="rs('normsp_1/D1')" onmouseout="rh()">NORMSP_1:def 1</a></span>
<br/>.= 
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c26">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">n</font></span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c25">R3</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c26">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">n</font></span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:26_12_4"><span class="lab"><font color="Green" title="E64">A83</font></span></a>, <a class="txt" href="cfdiff_2.html#E24"><span class="lab"><font color="Green" title="E24">Lm20</font></span></a></span>
<br/>.= 
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c26">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">n</font></span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c25">R3</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c26">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">n</font></span>)</span></span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T65" onmouseover="rs('complex1/T65')" onmouseout="rh()">COMPLEX1:65</a></span>
<br/>.= 
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c26">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">n</font></span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c25">R3</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c26">h</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">n</font></span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:26_12_4"><span class="lab"><font color="Green" title="E59">A78</font></span></a>, <a class="ref" href="funct_2.html#T108" onmouseover="rs('funct_2/T108')" onmouseout="rh()">FUNCT_2:108</a></span>
<br/>.= 
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c26">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">n</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c25">R3</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c26">h</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">n</font></span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#T10" onmouseover="rs('valued_1/T10')" onmouseout="rh()">VALUED_1:10</a></span>
<br/>.= 
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c26">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c25">R3</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c26">h</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">n</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#T5" onmouseover="rs('valued_1/T5')" onmouseout="rh()">VALUED_1:5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c26">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c25">R3</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c26">h</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">n</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#T18" onmouseover="rs('valued_1/T18')" onmouseout="rh()">VALUED_1:18</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="E9:26_12"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A84</font></span>: 
 <a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c26">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c25">R3</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c26">h</font></span>)</span></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1"><a href="normsp_0.html#K5" title="NORMSP_0:func.5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="normsp_0.html#K5" title="NORMSP_0:func.5">||.</a><span class="default"><font color="Maroon" title="c28">h1</font></span><a href="normsp_0.html#K5" title="NORMSP_0:func.5">.||</a></span> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="ndiff_1.html#K1" title="NDIFF_1:func.1">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">R</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c28">h1</font></span>)</span></span>)</span></span><a href="normsp_0.html#K5" title="NORMSP_0:func.5">.||</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T63" onmouseover="rs('funct_2/T63')" onmouseout="rh()">FUNCT_2:63</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="26_12_5"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  holds <font color="Maroon" title="c28">h1</font> <a href="normsp_0.html#K1" title="NORMSP_0:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> 2</span>)</span></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c30">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c28">h1</font> <a href="normsp_0.html#K1" title="NORMSP_0:func.1">.</a> <font color="Maroon" title="c30">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> 2</span>)</span></span><br/><a NAME="E1:26_12_5"/><span class="lab"><font color="Green" title="E60">A85</font></span>: 
<font color="Maroon" title="c26">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="seq_1.html#T5" onmouseover="rs('seq_1/T5')" onmouseout="rh()">SEQ_1:5</a></span>;<br/><a NAME="E2:26_12_5"/><span class="lab"><font color="Green" title="E61">A86</font></span>: 
<font color="Maroon" title="c28">h1</font> <a href="normsp_0.html#K1" title="NORMSP_0:func.1">.</a> <font color="Maroon" title="c30">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<span class="p2">(<span class="default"><font color="Maroon" title="c26">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">n</font></span>)</span></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:26_12"><span class="lab"><font color="Green" title="E57">A75</font></span></a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="26_12_5_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <font color="Maroon" title="c28">h1</font> <a href="normsp_0.html#K1" title="NORMSP_0:func.1">.</a> <font color="Maroon" title="c30">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> 2</span>)</span></span></a><div class="add"><span class="kw">assume </span><a NAME="E1:26_12_5_1"/>
<font color="Maroon" title="c28">h1</font> <a href="normsp_0.html#K1" title="NORMSP_0:func.1">.</a> <font color="Maroon" title="c30">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> 2</span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">then </span><font color="Maroon" title="c28">h1</font> <a href="normsp_0.html#K1" title="NORMSP_0:func.1">.</a> <font color="Maroon" title="c30">n</font> = 
 <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> 2
<span class="kw">by</span> <span class="lab"><a class="ref" href="real_ns1.html#D4" onmouseover="rs('real_ns1/D4')" onmouseout="rh()">REAL_NS1:def 4</a></span>
<br/>.= 
<span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T54" onmouseover="rs('euclid/T54')" onmouseout="rh()">EUCLID:54</a>, <a class="ref" href="euclid.html#T70" onmouseover="rs('euclid/T70')" onmouseout="rh()">EUCLID:70</a></span>
;<br/><span class="kw">hence </span><a NAME="E3:26_12_5_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:26_12_5"><span class="lab"><font color="Green" title="E61">A86</font></span></a>, <a class="txt" href="#E1:26_12_5"><span class="lab"><font color="Green" title="E60">A85</font></span></a>, <a class="ref" href="finseq_1.html#T77" onmouseover="rs('finseq_1/T77')" onmouseout="rh()">FINSEQ_1:77</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E4:26_12_5"/>
<font color="Maroon" title="c28">h1</font> <a href="normsp_0.html#K1" title="NORMSP_0:func.1">.</a> <font color="Maroon" title="c30">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> 2</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/></div><span class="kw">end;</span></div>
<a NAME="E11:26_12"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A87</font></span>: 
<font color="Maroon" title="c28">h1</font> is  <a href="struct_0.html#V12" title="STRUCT_0:attr.12">non-zero</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ndiff_1.html#T7" onmouseover="rs('ndiff_1/T7')" onmouseout="rh()">NDIFF_1:7</a></span>;<br/>
<a NAME="E12:26_12"/>
( <font color="Maroon" title="c26">h</font> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a>  &amp;  <a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c26">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 
;<br/>
<a NAME="E13:26_12"/><span class="kw">then </span>
(  <a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c26">h</font> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a>  &amp;  <a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p1">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c26">h</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cfdiff_2.html#T3" target="_self" onmouseover="rs('cfdiff_2/T3')" onmouseout="rh()">Th3</a></span>;<br/>
<a NAME="E14:26_12"/><span class="kw">then </span>
( <font color="Maroon" title="c28">h1</font> is  <a href="normsp_1.html#V3" title="NORMSP_1:attr.3">convergent</a>  &amp;  <a href="normsp_1.html#K5" title="NORMSP_1:func.5">lim</a> <font color="Maroon" title="c28">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> 2</span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:26_12"><span class="lab"><font color="Green" title="E58">A77</font></span></a>, <a class="ref" href="cfdiff_2.html#T4" target="_self" onmouseover="rs('cfdiff_2/T4')" onmouseout="rh()">Th4</a></span>;<br/>
<a NAME="E15:26_12"/><span class="kw">then </span>
( <font color="Maroon" title="c28">h1</font> is  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> 2</span>)</span> <a href="ndiff_1.html#V1" title="NDIFF_1:attr.1">-convergent</a>  &amp; <font color="Maroon" title="c28">h1</font> is  <a href="struct_0.html#V12" title="STRUCT_0:attr.12">non-zero</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:26_12"><span class="lab"><font color="Green" title="E60">A87</font></span></a>, <a class="ref" href="ndiff_1.html#D4" onmouseover="rs('ndiff_1/D4')" onmouseout="rh()">NDIFF_1:def 4</a></span>;<br/>
<a NAME="E16:26_12"/><span class="kw">then </span>
( <span class="p1">(<span class="default"><span class="p2"><a href="normsp_0.html#K5" title="NORMSP_0:func.5">||.</a><span class="default"><font color="Maroon" title="c28">h1</font></span><a href="normsp_0.html#K5" title="NORMSP_0:func.5">.||</a></span> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="ndiff_1.html#K1" title="NDIFF_1:func.1">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">R</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c28">h1</font></span>)</span> is  <a href="normsp_1.html#V3" title="NORMSP_1:attr.3">convergent</a>  &amp;  <a href="normsp_1.html#K5" title="NORMSP_1:func.5">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="normsp_0.html#K5" title="NORMSP_0:func.5">||.</a><span class="default"><font color="Maroon" title="c28">h1</font></span><a href="normsp_0.html#K5" title="NORMSP_0:func.5">.||</a></span> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="ndiff_1.html#K1" title="NDIFF_1:func.1">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">R</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c28">h1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> 1</span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ndiff_1.html#D5" onmouseover="rs('ndiff_1/D5')" onmouseout="rh()">NDIFF_1:def 5</a></span>;<br/>
<a NAME="E17:26_12"/><span class="kw">then </span>
( <span class="p1"><a href="normsp_0.html#K5" title="NORMSP_0:func.5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="normsp_0.html#K5" title="NORMSP_0:func.5">||.</a><span class="default"><font color="Maroon" title="c28">h1</font></span><a href="normsp_0.html#K5" title="NORMSP_0:func.5">.||</a></span> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="ndiff_1.html#K1" title="NDIFF_1:func.1">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">R</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c28">h1</font></span>)</span></span>)</span></span><a href="normsp_0.html#K5" title="NORMSP_0:func.5">.||</a></span> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a>  &amp;  <a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p1"><a href="normsp_0.html#K5" title="NORMSP_0:func.5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="normsp_0.html#K5" title="NORMSP_0:func.5">||.</a><span class="default"><font color="Maroon" title="c28">h1</font></span><a href="normsp_0.html#K5" title="NORMSP_0:func.5">.||</a></span> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="ndiff_1.html#K1" title="NDIFF_1:func.1">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">R</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c28">h1</font></span>)</span></span>)</span></span><a href="normsp_0.html#K5" title="NORMSP_0:func.5">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cfdiff_2.html#T4" target="_self" onmouseover="rs('cfdiff_2/T4')" onmouseout="rh()">Th4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E18:26_12"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c26">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">R3</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c26">h</font></span>)</span> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a>  &amp;  <a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c26">h</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c25">R3</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c26">h</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:26_12"><span class="lab"><font color="Green" title="E59">A84</font></span></a>, <a class="ref" href="cfdiff_2.html#T3" target="_self" onmouseover="rs('cfdiff_2/T3')" onmouseout="rh()">Th3</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">then </span><span class="kw">reconsider </span><font color="Maroon" title="c26">R3</font> = <font color="Maroon" title="c25">R3</font> as   <a href="fdiff_1.html#NM1" title="FDIFF_1:NM.1">RestFunc</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="fdiff_1.html#D2" onmouseover="rs('fdiff_1/D2')" onmouseout="rh()">FDIFF_1:def 2</a></span>;<br/>
<a NAME="E54:26"/>
<font color="Maroon" title="c1">ey0</font> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="real_ns1.html#D4" onmouseover="rs('real_ns1/D4')" onmouseout="rh()">REAL_NS1:def 4</a></span>;<br/>
<a NAME="E55:26"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="pdiff_1.html#K8" title="PDIFF_1:func.8">diff</a> (<span class="p2">(<span class="default"><a href="pdiff_1.html#K3" title="PDIFF_1:func.3">&lt;&gt;*</a> <font color="Maroon" title="c3">u</font></span>)</span>,<font color="Maroon" title="c6">xy0</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">ey0</font> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T5" onmouseover="rs('funct_2/T5')" onmouseout="rh()">FUNCT_2:5</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c27">Duy</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <span class="kw"> such that </span><br/><a NAME="E57:26"/><span class="lab"><font color="Green" title="E55">A88</font></span>: 
<span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c27">Duy</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K8" title="PDIFF_1:func.8">diff</a> (<span class="p2">(<span class="default"><a href="pdiff_1.html#K3" title="PDIFF_1:func.3">&lt;&gt;*</a> <font color="Maroon" title="c3">u</font></span>)</span>,<font color="Maroon" title="c6">xy0</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">ey0</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T97" onmouseover="rs('finseq_2/T97')" onmouseout="rh()">FINSEQ_2:97</a></span>;<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>2</sub></font>(  <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>) <span class="kw">-&gt; </span>   <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>  =  <a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c27">Duy</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> $1</span>)</span>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>);<br/>
<span class="kw">consider </span><font color="Maroon" title="c28">LD3</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</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="E59:26"/><span class="lab"><font color="Green" title="E56">A89</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  holds  <font color="Maroon" title="c28">LD3</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">x</font>)
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S4" onmouseover="rs('funct_2/S4')" onmouseout="rh()">FUNCT_2:sch 4</a>();<br/></span>
<a NAME="E60:26"/><span class="lab"><font color="Green" title="E57">A90</font></span>: 
 for <font color="Olive" title="b1">x</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> holds  <font color="Maroon" title="c28">LD3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">Duy</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="26_13"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c29">x</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"> <font color="Maroon" title="c28">LD3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">Duy</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c29">x</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c30">x</font> = <font color="Maroon" title="c29">x</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="E2:26_13"/>
<font color="Maroon" title="c28">LD3</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c30">x</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E59:26"><span class="lab"><font color="Green" title="E56">A89</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:26_13"/>
<font color="Maroon" title="c28">LD3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">Duy</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c29">x</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">reconsider </span><font color="Maroon" title="c29">LD3</font> = <font color="Maroon" title="c28">LD3</font> as   <a href="fdiff_1.html#NM2" title="FDIFF_1:NM.2">LinearFunc</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E60:26"><span class="lab"><font color="Green" title="E57">A90</font></span></a>, <a class="ref" href="fdiff_1.html#D3" onmouseover="rs('fdiff_1/D3')" onmouseout="rh()">FDIFF_1:def 3</a></span>;<br/>
<span class="lab"><font color="Green" title="E58">A91</font></span>: <font color="Maroon" title="c29">LD3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 = 
<font color="Maroon" title="c27">Duy</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E60:26"><span class="lab"><font color="Green" title="E57">A90</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c27">Duy</font>

;<br/>
<a NAME="E63:26"/><span class="lab"><font color="Green" title="E59">A92</font></span>: 
 for <font color="Olive" title="b1">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  holds  <span class="p1">(<span class="default"><font color="Maroon" title="c3">u</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="pdiff_1.html#K6" title="PDIFF_1:func.6">reproj</a> (2,<font color="Maroon" title="c6">xy0</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">u</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c7">x0</font>,<font color="Olive" title="b1">y</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="26_15"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c30">y</font> be    <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> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c3">u</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="pdiff_1.html#K6" title="PDIFF_1:func.6">reproj</a> (2,<font color="Maroon" title="c6">xy0</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">u</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c7">x0</font>,<font color="Maroon" title="c30">y</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span></span><br/>

<a NAME="E1:26_15"/>
<font color="Maroon" title="c30">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
;<br/>
<a NAME="E2:26_15"/><span class="kw">then </span>
<font color="Maroon" title="c30">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K6" title="PDIFF_1:func.6">reproj</a> (2,<font color="Maroon" title="c6">xy0</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pdiff_1.html#D5" onmouseover="rs('pdiff_1/D5')" onmouseout="rh()">PDIFF_1:def 5</a></span>;<br/>
<span class="kw">hence </span><span class="p1">(<span class="default"><font color="Maroon" title="c3">u</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="pdiff_1.html#K6" title="PDIFF_1:func.6">reproj</a> (2,<font color="Maroon" title="c6">xy0</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">y</font> = 
<font color="Maroon" title="c3">u</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K6" title="PDIFF_1:func.6">reproj</a> (2,<font color="Maroon" title="c6">xy0</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">y</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T13" onmouseover="rs('funct_1/T13')" onmouseout="rh()">FUNCT_1:13</a></span>
<br/>.= 
<font color="Maroon" title="c3">u</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> (<font color="Maroon" title="c6">xy0</font>,2,<font color="Maroon" title="c30">y</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="pdiff_1.html#D5" onmouseover="rs('pdiff_1/D5')" onmouseout="rh()">PDIFF_1:def 5</a></span>
<br/>.= 
<font color="Maroon" title="c3">u</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c7">x0</font>,<font color="Maroon" title="c30">y</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:26"><span class="lab"><font color="Green" title="E29">A3</font></span></a>, <a class="ref" href="finseq_7.html#T14" onmouseover="rs('finseq_7/T14')" onmouseout="rh()">FINSEQ_7:14</a></span>
;<br/> <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="E64:26"/><span class="lab"><font color="Green" title="E60">A93</font></span>: 
 for <font color="Olive" title="b1">y</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c24">Ny0</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">u</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="pdiff_1.html#K6" title="PDIFF_1:func.6">reproj</a> (2,<font color="Maroon" title="c6">xy0</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">u</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="pdiff_1.html#K6" title="PDIFF_1:func.6">reproj</a> (2,<font color="Maroon" title="c6">xy0</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">y0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c29">LD3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">y</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c8">y0</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">R3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">y</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c8">y0</font></span>)</span></span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="26_16"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:26_16"/>
<font color="Maroon" title="c1">ey0</font> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="real_ns1.html#D4" onmouseover="rs('real_ns1/D4')" onmouseout="rh()">REAL_NS1:def 4</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c30">D1</font> = <span class="p1">(<span class="default"><a href="pdiff_1.html#K8" title="PDIFF_1:func.8">diff</a> (<span class="p2">(<span class="default"><a href="pdiff_1.html#K3" title="PDIFF_1:func.3">&lt;&gt;*</a> <font color="Maroon" title="c3">u</font></span>)</span>,<font color="Maroon" title="c6">xy0</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">ey0</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> 1 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T5" onmouseover="rs('funct_2/T5')" onmouseout="rh()">FUNCT_2:5</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c31">y</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"> ( <font color="Maroon" title="c31">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c24">Ny0</font> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">u</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="pdiff_1.html#K6" title="PDIFF_1:func.6">reproj</a> (2,<font color="Maroon" title="c6">xy0</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">y</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">u</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="pdiff_1.html#K6" title="PDIFF_1:func.6">reproj</a> (2,<font color="Maroon" title="c6">xy0</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">y0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c29">LD3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c31">y</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c8">y0</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">R3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c31">y</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c8">y0</font></span>)</span></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E3:26_16"/><span class="lab"><font color="Green" title="E61">A94</font></span>: 
<font color="Maroon" title="c31">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c24">Ny0</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">u</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="pdiff_1.html#K6" title="PDIFF_1:func.6">reproj</a> (2,<font color="Maroon" title="c6">xy0</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">y</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">u</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="pdiff_1.html#K6" title="PDIFF_1:func.6">reproj</a> (2,<font color="Maroon" title="c6">xy0</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">y0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c29">LD3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c31">y</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c8">y0</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">R3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c31">y</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c8">y0</font></span>)</span></span>)</span></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c32">yy</font> = <font color="Maroon" title="c31">y</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/>
<span class="kw">reconsider </span><font color="Maroon" title="c33">xy</font> = <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c7">x0</font>,<font color="Maroon" title="c32">yy</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> 2 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T101" onmouseover="rs('finseq_2/T101')" onmouseout="rh()">FINSEQ_2:101</a></span>;<br/>
<a NAME="E6:26_16"/><span class="lab"><font color="Green" title="E62">A95</font></span>: 
<font color="Maroon" title="c29">LD3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">y</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c8">y0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">Duy</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">y</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c8">y0</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E60:26"><span class="lab"><font color="Green" title="E57">A90</font></span></a></span>;<br/>
<span class="lab"><font color="Green" title="E63">A96</font></span>: <font color="Maroon" title="c33">xy</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">xy0</font> = 
<span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c7">x0</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c31">y</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c8">y0</font></span>)</span></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:26"><span class="lab"><font color="Green" title="E29">A3</font></span></a>, <a class="txt" href="cfdiff_2.html#E9"><span class="lab"><font color="Green" title="E9">Lm7</font></span></a></span>
<br/>.= 
<span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c31">y</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c8">y0</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c31">y</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c8">y0</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 1</span>)</span></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c31">y</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c8">y0</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c1">ey0</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="cfdiff_2.html#E13"><span class="lab"><font color="Green" title="E13">Lm11</font></span></a></span>
;<br/>
<a NAME="E8:26_16"/><span class="lab"><font color="Green" title="E64">A97</font></span>: 
 <a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> (<font color="Maroon" title="c15">f</font>,<font color="Maroon" title="c16">fxy0</font>) is   <a href="lopban_1.html#NM1" title="LOPBAN_1:NM.1">LinearOperator</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lopban_1.html#D9" onmouseover="rs('lopban_1/D9')" onmouseout="rh()">LOPBAN_1:def 9</a></span>;<br/>
<span class="p1">(<span class="default"><font color="Maroon" title="c31">y</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c8">y0</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c30">D1</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c31">y</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c8">y0</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> (<font color="Maroon" title="c15">f</font>,<font color="Maroon" title="c16">fxy0</font>)</span>)</span> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Maroon" title="c1">ey0</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:26"><span class="lab"><font color="Green" title="E39">A13</font></span></a>, <a class="ref" href="real_ns1.html#T3" onmouseover="rs('real_ns1/T3')" onmouseout="rh()">REAL_NS1:3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="pdiff_1.html#K8" title="PDIFF_1:func.8">diff</a> (<span class="p2">(<span class="default"><a href="pdiff_1.html#K3" title="PDIFF_1:func.3">&lt;&gt;*</a> <font color="Maroon" title="c3">u</font></span>)</span>,<font color="Maroon" title="c6">xy0</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c33">xy</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">xy0</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:26"><span class="lab"><font color="Green" title="E39">A13</font></span></a>, <a class="txt" href="#E7:26_16"><span class="lab"><font color="Green" title="E63">A96</font></span></a>, <a class="txt" href="#E8:26_16"><span class="lab"><font color="Green" title="E64">A97</font></span></a>, <a class="ref" href="lopban_1.html#D5" onmouseover="rs('lopban_1/D5')" onmouseout="rh()">LOPBAN_1:def 5</a></span>
;<br/>
<a NAME="E10:26_16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A98</font></span>: 
<span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c29">LD3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c31">y</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c8">y0</font></span>)</span></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K8" title="PDIFF_1:func.8">diff</a> (<span class="p2">(<span class="default"><a href="pdiff_1.html#K3" title="PDIFF_1:func.3">&lt;&gt;*</a> <font color="Maroon" title="c3">u</font></span>)</span>,<font color="Maroon" title="c6">xy0</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c33">xy</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">xy0</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E57:26"><span class="lab"><font color="Green" title="E55">A88</font></span></a>, <a class="txt" href="#E6:26_16"><span class="lab"><font color="Green" title="E62">A95</font></span></a>, <a class="ref" href="rvsum_1.html#T47" onmouseover="rs('rvsum_1/T47')" onmouseout="rh()">RVSUM_1:47</a></span>;<br/>
<span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<span class="p2">(<span class="default"><font color="Maroon" title="c31">y</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c8">y0</font></span>)</span></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> = 
<span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c7">x0</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c31">y</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c8">y0</font></span>)</span></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>

<br/>.= 
<font color="Maroon" title="c33">xy</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">xy0</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:26"><span class="lab"><font color="Green" title="E29">A3</font></span></a>, <a class="txt" href="cfdiff_2.html#E9"><span class="lab"><font color="Green" title="E9">Lm7</font></span></a></span>
;<br/>
<a NAME="E12:26_16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A99</font></span>: 
<span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c26">R3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c31">y</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c8">y0</font></span>)</span></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c33">xy</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">xy0</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E51:26"><span class="lab"><font color="Green" title="E54">A71</font></span></a></span>;<br/>
<span class="kw">thus </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">u</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="pdiff_1.html#K6" title="PDIFF_1:func.6">reproj</a> (2,<font color="Maroon" title="c6">xy0</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">y</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">u</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="pdiff_1.html#K6" title="PDIFF_1:func.6">reproj</a> (2,<font color="Maroon" title="c6">xy0</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">y0</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">u</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c7">x0</font>,<font color="Maroon" title="c32">yy</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">u</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="pdiff_1.html#K6" title="PDIFF_1:func.6">reproj</a> (2,<font color="Maroon" title="c6">xy0</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">y0</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E63:26"><span class="lab"><font color="Green" title="E59">A92</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">u</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c33">xy</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">u</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">xy0</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:26"><span class="lab"><font color="Green" title="E29">A3</font></span></a>, <a class="txt" href="#E63:26"><span class="lab"><font color="Green" title="E59">A92</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c29">LD3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c31">y</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c8">y0</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">R3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c31">y</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c8">y0</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:26"><span class="lab"><font color="Green" title="E41">A15</font></span></a>, <a class="txt" href="#E44:26"><span class="lab"><font color="Green" title="E50">A59</font></span></a>, <a class="txt" href="#E3:26_16"><span class="lab"><font color="Green" title="E61">A94</font></span></a>, <a class="txt" href="#E10:26_16"><span class="lab"><font color="Green" title="E65">A98</font></span></a>, <a class="txt" href="#E12:26_16"><span class="lab"><font color="Green" title="E66">A99</font></span></a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">reconsider </span><font color="Maroon" title="c30">LD1</font> = <font color="Maroon" title="c22">LD1</font> as   <a href="fdiff_1.html#NM2" title="FDIFF_1:NM.2">LinearFunc</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E42:26"><span class="lab"><font color="Green" title="E49">A58</font></span></a>, <a class="ref" href="fdiff_1.html#D3" onmouseover="rs('fdiff_1/D3')" onmouseout="rh()">FDIFF_1:def 3</a></span>;<br/>
<span class="lab"><font color="Green" title="E61">A100</font></span>: <font color="Maroon" title="c30">LD1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 = 
<font color="Maroon" title="c21">Dux</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E42:26"><span class="lab"><font color="Green" title="E49">A58</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c21">Dux</font>

;<br/>
<a NAME="E67:26"/><span class="lab"><font color="Green" title="E62">A101</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  holds  <span class="p1">(<span class="default"><font color="Maroon" title="c3">u</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="pdiff_1.html#K6" title="PDIFF_1:func.6">reproj</a> (1,<font color="Maroon" title="c6">xy0</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">u</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Maroon" title="c8">y0</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="26_18"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c31">x</font> be    <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> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c3">u</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="pdiff_1.html#K6" title="PDIFF_1:func.6">reproj</a> (1,<font color="Maroon" title="c6">xy0</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">u</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c31">x</font>,<font color="Maroon" title="c8">y0</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span></span><br/>

<a NAME="E1:26_18"/>
<font color="Maroon" title="c31">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
;<br/>
<a NAME="E2:26_18"/><span class="kw">then </span>
<font color="Maroon" title="c31">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K6" title="PDIFF_1:func.6">reproj</a> (1,<font color="Maroon" title="c6">xy0</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pdiff_1.html#D5" onmouseover="rs('pdiff_1/D5')" onmouseout="rh()">PDIFF_1:def 5</a></span>;<br/>
<span class="kw">hence </span><span class="p1">(<span class="default"><font color="Maroon" title="c3">u</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="pdiff_1.html#K6" title="PDIFF_1:func.6">reproj</a> (1,<font color="Maroon" title="c6">xy0</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">x</font> = 
<font color="Maroon" title="c3">u</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K6" title="PDIFF_1:func.6">reproj</a> (1,<font color="Maroon" title="c6">xy0</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c31">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T13" onmouseover="rs('funct_1/T13')" onmouseout="rh()">FUNCT_1:13</a></span>
<br/>.= 
<font color="Maroon" title="c3">u</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> (<font color="Maroon" title="c6">xy0</font>,1,<font color="Maroon" title="c31">x</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="pdiff_1.html#D5" onmouseover="rs('pdiff_1/D5')" onmouseout="rh()">PDIFF_1:def 5</a></span>
<br/>.= 
<font color="Maroon" title="c3">u</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c31">x</font>,<font color="Maroon" title="c8">y0</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:26"><span class="lab"><font color="Green" title="E29">A3</font></span></a>, <a class="ref" href="finseq_7.html#T13" onmouseover="rs('finseq_7/T13')" onmouseout="rh()">FINSEQ_7:13</a></span>
;<br/> <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="E68:26"/><span class="lab"><font color="Green" title="E63">A102</font></span>: 
 for <font color="Olive" title="b1">x</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">Nx0</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">u</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="pdiff_1.html#K6" title="PDIFF_1:func.6">reproj</a> (1,<font color="Maroon" title="c6">xy0</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">u</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="pdiff_1.html#K6" title="PDIFF_1:func.6">reproj</a> (1,<font color="Maroon" title="c6">xy0</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c30">LD1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">x</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c7">x0</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">R1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">x</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c7">x0</font></span>)</span></span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="26_19"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:26_19"/>
<font color="Maroon" title="c2">ex0</font> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="real_ns1.html#D4" onmouseover="rs('real_ns1/D4')" onmouseout="rh()">REAL_NS1:def 4</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c31">D1</font> = <span class="p1">(<span class="default"><a href="pdiff_1.html#K8" title="PDIFF_1:func.8">diff</a> (<span class="p2">(<span class="default"><a href="pdiff_1.html#K3" title="PDIFF_1:func.3">&lt;&gt;*</a> <font color="Maroon" title="c3">u</font></span>)</span>,<font color="Maroon" title="c6">xy0</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">ex0</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> 1 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T5" onmouseover="rs('funct_2/T5')" onmouseout="rh()">FUNCT_2:5</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c32">x</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"> ( <font color="Maroon" title="c32">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">Nx0</font> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">u</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="pdiff_1.html#K6" title="PDIFF_1:func.6">reproj</a> (1,<font color="Maroon" title="c6">xy0</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c32">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">u</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="pdiff_1.html#K6" title="PDIFF_1:func.6">reproj</a> (1,<font color="Maroon" title="c6">xy0</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c30">LD1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c32">x</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c7">x0</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">R1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c32">x</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c7">x0</font></span>)</span></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E3:26_19"/><span class="lab"><font color="Green" title="E64">A103</font></span>: 
<font color="Maroon" title="c32">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">Nx0</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">u</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="pdiff_1.html#K6" title="PDIFF_1:func.6">reproj</a> (1,<font color="Maroon" title="c6">xy0</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c32">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">u</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="pdiff_1.html#K6" title="PDIFF_1:func.6">reproj</a> (1,<font color="Maroon" title="c6">xy0</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c30">LD1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c32">x</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c7">x0</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">R1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c32">x</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c7">x0</font></span>)</span></span>)</span></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c33">xx</font> = <font color="Maroon" title="c32">x</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/>
<span class="kw">reconsider </span><font color="Maroon" title="c34">xy</font> = <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c33">xx</font>,<font color="Maroon" title="c8">y0</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> 2 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T101" onmouseover="rs('finseq_2/T101')" onmouseout="rh()">FINSEQ_2:101</a></span>;<br/>
<a NAME="E6:26_19"/><span class="lab"><font color="Green" title="E65">A104</font></span>: 
<font color="Maroon" title="c30">LD1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c32">x</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c7">x0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">Dux</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c32">x</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c7">x0</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E42:26"><span class="lab"><font color="Green" title="E49">A58</font></span></a></span>;<br/>
<span class="lab"><font color="Green" title="E66">A105</font></span>: <font color="Maroon" title="c34">xy</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">xy0</font> = 
<span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c32">x</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c7">x0</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">y0</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c8">y0</font></span>)</span></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:26"><span class="lab"><font color="Green" title="E29">A3</font></span></a>, <a class="txt" href="cfdiff_2.html#E9"><span class="lab"><font color="Green" title="E9">Lm7</font></span></a></span>
<br/>.= 
<span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c32">x</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c7">x0</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 1</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c32">x</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c7">x0</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c32">x</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c7">x0</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c2">ex0</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="cfdiff_2.html#E13"><span class="lab"><font color="Green" title="E13">Lm11</font></span></a></span>
;<br/>
<a NAME="E8:26_19"/><span class="lab"><font color="Green" title="E67">A106</font></span>: 
 <a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> (<font color="Maroon" title="c15">f</font>,<font color="Maroon" title="c16">fxy0</font>) is   <a href="lopban_1.html#NM1" title="LOPBAN_1:NM.1">LinearOperator</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lopban_1.html#D9" onmouseover="rs('lopban_1/D9')" onmouseout="rh()">LOPBAN_1:def 9</a></span>;<br/>
<span class="p1">(<span class="default"><font color="Maroon" title="c32">x</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c7">x0</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c31">D1</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c32">x</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c7">x0</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> (<font color="Maroon" title="c15">f</font>,<font color="Maroon" title="c16">fxy0</font>)</span>)</span> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Maroon" title="c2">ex0</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:26"><span class="lab"><font color="Green" title="E39">A13</font></span></a>, <a class="ref" href="real_ns1.html#T3" onmouseover="rs('real_ns1/T3')" onmouseout="rh()">REAL_NS1:3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="pdiff_1.html#K8" title="PDIFF_1:func.8">diff</a> (<span class="p2">(<span class="default"><a href="pdiff_1.html#K3" title="PDIFF_1:func.3">&lt;&gt;*</a> <font color="Maroon" title="c3">u</font></span>)</span>,<font color="Maroon" title="c6">xy0</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c34">xy</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">xy0</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:26"><span class="lab"><font color="Green" title="E39">A13</font></span></a>, <a class="txt" href="#E7:26_19"><span class="lab"><font color="Green" title="E66">A105</font></span></a>, <a class="txt" href="#E8:26_19"><span class="lab"><font color="Green" title="E67">A106</font></span></a>, <a class="ref" href="lopban_1.html#D5" onmouseover="rs('lopban_1/D5')" onmouseout="rh()">LOPBAN_1:def 5</a></span>
;<br/>
<a NAME="E10:26_19"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A107</font></span>: 
<span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c30">LD1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c32">x</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c7">x0</font></span>)</span></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K8" title="PDIFF_1:func.8">diff</a> (<span class="p2">(<span class="default"><a href="pdiff_1.html#K3" title="PDIFF_1:func.3">&lt;&gt;*</a> <font color="Maroon" title="c3">u</font></span>)</span>,<font color="Maroon" title="c6">xy0</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c34">xy</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">xy0</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:26"><span class="lab"><font color="Green" title="E47">A56</font></span></a>, <a class="txt" href="#E6:26_19"><span class="lab"><font color="Green" title="E65">A104</font></span></a>, <a class="ref" href="rvsum_1.html#T47" onmouseover="rs('rvsum_1/T47')" onmouseout="rh()">RVSUM_1:47</a></span>;<br/>
<span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c32">x</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c7">x0</font></span>)</span>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> = 
<span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c32">x</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c7">x0</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">y0</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c8">y0</font></span>)</span></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>

<br/>.= 
<font color="Maroon" title="c34">xy</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">xy0</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:26"><span class="lab"><font color="Green" title="E29">A3</font></span></a>, <a class="txt" href="cfdiff_2.html#E9"><span class="lab"><font color="Green" title="E9">Lm7</font></span></a></span>
;<br/>
<a NAME="E12:26_19"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A108</font></span>: 
<span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">R1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c32">x</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c7">x0</font></span>)</span></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c34">xy</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">xy0</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:26"><span class="lab"><font color="Green" title="E46">A39</font></span></a></span>;<br/>
<span class="kw">thus </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">u</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="pdiff_1.html#K6" title="PDIFF_1:func.6">reproj</a> (1,<font color="Maroon" title="c6">xy0</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c32">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">u</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="pdiff_1.html#K6" title="PDIFF_1:func.6">reproj</a> (1,<font color="Maroon" title="c6">xy0</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x0</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">u</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c33">xx</font>,<font color="Maroon" title="c8">y0</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">u</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="pdiff_1.html#K6" title="PDIFF_1:func.6">reproj</a> (1,<font color="Maroon" title="c6">xy0</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x0</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E67:26"><span class="lab"><font color="Green" title="E62">A101</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">u</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c34">xy</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">u</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">xy0</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:26"><span class="lab"><font color="Green" title="E29">A3</font></span></a>, <a class="txt" href="#E67:26"><span class="lab"><font color="Green" title="E62">A101</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c30">LD1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c32">x</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c7">x0</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">R1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c32">x</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c7">x0</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:26"><span class="lab"><font color="Green" title="E41">A15</font></span></a>, <a class="txt" href="#E26:26"><span class="lab"><font color="Green" title="E42">A27</font></span></a>, <a class="txt" href="#E3:26_19"><span class="lab"><font color="Green" title="E64">A103</font></span></a>, <a class="txt" href="#E10:26_19"><span class="lab"><font color="Green" title="E68">A107</font></span></a>, <a class="txt" href="#E12:26_19"><span class="lab"><font color="Green" title="E69">A108</font></span></a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="lab"><font color="Green" title="E64">A109</font></span>: <span class="p1">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> (2,2)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">xy0</font> = 
<font color="Maroon" title="c6">xy0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2
<span class="kw">by</span> <span class="lab"><a class="ref" href="pdiff_1.html#D1" onmouseover="rs('pdiff_1/D1')" onmouseout="rh()">PDIFF_1:def 1</a></span>
<br/>.= 
<font color="Maroon" title="c8">y0</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:26"><span class="lab"><font color="Green" title="E29">A3</font></span></a>, <a class="ref" href="finseq_1.html#T44" onmouseover="rs('finseq_1/T44')" onmouseout="rh()">FINSEQ_1:44</a></span>
;<br/>
<a NAME="E70:26"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A110</font></span>: 
<font color="Maroon" title="c3">u</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K6" title="PDIFF_1:func.6">reproj</a> (2,<font color="Maroon" title="c6">xy0</font>)</span>)</span> <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> (2,2)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">xy0</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E64:26"><span class="lab"><font color="Green" title="E60">A93</font></span></a>, <a class="txt" href="#E47:26"><span class="lab"><font color="Green" title="E51">A66</font></span></a>, <a class="ref" href="fdiff_1.html#D4" onmouseover="rs('fdiff_1/D4')" onmouseout="rh()">FDIFF_1:def 4</a></span>;<br/>
<span class="lab"><font color="Green" title="E66">A111</font></span>: <span class="p1">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> (1,2)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">xy0</font> = 
<font color="Maroon" title="c6">xy0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
<span class="kw">by</span> <span class="lab"><a class="ref" href="pdiff_1.html#D1" onmouseover="rs('pdiff_1/D1')" onmouseout="rh()">PDIFF_1:def 1</a></span>
<br/>.= 
<font color="Maroon" title="c7">x0</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:26"><span class="lab"><font color="Green" title="E29">A3</font></span></a>, <a class="ref" href="finseq_1.html#T44" onmouseover="rs('finseq_1/T44')" onmouseout="rh()">FINSEQ_1:44</a></span>
;<br/>
<a NAME="E72:26"/><span class="kw">then </span>
<font color="Maroon" title="c3">u</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K6" title="PDIFF_1:func.6">reproj</a> (1,<font color="Maroon" title="c6">xy0</font>)</span>)</span> <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> (1,2)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">xy0</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E68:26"><span class="lab"><font color="Green" title="E63">A102</font></span></a>, <a class="txt" href="#E29:26"><span class="lab"><font color="Green" title="E43">A34</font></span></a>, <a class="ref" href="fdiff_1.html#D4" onmouseover="rs('fdiff_1/D4')" onmouseout="rh()">FDIFF_1:def 4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E73:26"/>
( <font color="Maroon" title="c3">u</font> <a href="pdiff_1.html#R3" title="PDIFF_1:pred.3">is_partial_differentiable_in</a> <font color="Maroon" title="c6">xy0</font>,1 &amp; <font color="Maroon" title="c3">u</font> <a href="pdiff_1.html#R3" title="PDIFF_1:pred.3">is_partial_differentiable_in</a> <font color="Maroon" title="c6">xy0</font>,2 &amp; <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K10" title="PDIFF_1:func.10">partdiff</a> (<font color="Maroon" title="c3">u</font>,<font color="Maroon" title="c6">xy0</font>,1)</span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K8" title="PDIFF_1:func.8">diff</a> (<span class="p2">(<span class="default"><a href="pdiff_1.html#K3" title="PDIFF_1:func.3">&lt;&gt;*</a> <font color="Maroon" title="c3">u</font></span>)</span>,<font color="Maroon" title="c6">xy0</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default">1,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> &amp; <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K10" title="PDIFF_1:func.10">partdiff</a> (<font color="Maroon" title="c3">u</font>,<font color="Maroon" title="c6">xy0</font>,2)</span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K8" title="PDIFF_1:func.8">diff</a> (<span class="p2">(<span class="default"><a href="pdiff_1.html#K3" title="PDIFF_1:func.3">&lt;&gt;*</a> <font color="Maroon" title="c3">u</font></span>)</span>,<font color="Maroon" title="c6">xy0</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1</span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E71:26"><span class="lab"><font color="Green" title="E66">A111</font></span></a>, <a class="txt" href="#E69:26"><span class="lab"><font color="Green" title="E64">A109</font></span></a>, <a class="txt" href="#E39:26"><span class="lab"><font color="Green" title="E47">A56</font></span></a>, <a class="txt" href="#E57:26"><span class="lab"><font color="Green" title="E55">A88</font></span></a>, <a class="txt" href="#E68:26"><span class="lab"><font color="Green" title="E63">A102</font></span></a>, <a class="txt" href="#E64:26"><span class="lab"><font color="Green" title="E60">A93</font></span></a>, <a class="txt" href="#E66:26"><span class="lab"><font color="Green" title="E61">A100</font></span></a>, <a class="txt" href="#E29:26"><span class="lab"><font color="Green" title="E43">A34</font></span></a>, <a class="txt" href="#E62:26"><span class="lab"><font color="Green" title="E58">A91</font></span></a>, <a class="txt" href="#E47:26"><span class="lab"><font color="Green" title="E51">A66</font></span></a>, <a class="txt" href="#E70:26"><span class="lab"><font color="Green" title="E65">A110</font></span></a>, <a class="ref" href="fdiff_1.html#D5" onmouseover="rs('fdiff_1/D5')" onmouseout="rh()">FDIFF_1:def 5</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
