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<span class="kw">let </span><font color="Maroon" title="c3">n</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/>  for <font color="Olive" title="b2">r</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><br/>  for <font color="Olive" title="b3">g</font> being   <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> <font color="Maroon" title="c3">n</font></span>)</span>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a><br/>  for <font color="Olive" title="b4">y</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> <font color="Maroon" title="c3">n</font>  st <font color="Olive" title="b3">g</font> <a href="pdiff_1.html#R3" title="PDIFF_1:pred.3">is_partial_differentiable_in</a> <font color="Olive" title="b4">y</font>,<font color="Olive" title="b1">i</font> holds <br/>( <font color="Olive" title="b2">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Olive" title="b3">g</font> <a href="pdiff_1.html#R3" title="PDIFF_1:pred.3">is_partial_differentiable_in</a> <font color="Olive" title="b4">y</font>,<font color="Olive" title="b1">i</font> &amp;  <a href="pdiff_1.html#K11" title="PDIFF_1:func.11">partdiff</a> (<span class="p1">(<span class="default"><font color="Olive" title="b2">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Olive" title="b3">g</font></span>)</span>,<font color="Olive" title="b4">y</font>,<font color="Olive" title="b1">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K11" title="PDIFF_1:func.11">partdiff</a> (<font color="Olive" title="b3">g</font>,<font color="Olive" title="b4">y</font>,<font color="Olive" title="b1">i</font>)</span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">i</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">r</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><br/>  for <font color="Olive" title="b2">g</font> being   <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> <font color="Maroon" title="c3">n</font></span>)</span>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a><br/>  for <font color="Olive" title="b3">y</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> <font color="Maroon" title="c3">n</font>  st <font color="Olive" title="b2">g</font> <a href="pdiff_1.html#R3" title="PDIFF_1:pred.3">is_partial_differentiable_in</a> <font color="Olive" title="b3">y</font>,<font color="Maroon" title="c4">i</font> holds <br/>( <font color="Olive" title="b1">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Olive" title="b2">g</font> <a href="pdiff_1.html#R3" title="PDIFF_1:pred.3">is_partial_differentiable_in</a> <font color="Olive" title="b3">y</font>,<font color="Maroon" title="c4">i</font> &amp;  <a href="pdiff_1.html#K11" title="PDIFF_1:func.11">partdiff</a> (<span class="p1">(<span class="default"><font color="Olive" title="b1">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Olive" title="b2">g</font></span>)</span>,<font color="Olive" title="b3">y</font>,<font color="Maroon" title="c4">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K11" title="PDIFF_1:func.11">partdiff</a> (<font color="Olive" title="b2">g</font>,<font color="Olive" title="b3">y</font>,<font color="Maroon" title="c4">i</font>)</span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c5">r</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">g</font> being   <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> <font color="Maroon" title="c3">n</font></span>)</span>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a><br/>  for <font color="Olive" title="b2">y</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> <font color="Maroon" title="c3">n</font>  st <font color="Olive" title="b1">g</font> <a href="pdiff_1.html#R3" title="PDIFF_1:pred.3">is_partial_differentiable_in</a> <font color="Olive" title="b2">y</font>,<font color="Maroon" title="c4">i</font> holds <br/>( <font color="Maroon" title="c5">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Olive" title="b1">g</font> <a href="pdiff_1.html#R3" title="PDIFF_1:pred.3">is_partial_differentiable_in</a> <font color="Olive" title="b2">y</font>,<font color="Maroon" title="c4">i</font> &amp;  <a href="pdiff_1.html#K11" title="PDIFF_1:func.11">partdiff</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Olive" title="b1">g</font></span>)</span>,<font color="Olive" title="b2">y</font>,<font color="Maroon" title="c4">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K11" title="PDIFF_1:func.11">partdiff</a> (<font color="Olive" title="b1">g</font>,<font color="Olive" title="b2">y</font>,<font color="Maroon" title="c4">i</font>)</span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c6">g</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> <font color="Maroon" title="c3">n</font></span>)</span>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>; <a class="txt" 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="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c3">n</font>  st <font color="Maroon" title="c6">g</font> <a href="pdiff_1.html#R3" title="PDIFF_1:pred.3">is_partial_differentiable_in</a> <font color="Olive" title="b1">y</font>,<font color="Maroon" title="c4">i</font> holds <br/>( <font color="Maroon" title="c5">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c6">g</font> <a href="pdiff_1.html#R3" title="PDIFF_1:pred.3">is_partial_differentiable_in</a> <font color="Olive" title="b1">y</font>,<font color="Maroon" title="c4">i</font> &amp;  <a href="pdiff_1.html#K11" title="PDIFF_1:func.11">partdiff</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c6">g</font></span>)</span>,<font color="Olive" title="b1">y</font>,<font color="Maroon" title="c4">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K11" title="PDIFF_1:func.11">partdiff</a> (<font color="Maroon" title="c6">g</font>,<font color="Olive" title="b1">y</font>,<font color="Maroon" title="c4">i</font>)</span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c7">y</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> <font color="Maroon" title="c3">n</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">g</font> <a href="pdiff_1.html#R3" title="PDIFF_1:pred.3">is_partial_differentiable_in</a> <font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c4">i</font> implies ( <font color="Maroon" title="c5">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c6">g</font> <a href="pdiff_1.html#R3" title="PDIFF_1:pred.3">is_partial_differentiable_in</a> <font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c4">i</font> &amp;  <a href="pdiff_1.html#K11" title="PDIFF_1:func.11">partdiff</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c6">g</font></span>)</span>,<font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c4">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K11" title="PDIFF_1:func.11">partdiff</a> (<font color="Maroon" title="c6">g</font>,<font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c4">i</font>)</span>)</span> ) )</span><br/>









<a NAME="E1:57"/><span class="lab"><font color="Green" title="E39">A1</font></span>: 
 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> <font color="Maroon" title="c3">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c3">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="real_ns1.html#D4" title="REAL_NS1:def.4">REAL_NS1:def 4</a></span>;<br/>
<a NAME="E2:57"/><span class="lab"><font color="Green" title="E40">A2</font></span>: 
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c6">g</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">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" target="_self" title="PDIFF_1:th.2">Th2</a></span>;<br/>
<a NAME="E3:57"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A3</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c6">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c6">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T27" title="RELAT_1:th.27">RELAT_1:27</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">Pd</font> = <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"><a href="pdiff_1.html#K11" title="PDIFF_1:func.11">partdiff</a> (<font color="Maroon" title="c6">g</font>,<font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c4">i</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&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> 1 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T98" title="FINSEQ_2:th.98">FINSEQ_2:98</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">x</font> = <font color="Maroon" title="c7">y</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> <font color="Maroon" title="c3">n</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="real_ns1.html#D4" title="REAL_NS1:def.4">REAL_NS1:def 4</a></span>;<br/>
<a NAME="E6:57"/><span class="lab"><font color="Green" title="E42">A4</font></span>: 
 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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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" title="REAL_NS1:def.4">REAL_NS1:def 4</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">f</font> =  <a href="pdiff_1.html#K3" title="PDIFF_1:func.3">&lt;&gt;*</a> <font color="Maroon" title="c6">g</font> as   <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> <font color="Maroon" title="c3">n</font></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="real_ns1.html#D4" title="REAL_NS1:def.4">REAL_NS1:def 4</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">One</font> = <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> as   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</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="#E6:57"><span class="lab"><font color="Green" title="E42">A4</font></span></a>, <a class="ref" href="finseq_2.html#T98" title="FINSEQ_2:th.98">FINSEQ_2:98</a></span>;<br/>
<span class="kw">assume </span><a NAME="E9:57"/>
<font color="Maroon" title="c6">g</font> <a href="pdiff_1.html#R3" title="PDIFF_1:pred.3">is_partial_differentiable_in</a> <font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c4">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c6">g</font> <a href="pdiff_1.html#R3" title="PDIFF_1:pred.3">is_partial_differentiable_in</a> <font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c4">i</font> &amp;  <a href="pdiff_1.html#K11" title="PDIFF_1:func.11">partdiff</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c6">g</font></span>)</span>,<font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c4">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K11" title="PDIFF_1:func.11">partdiff</a> (<font color="Maroon" title="c6">g</font>,<font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c4">i</font>)</span>)</span> )</span><br/>

<a NAME="E10:57"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A5</font></span>: 
<font color="Maroon" title="c10">f</font> <a href="pdiff_1.html#R2" title="PDIFF_1:pred.2">is_partial_differentiable_in</a> <font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c4">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pdiff_1.html#T14" target="_self" title="PDIFF_1:th.14">Th14</a></span>;<br/>
<a NAME="E11:57"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A6</font></span>: 
<font color="Maroon" title="c5">r</font> <a href="vfunct_1.html#K4" title="VFUNCT_1:func.4">(#)</a> <font color="Maroon" title="c10">f</font> <a href="pdiff_1.html#R2" title="PDIFF_1:pred.2">is_partial_differentiable_in</a> <font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c4">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pdiff_1.html#T32" target="_self" title="PDIFF_1:th.32">Th32</a></span>;<br/>
<a NAME="E12:57"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="vfunct_1.html#K4" title="VFUNCT_1:func.4">(#)</a> <font color="Maroon" title="c10">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c6">g</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vfunct_1.html#D4" title="VFUNCT_1:def.4">VFUNCT_1:def 4</a></span>;<br/>
<a NAME="E13:57"/><span class="kw">then </span>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="vfunct_1.html#K4" title="VFUNCT_1:func.4">(#)</a> <font color="Maroon" title="c10">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c6">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:57"><span class="lab"><font color="Green" title="E40">A2</font></span></a>, <a class="ref" href="relat_1.html#T27" title="RELAT_1:th.27">RELAT_1:27</a></span>;<br/>
<a NAME="E14:57"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A7</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="vfunct_1.html#K4" title="VFUNCT_1:func.4">(#)</a> <font color="Maroon" title="c10">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c6">g</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#D5" title="VALUED_1:def.5">VALUED_1:def 5</a></span>;<br/>
<a NAME="E15:57"/><span class="lab"><font color="Green" title="E46">A8</font></span>: 
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c6">g</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">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" target="_self" title="PDIFF_1:th.2">Th2</a></span>;<br/>
<a NAME="E16:57"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A9</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c6">g</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c6">g</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T27" title="RELAT_1:th.27">RELAT_1:27</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E48">A10</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c12">x</font> be   <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> <font color="Maroon" title="c3">n</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="vfunct_1.html#K4" title="VFUNCT_1:func.4">(#)</a> <font color="Maroon" title="c10">f</font></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="vfunct_1.html#K4" title="VFUNCT_1:func.4">(#)</a> <font color="Maroon" title="c10">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K3" title="PDIFF_1:func.3">&lt;&gt;*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c6">g</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font> )</span><br/><span class="kw">consider </span><font color="Maroon" title="c13">I</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>,<span class="p1">(<span class="default"><a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> 1</span>)</span><span class="kw"> such that </span><br/><a NAME="E2:57_1"/>
<font color="Maroon" title="c13">I</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a> 
 <span class="kw">and </span><br/><a NAME="E3:57_1"/><span class="lab"><font color="Green" title="E49">A11</font></span>: 
<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>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">I</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pdiff_1.html#T2" target="_self" title="PDIFF_1:th.2">Th2</a></span>;<br/><span class="kw">assume </span><a NAME="E4:57_1"/><span class="lab"><font color="Green" title="E50">A12</font></span>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="vfunct_1.html#K4" title="VFUNCT_1:func.4">(#)</a> <font color="Maroon" title="c10">f</font></span>)</span>
 ; <a class="txt" 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="c5">r</font> <a href="vfunct_1.html#K4" title="VFUNCT_1:func.4">(#)</a> <font color="Maroon" title="c10">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K3" title="PDIFF_1:func.3">&lt;&gt;*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c6">g</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font></span><br/><a NAME="E5:57_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A13</font></span>: 
<span class="p1">(<span class="default"><a href="pdiff_1.html#K3" title="PDIFF_1:func.3">&lt;&gt;*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c6">g</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="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> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c6">g</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">x</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:57"><span class="lab"><font color="Green" title="E45">A7</font></span></a>, <a class="txt" href="#E16:57"><span class="lab"><font color="Green" title="E47">A9</font></span></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>;<br/><a NAME="E6:57_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="vfunct_1.html#K4" title="VFUNCT_1:func.4">(#)</a> <font color="Maroon" title="c10">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="vfunct_1.html#K4" title="VFUNCT_1:func.4">(#)</a> <font color="Maroon" title="c10">f</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:57_1"><span class="lab"><font color="Green" title="E50">A12</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>;<br/><a NAME="E7:57_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A14</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="vfunct_1.html#K4" title="VFUNCT_1:func.4">(#)</a> <font color="Maroon" title="c10">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">x</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:57_1"><span class="lab"><font color="Green" title="E50">A12</font></span></a>, <a class="ref" href="vfunct_1.html#D4" title="VFUNCT_1:def.4">VFUNCT_1:def 4</a></span>;<br/><a NAME="E8:57_1"/><span class="lab"><font color="Green" title="E53">A15</font></span>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c6">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:57"><span class="lab"><font color="Green" title="E41">A3</font></span></a>, <a class="txt" href="#E4:57_1"><span class="lab"><font color="Green" title="E50">A12</font></span></a>, <a class="ref" href="vfunct_1.html#D4" title="VFUNCT_1:def.4">VFUNCT_1:def 4</a></span>;<br/><a NAME="E9:57_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A16</font></span>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c6">g</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#D5" title="VALUED_1:def.5">VALUED_1:def 5</a></span>;<br/><a NAME="E10:57_1"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c10">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:57_1"><span class="lab"><font color="Green" title="E50">A12</font></span></a>, <a class="ref" href="vfunct_1.html#D4" title="VFUNCT_1:def.4">VFUNCT_1:def 4</a></span>;<br/><a NAME="E11:57_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c6">g</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>;<br/><a NAME="E12:57_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="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> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">x</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:57_1"><span class="lab"><font color="Green" title="E53">A15</font></span></a>, <a class="ref" href="funct_1.html#T13" title="FUNCT_1:th.13">FUNCT_1:13</a></span>;<br/><a NAME="E13:57_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">x</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:57_1"><span class="lab"><font color="Green" title="E49">A11</font></span></a>, <a class="ref" href="pdiff_1.html#T3" target="_self" title="PDIFF_1:th.3">Th3</a></span>;<br/><span class="kw">hence </span><a NAME="E14:57_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="vfunct_1.html#K4" title="VFUNCT_1:func.4">(#)</a> <font color="Maroon" title="c10">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K3" title="PDIFF_1:func.3">&lt;&gt;*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c6">g</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:57_1"><span class="lab"><font color="Green" title="E54">A16</font></span></a>, <a class="txt" href="#E7:57_1"><span class="lab"><font color="Green" title="E52">A14</font></span></a>, <a class="txt" href="#E5:57_1"><span class="lab"><font color="Green" title="E51">A13</font></span></a>, <a class="txt" href="#E3:57_1"><span class="lab"><font color="Green" title="E49">A11</font></span></a>, <a class="ref" href="valued_1.html#D5" title="VALUED_1:def.5">VALUED_1:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E18:57"/><span class="lab"><font color="Green" title="E49">A17</font></span>: 
<span class="p1">(<span class="default"><a href="pdiff_1.html#K9" title="PDIFF_1:func.9">partdiff</a> (<font color="Maroon" title="c10">f</font>,<font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c4">i</font>)</span>)</span> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Maroon" title="c11">One</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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"><a href="pdiff_1.html#K11" title="PDIFF_1:func.11">partdiff</a> (<font color="Maroon" title="c6">g</font>,<font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c4">i</font>)</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="#E10:57"><span class="lab"><font color="Green" title="E43">A5</font></span></a>, <a class="ref" href="pdiff_1.html#T15" target="_self" title="PDIFF_1:th.15">Th15</a></span>;<br/>
<a NAME="E19:57"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="vfunct_1.html#K4" title="VFUNCT_1:func.4">(#)</a> <font color="Maroon" title="c10">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K3" title="PDIFF_1:func.3">&lt;&gt;*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c6">g</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:57"><span class="lab"><font color="Green" title="E45">A7</font></span></a>, <a class="txt" href="#E15:57"><span class="lab"><font color="Green" title="E46">A8</font></span></a>, <a class="ref" href="relat_1.html#T27" title="RELAT_1:th.27">RELAT_1:27</a></span>;<br/>
<a NAME="E20:57"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A18</font></span>: 
<font color="Maroon" title="c5">r</font> <a href="vfunct_1.html#K4" title="VFUNCT_1:func.4">(#)</a> <font color="Maroon" title="c10">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pdiff_1.html#K3" title="PDIFF_1:func.3">&lt;&gt;*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c6">g</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:57"><span class="lab"><font color="Green" title="E39">A1</font></span></a>, <a class="txt" href="#E6:57"><span class="lab"><font color="Green" title="E42">A4</font></span></a>, <a class="txt" href="#E17:57"><span class="lab"><font color="Green" title="E48">A10</font></span></a>, <a class="ref" href="partfun1.html#T5" title="PARTFUN1:th.5">PARTFUN1:5</a></span>;<br/>
<span class="kw">then </span><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"><a href="pdiff_1.html#K11" title="PDIFF_1:func.11">partdiff</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c6">g</font></span>)</span>,<font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c4">i</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> = 
<span class="p1">(<span class="default"><a href="pdiff_1.html#K9" title="PDIFF_1:func.9">partdiff</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="vfunct_1.html#K4" title="VFUNCT_1:func.4">(#)</a> <font color="Maroon" title="c10">f</font></span>)</span>,<font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c4">i</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">1</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="#E10:57"><span class="lab"><font color="Green" title="E43">A5</font></span></a>, <a class="ref" href="pdiff_1.html#T15" target="_self" title="PDIFF_1:th.15">Th15</a>, <a class="ref" href="pdiff_1.html#T32" target="_self" title="PDIFF_1:th.32">Th32</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><a href="pdiff_1.html#K9" title="PDIFF_1:func.9">partdiff</a> (<font color="Maroon" title="c10">f</font>,<font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c4">i</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">1</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="#E10:57"><span class="lab"><font color="Green" title="E43">A5</font></span></a>, <a class="ref" href="pdiff_1.html#T32" target="_self" title="PDIFF_1:th.32">Th32</a></span>
<br/>.= 
<font color="Maroon" title="c5">r</font> <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="pdiff_1.html#K9" title="PDIFF_1:func.9">partdiff</a> (<font color="Maroon" title="c10">f</font>,<font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c4">i</font>)</span>)</span> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Maroon" title="c11">One</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lopban_1.html#T36" title="LOPBAN_1:th.36">LOPBAN_1:36</a></span>
<br/>.= 
<font color="Maroon" title="c5">r</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c8">Pd</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:57"><span class="lab"><font color="Green" title="E49">A17</font></span></a>, <a class="ref" href="real_ns1.html#T3" title="REAL_NS1:th.3">REAL_NS1:3</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="c5">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><a href="pdiff_1.html#K11" title="PDIFF_1:func.11">partdiff</a> (<font color="Maroon" title="c6">g</font>,<font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c4">i</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="ref" href="rvsum_1.html#T47" title="RVSUM_1:th.47">RVSUM_1:47</a></span>
;<br/>
<a NAME="E22:57"/><span class="kw">then </span>
 <a href="pdiff_1.html#K11" title="PDIFF_1:func.11">partdiff</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c6">g</font></span>)</span>,<font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c4">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="c5">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><a href="pdiff_1.html#K11" title="PDIFF_1:func.11">partdiff</a> (<font color="Maroon" title="c6">g</font>,<font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c4">i</font>)</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T40" title="FINSEQ_1:th.40">FINSEQ_1:40</a></span>;<br/>
<span class="kw">hence </span><a NAME="E23:57"/>
( <font color="Maroon" title="c5">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c6">g</font> <a href="pdiff_1.html#R3" title="PDIFF_1:pred.3">is_partial_differentiable_in</a> <font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c4">i</font> &amp;  <a href="pdiff_1.html#K11" title="PDIFF_1:func.11">partdiff</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c6">g</font></span>)</span>,<font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c4">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K11" title="PDIFF_1:func.11">partdiff</a> (<font color="Maroon" title="c6">g</font>,<font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c4">i</font>)</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:57"><span class="lab"><font color="Green" title="E50">A18</font></span></a>, <a class="txt" href="#E11:57"><span class="lab"><font color="Green" title="E44">A6</font></span></a>, <a class="ref" href="pdiff_1.html#T14" target="_self" title="PDIFF_1:th.14">Th14</a>, <a class="ref" href="finseq_1.html#T40" title="FINSEQ_1:th.40">FINSEQ_1:40</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
