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<span class="kw">let </span><font color="Maroon" title="c4">n</font> be   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>  <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">  for <font color="Olive" title="b1">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><br/>  for <font color="Olive" title="b2">g1</font>, <font color="Olive" title="b3">g2</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="c4">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="c4">n</font>  st <font color="Olive" title="b2">g1</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; <font color="Olive" title="b3">g2</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">g1</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Olive" title="b3">g2</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#K10" title="PDIFF_1:func.10">partdiff</a> (<span class="p1">(<span class="default"><font color="Olive" title="b2">g1</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Olive" title="b3">g2</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> <span class="p1">(<span class="default"><a href="pdiff_1.html#K10" title="PDIFF_1:func.10">partdiff</a> (<font color="Olive" title="b2">g1</font>,<font color="Olive" title="b4">y</font>,<font color="Olive" title="b1">i</font>)</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K10" title="PDIFF_1:func.10">partdiff</a> (<font color="Olive" title="b3">g2</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="c5">i</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">  for <font color="Olive" title="b1">g1</font>, <font color="Olive" title="b2">g2</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="c4">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="c4">n</font>  st <font color="Olive" title="b1">g1</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="c5">i</font> &amp; <font color="Olive" title="b2">g2</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="c5">i</font> holds <br/>( <font color="Olive" title="b1">g1</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Olive" title="b2">g2</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="c5">i</font> &amp;  <a href="pdiff_1.html#K10" title="PDIFF_1:func.10">partdiff</a> (<span class="p1">(<span class="default"><font color="Olive" title="b1">g1</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Olive" title="b2">g2</font></span>)</span>,<font color="Olive" title="b3">y</font>,<font color="Maroon" title="c5">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K10" title="PDIFF_1:func.10">partdiff</a> (<font color="Olive" title="b1">g1</font>,<font color="Olive" title="b3">y</font>,<font color="Maroon" title="c5">i</font>)</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K10" title="PDIFF_1:func.10">partdiff</a> (<font color="Olive" title="b2">g2</font>,<font color="Olive" title="b3">y</font>,<font color="Maroon" title="c5">i</font>)</span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c6">g1</font>, <font color="Maroon" title="c7">g2</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="c4">n</font></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">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="c4">n</font>  st <font color="Maroon" title="c6">g1</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="c5">i</font> &amp; <font color="Maroon" title="c7">g2</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="c5">i</font> holds <br/>( <font color="Maroon" title="c6">g1</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c7">g2</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="c5">i</font> &amp;  <a href="pdiff_1.html#K10" title="PDIFF_1:func.10">partdiff</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c6">g1</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c7">g2</font></span>)</span>,<font color="Olive" title="b1">y</font>,<font color="Maroon" title="c5">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K10" title="PDIFF_1:func.10">partdiff</a> (<font color="Maroon" title="c6">g1</font>,<font color="Olive" title="b1">y</font>,<font color="Maroon" title="c5">i</font>)</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K10" title="PDIFF_1:func.10">partdiff</a> (<font color="Maroon" title="c7">g2</font>,<font color="Olive" title="b1">y</font>,<font color="Maroon" title="c5">i</font>)</span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c8">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="c4">n</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">g1</font> <a href="pdiff_1.html#R3" title="PDIFF_1:pred.3">is_partial_differentiable_in</a> <font color="Maroon" title="c8">y</font>,<font color="Maroon" title="c5">i</font> &amp; <font color="Maroon" title="c7">g2</font> <a href="pdiff_1.html#R3" title="PDIFF_1:pred.3">is_partial_differentiable_in</a> <font color="Maroon" title="c8">y</font>,<font color="Maroon" title="c5">i</font> implies ( <font color="Maroon" title="c6">g1</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c7">g2</font> <a href="pdiff_1.html#R3" title="PDIFF_1:pred.3">is_partial_differentiable_in</a> <font color="Maroon" title="c8">y</font>,<font color="Maroon" title="c5">i</font> &amp;  <a href="pdiff_1.html#K10" title="PDIFF_1:func.10">partdiff</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c6">g1</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c7">g2</font></span>)</span>,<font color="Maroon" title="c8">y</font>,<font color="Maroon" title="c5">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K10" title="PDIFF_1:func.10">partdiff</a> (<font color="Maroon" title="c6">g1</font>,<font color="Maroon" title="c8">y</font>,<font color="Maroon" title="c5">i</font>)</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K10" title="PDIFF_1:func.10">partdiff</a> (<font color="Maroon" title="c7">g2</font>,<font color="Maroon" title="c8">y</font>,<font color="Maroon" title="c5">i</font>)</span>)</span> ) )</span><br/>









<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:49"/><span class="lab"><font color="Green" title="E28">A1</font></span>: 
<font color="Maroon" title="c6">g1</font> <a href="pdiff_1.html#R3" title="PDIFF_1:pred.3">is_partial_differentiable_in</a> <font color="Maroon" title="c8">y</font>,<font color="Maroon" title="c5">i</font>
 <span class="kw">and </span><br/><a NAME="E2:49"/><span class="lab"><font color="Green" title="E29">A2</font></span>: 
<font color="Maroon" title="c7">g2</font> <a href="pdiff_1.html#R3" title="PDIFF_1:pred.3">is_partial_differentiable_in</a> <font color="Maroon" title="c8">y</font>,<font color="Maroon" title="c5">i</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="c6">g1</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c7">g2</font> <a href="pdiff_1.html#R3" title="PDIFF_1:pred.3">is_partial_differentiable_in</a> <font color="Maroon" title="c8">y</font>,<font color="Maroon" title="c5">i</font> &amp;  <a href="pdiff_1.html#K10" title="PDIFF_1:func.10">partdiff</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c6">g1</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c7">g2</font></span>)</span>,<font color="Maroon" title="c8">y</font>,<font color="Maroon" title="c5">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K10" title="PDIFF_1:func.10">partdiff</a> (<font color="Maroon" title="c6">g1</font>,<font color="Maroon" title="c8">y</font>,<font color="Maroon" title="c5">i</font>)</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K10" title="PDIFF_1:func.10">partdiff</a> (<font color="Maroon" title="c7">g2</font>,<font color="Maroon" title="c8">y</font>,<font color="Maroon" title="c5">i</font>)</span>)</span> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c9">x</font> = <font color="Maroon" title="c8">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="c4">n</font></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="E4:49"/><span class="lab"><font color="Green" title="E30">A3</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" 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="c10">f1</font> =  <a href="pdiff_1.html#K3" title="PDIFF_1:func.3">&lt;&gt;*</a> <font color="Maroon" title="c6">g1</font>, <font color="Maroon" title="c11">f2</font> =  <a href="pdiff_1.html#K3" title="PDIFF_1:func.3">&lt;&gt;*</a> <font color="Maroon" title="c7">g2</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="c4">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" onmouseover="rs('real_ns1/D4')" onmouseout="rh()">REAL_NS1:def 4</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c12">One</font> = <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c3">jj</font></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="#E4:49"><span class="lab"><font color="Green" title="E30">A3</font></span></a>, <a class="ref" href="finseq_2.html#T98" onmouseover="rs('finseq_2/T98')" onmouseout="rh()">FINSEQ_2:98</a></span>;<br/>
<a NAME="E7:49"/><span class="lab"><font color="Green" title="E31">A4</font></span>: 
<font color="Maroon" title="c10">f1</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="c5">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:49"><span class="lab"><font color="Green" title="E28">A1</font></span></a>, <a class="ref" href="pdiff_1.html#T14" target="_self" onmouseover="rs('pdiff_1/T14')" onmouseout="rh()">Th14</a></span>;<br/>
<a NAME="E8:49"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A5</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">f1</font>,<font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c5">i</font>)</span>)</span> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Maroon" title="c12">One</font> <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"><a href="pdiff_1.html#K10" title="PDIFF_1:func.10">partdiff</a> (<font color="Maroon" title="c6">g1</font>,<font color="Maroon" title="c8">y</font>,<font color="Maroon" title="c5">i</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="ref" href="pdiff_1.html#T15" target="_self" onmouseover="rs('pdiff_1/T15')" onmouseout="rh()">Th15</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c13">d2</font> =  <a href="pdiff_1.html#K10" title="PDIFF_1:func.10">partdiff</a> (<font color="Maroon" title="c7">g2</font>,<font color="Maroon" title="c8">y</font>,<font color="Maroon" title="c5">i</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="c14">Pd2</font> = <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c13">d2</font></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" onmouseover="rs('finseq_2/T98')" onmouseout="rh()">FINSEQ_2:98</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c15">d1</font> =  <a href="pdiff_1.html#K10" title="PDIFF_1:func.10">partdiff</a> (<font color="Maroon" title="c6">g1</font>,<font color="Maroon" title="c8">y</font>,<font color="Maroon" title="c5">i</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="c16">Pd1</font> = <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c15">d1</font></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" onmouseover="rs('finseq_2/T98')" onmouseout="rh()">FINSEQ_2:98</a></span>;<br/>
<a NAME="E13:49"/><span class="lab"><font color="Green" title="E33">A6</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="c4">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="c4">n</font>
 
<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="E14:49"/>
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c7">g2</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" target="_self" onmouseover="rs('pdiff_1/T2')" onmouseout="rh()">Th2</a></span>;<br/>
<a NAME="E15:49"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A7</font></span>: 
 <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"><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="c7">g2</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="c7">g2</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="E16:49"/>
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c6">g1</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" target="_self" onmouseover="rs('pdiff_1/T2')" onmouseout="rh()">Th2</a></span>;<br/>
<a NAME="E17:49"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A8</font></span>: 
 <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"><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">g1</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">g1</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="E18:49"/><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="c10">f1</font> <a href="vfunct_1.html#K6" title="VFUNCT_1:func.6">+</a> <font color="Maroon" title="c11">f2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c7">g2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:49"><span class="lab"><font color="Green" title="E34">A7</font></span></a>, <a class="ref" href="vfunct_1.html#D1" onmouseover="rs('vfunct_1/D1')" onmouseout="rh()">VFUNCT_1:def 1</a></span>;<br/>
<a NAME="E19:49"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A9</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">f1</font> <a href="vfunct_1.html#K6" title="VFUNCT_1:func.6">+</a> <font color="Maroon" title="c11">f2</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="c6">g1</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c7">g2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#D1" onmouseover="rs('valued_1/D1')" onmouseout="rh()">VALUED_1:def 1</a></span>;<br/>
<a NAME="E20:49"/><span class="lab"><font color="Green" title="E37">A10</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="c6">g1</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c7">g2</font></span>)</span> <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" target="_self" onmouseover="rs('pdiff_1/T2')" onmouseout="rh()">Th2</a></span>;<br/>
<a NAME="E21:49"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A11</font></span>: 
 <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"><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="c6">g1</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c7">g2</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="c6">g1</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c7">g2</font></span>)</span>
 
<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/>
<div><span class="lab"><font color="Green" title="E39">A12</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="49_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">x</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> <font color="Maroon" title="c4">n</font></span>)</span>  st <font color="Olive" title="b1">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"><font color="Maroon" title="c10">f1</font> <a href="vfunct_1.html#K6" title="VFUNCT_1:func.6">+</a> <font color="Maroon" title="c11">f2</font></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c10">f1</font> <a href="vfunct_1.html#K6" title="VFUNCT_1:func.6">+</a> <font color="Maroon" title="c11">f2</font></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> <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="c6">g1</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c7">g2</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></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c17">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="c4">n</font></span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c17">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"><font color="Maroon" title="c10">f1</font> <a href="vfunct_1.html#K6" title="VFUNCT_1:func.6">+</a> <font color="Maroon" title="c11">f2</font></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c10">f1</font> <a href="vfunct_1.html#K6" title="VFUNCT_1:func.6">+</a> <font color="Maroon" title="c11">f2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">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="c6">g1</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c7">g2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x</font> )</span><br/><span class="kw">assume </span><a NAME="E1:49_1"/><span class="lab"><font color="Green" title="E40">A13</font></span>: 
<font color="Maroon" title="c17">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"><font color="Maroon" title="c10">f1</font> <a href="vfunct_1.html#K6" title="VFUNCT_1:func.6">+</a> <font color="Maroon" title="c11">f2</font></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c10">f1</font> <a href="vfunct_1.html#K6" title="VFUNCT_1:func.6">+</a> <font color="Maroon" title="c11">f2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">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="c6">g1</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c7">g2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x</font></span><br/><a NAME="E2:49_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c10">f1</font> <a href="vfunct_1.html#K6" title="VFUNCT_1:func.6">+</a> <font color="Maroon" title="c11">f2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">f1</font> <a href="vfunct_1.html#K6" title="VFUNCT_1:func.6">+</a> <font color="Maroon" title="c11">f2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D6" onmouseover="rs('partfun1/D6')" onmouseout="rh()">PARTFUN1:def 6</a></span>;<br/><a NAME="E3:49_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A14</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">f1</font> <a href="vfunct_1.html#K6" title="VFUNCT_1:func.6">+</a> <font color="Maroon" title="c11">f2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">f2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">x</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:49_1"><span class="lab"><font color="Green" title="E40">A13</font></span></a>, <a class="ref" href="vfunct_1.html#D1" onmouseover="rs('vfunct_1/D1')" onmouseout="rh()">VFUNCT_1:def 1</a></span>;<br/><a NAME="E4:49_1"/><span class="lab"><font color="Green" title="E42">A15</font></span>: 
<font color="Maroon" title="c17">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c10">f1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c11">f2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:49_1"><span class="lab"><font color="Green" title="E40">A13</font></span></a>, <a class="ref" href="vfunct_1.html#D1" onmouseover="rs('vfunct_1/D1')" onmouseout="rh()">VFUNCT_1:def 1</a></span>;<br/><a NAME="E5:49_1"/><span class="kw">then </span>
<font color="Maroon" title="c17">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> <font color="Maroon" title="c10">f1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/><a NAME="E6:49_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A16</font></span>: 
<font color="Maroon" title="c10">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">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">g1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D6" onmouseover="rs('partfun1/D6')" onmouseout="rh()">PARTFUN1:def 6</a></span>;<br/><a NAME="E7:49_1"/>
<font color="Maroon" title="c17">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> <font color="Maroon" title="c11">f2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:49_1"><span class="lab"><font color="Green" title="E42">A15</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/><a NAME="E8:49_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A17</font></span>: 
<font color="Maroon" title="c11">f2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">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="c7">g2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D6" onmouseover="rs('partfun1/D6')" onmouseout="rh()">PARTFUN1:def 6</a></span>;<br/><a NAME="E9:49_1"/>
<font color="Maroon" title="c17">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> <font color="Maroon" title="c7">g2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:49"><span class="lab"><font color="Green" title="E34">A7</font></span></a>, <a class="txt" href="#E4:49_1"><span class="lab"><font color="Green" title="E42">A15</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/><a NAME="E10:49_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A18</font></span>: 
<font color="Maroon" title="c11">f2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">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="c7">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:49_1"><span class="lab"><font color="Green" title="E44">A17</font></span></a>, <a class="ref" href="funct_1.html#T13" onmouseover="rs('funct_1/T13')" onmouseout="rh()">FUNCT_1:13</a></span>;<br/><a NAME="E11:49_1"/>
<font color="Maroon" title="c17">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> <font color="Maroon" title="c6">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:49"><span class="lab"><font color="Green" title="E35">A8</font></span></a>, <a class="txt" href="#E4:49_1"><span class="lab"><font color="Green" title="E42">A15</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/><a NAME="E12:49_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A19</font></span>: 
<font color="Maroon" title="c10">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">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">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:49_1"><span class="lab"><font color="Green" title="E43">A16</font></span></a>, <a class="ref" href="funct_1.html#T13" onmouseover="rs('funct_1/T13')" onmouseout="rh()">FUNCT_1:13</a></span>;<br/><a NAME="E13:49_1"/>
<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="c6">g1</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c7">g2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">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="c6">g1</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c7">g2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:49"><span class="lab"><font color="Green" title="E36">A9</font></span></a>, <a class="txt" href="#E21:49"><span class="lab"><font color="Green" title="E38">A11</font></span></a>, <a class="txt" href="#E1:49_1"><span class="lab"><font color="Green" title="E40">A13</font></span></a>, <a class="ref" href="funct_1.html#T12" onmouseover="rs('funct_1/T12')" onmouseout="rh()">FUNCT_1:12</a></span>;<br/><a NAME="E14:49_1"/><span class="kw">then </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="c6">g1</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c7">g2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">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="c6">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:49"><span class="lab"><font color="Green" title="E36">A9</font></span></a>, <a class="txt" href="#E1:49_1"><span class="lab"><font color="Green" title="E40">A13</font></span></a>, <a class="ref" href="valued_1.html#D1" onmouseover="rs('valued_1/D1')" onmouseout="rh()">VALUED_1:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E15:49_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c10">f1</font> <a href="vfunct_1.html#K6" title="VFUNCT_1:func.6">+</a> <font color="Maroon" title="c11">f2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">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="c6">g1</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c7">g2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:49_1"><span class="lab"><font color="Green" title="E41">A14</font></span></a>, <a class="txt" href="#E12:49_1"><span class="lab"><font color="Green" title="E46">A19</font></span></a>, <a class="txt" href="#E10:49_1"><span class="lab"><font color="Green" title="E45">A18</font></span></a>, <a class="ref" href="pdiff_1.html#T2" target="_self" onmouseover="rs('pdiff_1/T2')" onmouseout="rh()">Th2</a>, <a class="ref" href="pdiff_1.html#T3" target="_self" onmouseover="rs('pdiff_1/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>
<a NAME="E23:49"/><span class="lab"><font color="Green" title="E40">A20</font></span>: 
<font color="Maroon" title="c11">f2</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="c5">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:49"><span class="lab"><font color="Green" title="E29">A2</font></span></a>, <a class="ref" href="pdiff_1.html#T14" target="_self" onmouseover="rs('pdiff_1/T14')" onmouseout="rh()">Th14</a></span>;<br/>
<a NAME="E24:49"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A21</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="c11">f2</font>,<font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c5">i</font>)</span>)</span> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Maroon" title="c12">One</font> <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"><a href="pdiff_1.html#K10" title="PDIFF_1:func.10">partdiff</a> (<font color="Maroon" title="c7">g2</font>,<font color="Maroon" title="c8">y</font>,<font color="Maroon" title="c5">i</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="ref" href="pdiff_1.html#T15" target="_self" onmouseover="rs('pdiff_1/T15')" onmouseout="rh()">Th15</a></span>;<br/>
<a NAME="E25:49"/><span class="lab"><font color="Green" title="E42">A22</font></span>: 
<font color="Maroon" title="c10">f1</font> <a href="vfunct_1.html#K6" title="VFUNCT_1:func.6">+</a> <font color="Maroon" title="c11">f2</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="c5">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:49"><span class="lab"><font color="Green" title="E31">A4</font></span></a>, <a class="txt" href="#E23:49"><span class="lab"><font color="Green" title="E40">A20</font></span></a>, <a class="ref" href="pdiff_1.html#T28" target="_self" onmouseover="rs('pdiff_1/T28')" onmouseout="rh()">Th28</a></span>;<br/>
<a NAME="E26:49"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">f1</font> <a href="vfunct_1.html#K6" title="VFUNCT_1:func.6">+</a> <font color="Maroon" title="c11">f2</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="c6">g1</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c7">g2</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:49"><span class="lab"><font color="Green" title="E36">A9</font></span></a>, <a class="txt" href="#E20:49"><span class="lab"><font color="Green" title="E37">A10</font></span></a>, <a class="ref" href="relat_1.html#T27" onmouseover="rs('relat_1/T27')" onmouseout="rh()">RELAT_1:27</a></span>;<br/>
<a NAME="E27:49"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A23</font></span>: 
<font color="Maroon" title="c10">f1</font> <a href="vfunct_1.html#K6" title="VFUNCT_1:func.6">+</a> <font color="Maroon" title="c11">f2</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="c6">g1</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c7">g2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:49"><span class="lab"><font color="Green" title="E33">A6</font></span></a>, <a class="txt" href="#E4:49"><span class="lab"><font color="Green" title="E30">A3</font></span></a>, <a class="txt" href="#E22:49"><span class="lab"><font color="Green" title="E39">A12</font></span></a>, <a class="ref" href="partfun1.html#T5" onmouseover="rs('partfun1/T5')" onmouseout="rh()">PARTFUN1:5</a></span>;<br/>
<span class="kw">then </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"><a href="pdiff_1.html#K10" title="PDIFF_1:func.10">partdiff</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c6">g1</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c7">g2</font></span>)</span>,<font color="Maroon" title="c8">y</font>,<font color="Maroon" title="c5">i</font>)</span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&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="c10">f1</font> <a href="vfunct_1.html#K6" title="VFUNCT_1:func.6">+</a> <font color="Maroon" title="c11">f2</font></span>)</span>,<font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c5">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="#E7:49"><span class="lab"><font color="Green" title="E31">A4</font></span></a>, <a class="txt" href="#E23:49"><span class="lab"><font color="Green" title="E40">A20</font></span></a>, <a class="ref" href="pdiff_1.html#T15" target="_self" onmouseover="rs('pdiff_1/T15')" onmouseout="rh()">Th15</a>, <a class="ref" href="pdiff_1.html#T28" target="_self" onmouseover="rs('pdiff_1/T28')" onmouseout="rh()">Th28</a></span>
<br/>.= 
<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">f1</font>,<font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c5">i</font>)</span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><a href="pdiff_1.html#K9" title="PDIFF_1:func.9">partdiff</a> (<font color="Maroon" title="c11">f2</font>,<font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c5">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="#E7:49"><span class="lab"><font color="Green" title="E31">A4</font></span></a>, <a class="txt" href="#E23:49"><span class="lab"><font color="Green" title="E40">A20</font></span></a>, <a class="ref" href="pdiff_1.html#T28" target="_self" onmouseover="rs('pdiff_1/T28')" onmouseout="rh()">Th28</a></span>
<br/>.= 
<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">f1</font>,<font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c5">i</font>)</span>)</span> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Maroon" title="c12">One</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</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="c11">f2</font>,<font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c5">i</font>)</span>)</span> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Maroon" title="c12">One</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lopban_1.html#T35" onmouseover="rs('lopban_1/T35')" onmouseout="rh()">LOPBAN_1:35</a></span>
<br/>.= 
<font color="Maroon" title="c16">Pd1</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c14">Pd2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:49"><span class="lab"><font color="Green" title="E32">A5</font></span></a>, <a class="txt" href="#E24:49"><span class="lab"><font color="Green" title="E41">A21</font></span></a>, <a class="ref" href="real_ns1.html#T2" onmouseover="rs('real_ns1/T2')" onmouseout="rh()">REAL_NS1:2</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"><a href="pdiff_1.html#K10" title="PDIFF_1:func.10">partdiff</a> (<font color="Maroon" title="c6">g1</font>,<font color="Maroon" title="c8">y</font>,<font color="Maroon" title="c5">i</font>)</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><a href="pdiff_1.html#K10" title="PDIFF_1:func.10">partdiff</a> (<font color="Maroon" title="c7">g2</font>,<font color="Maroon" title="c8">y</font>,<font color="Maroon" title="c5">i</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/>
<a NAME="E29:49"/><span class="kw">then </span>
 <a href="pdiff_1.html#K10" title="PDIFF_1:func.10">partdiff</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c6">g1</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c7">g2</font></span>)</span>,<font color="Maroon" title="c8">y</font>,<font color="Maroon" title="c5">i</font>) <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"><span class="p3">(<span class="default"><a href="pdiff_1.html#K10" title="PDIFF_1:func.10">partdiff</a> (<font color="Maroon" title="c6">g1</font>,<font color="Maroon" title="c8">y</font>,<font color="Maroon" title="c5">i</font>)</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><a href="pdiff_1.html#K10" title="PDIFF_1:func.10">partdiff</a> (<font color="Maroon" title="c7">g2</font>,<font color="Maroon" title="c8">y</font>,<font color="Maroon" title="c5">i</font>)</span>)</span></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
 
;<br/>
<span class="kw">hence </span><a NAME="E30:49"/>
( <font color="Maroon" title="c6">g1</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c7">g2</font> <a href="pdiff_1.html#R3" title="PDIFF_1:pred.3">is_partial_differentiable_in</a> <font color="Maroon" title="c8">y</font>,<font color="Maroon" title="c5">i</font> &amp;  <a href="pdiff_1.html#K10" title="PDIFF_1:func.10">partdiff</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c6">g1</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c7">g2</font></span>)</span>,<font color="Maroon" title="c8">y</font>,<font color="Maroon" title="c5">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K10" title="PDIFF_1:func.10">partdiff</a> (<font color="Maroon" title="c6">g1</font>,<font color="Maroon" title="c8">y</font>,<font color="Maroon" title="c5">i</font>)</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K10" title="PDIFF_1:func.10">partdiff</a> (<font color="Maroon" title="c7">g2</font>,<font color="Maroon" title="c8">y</font>,<font color="Maroon" title="c5">i</font>)</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:49"><span class="lab"><font color="Green" title="E43">A23</font></span></a>, <a class="txt" href="#E25:49"><span class="lab"><font color="Green" title="E42">A22</font></span></a>, <a class="ref" href="pdiff_1.html#T14" target="_self" onmouseover="rs('pdiff_1/T14')" onmouseout="rh()">Th14</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/>


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