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<span class="kw">let </span><font color="Maroon" title="c1">m</font>, <font color="Maroon" title="c2">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">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">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">m</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c2">n</font></span>)</span><br/>  for <font color="Olive" title="b2">g</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</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="c1">m</font></span>)</span>,<span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c2">n</font></span>)</span>  st <font color="Olive" title="b1">f</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Olive" title="b2">g</font> holds <br/>( <font color="Olive" title="b1">f</font> is  <a href="pdiff_6.html#V1" title="PDIFF_6:attr.1">additive</a>  iff <font color="Olive" title="b2">g</font> is  <a href="grcat_1.html#V1" title="GRCAT_1:attr.1">additive</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">f</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">m</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c2">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">  for <font color="Olive" title="b1">g</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</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="c1">m</font></span>)</span>,<span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c2">n</font></span>)</span>  st <font color="Maroon" title="c3">f</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Olive" title="b1">g</font> holds <br/>( <font color="Maroon" title="c3">f</font> is  <a href="pdiff_6.html#V1" title="PDIFF_6:attr.1">additive</a>  iff <font color="Olive" title="b1">g</font> is  <a href="grcat_1.html#V1" title="GRCAT_1:attr.1">additive</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">g</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</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="c1">m</font></span>)</span>,<span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c2">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="c3">f</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c4">g</font> implies ( <font color="Maroon" title="c3">f</font> is  <a href="pdiff_6.html#V1" title="PDIFF_6:attr.1">additive</a>  iff <font color="Maroon" title="c4">g</font> is  <a href="grcat_1.html#V1" title="GRCAT_1:attr.1">additive</a>  ) )</span><br/>







<span class="kw">assume </span><a NAME="E1:13"/><span class="lab"><font color="Green" title="E15">A1</font></span>: 
<font color="Maroon" title="c3">f</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c4">g</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="c3">f</font> is  <a href="pdiff_6.html#V1" title="PDIFF_6:attr.1">additive</a>  iff <font color="Maroon" title="c4">g</font> is  <a href="grcat_1.html#V1" title="GRCAT_1:attr.1">additive</a>  )</span><br/>

<div><span class="kw">hereby </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="c4">g</font> is  <a href="grcat_1.html#V1" title="GRCAT_1:attr.1">additive</a>  implies <font color="Maroon" title="c3">f</font> is  <a href="pdiff_6.html#V1" title="PDIFF_6:attr.1">additive</a>  )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:13_1"/><span class="lab"><font color="Green" title="E16">A2</font></span>: 
<font color="Maroon" title="c3">f</font> is  <a href="pdiff_6.html#V1" title="PDIFF_6:attr.1">additive</a> 
 ; <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="c4">g</font> is  <a href="grcat_1.html#V1" title="GRCAT_1:attr.1">additive</a> </span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c6">y</font> be   <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="c1">m</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="c4">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c6">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">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="c4">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">y</font></span>)</span></span><br/><span class="kw">reconsider </span><font color="Maroon" title="c7">x1</font> = <font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c8">y1</font> = <font color="Maroon" title="c6">y</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">m</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/><font color="Maroon" title="c4">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c6">y</font></span>)</span> = 
<font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">x1</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c8">y1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:13"><span class="lab"><font color="Green" title="E15">A1</font></span></a>, <a class="ref" href="real_ns1.html#T2" title="REAL_NS1:th.2">REAL_NS1:2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">x1</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">y1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:13_1"><span class="lab"><font color="Green" title="E16">A2</font></span></a>, <a class="ref" href="pdiff_6.html#D1" target="_self" title="PDIFF_6:def.1">Def1</a></span>
;<br/><span class="kw">hence </span><a NAME="E3:13_1_1"/>
<font color="Maroon" title="c4">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c6">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">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="c4">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">y</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:13"><span class="lab"><font color="Green" title="E15">A1</font></span></a>, <a class="ref" href="real_ns1.html#T2" title="REAL_NS1:th.2">REAL_NS1:2</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><span class="kw">hence </span><a NAME="E3:13_1"/>
<font color="Maroon" title="c4">g</font> is  <a href="grcat_1.html#V1" title="GRCAT_1:attr.1">additive</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="grcat_1.html#D8" title="GRCAT_1:def.8">GRCAT_1:def 8</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>

<span class="kw">assume </span><a NAME="E3:13"/><span class="lab"><font color="Green" title="E16">A3</font></span>: 
<font color="Maroon" title="c4">g</font> is  <a href="grcat_1.html#V1" title="GRCAT_1:attr.1">additive</a> 
 ; <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="c3">f</font> is  <a href="pdiff_6.html#V1" title="PDIFF_6:attr.1">additive</a> </span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c6">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="c1">m</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="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c6">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">y</font></span>)</span></span><br/><span class="kw">reconsider </span><font color="Maroon" title="c7">x1</font> = <font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c8">y1</font> = <font color="Maroon" title="c6">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="c1">m</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/><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c6">y</font></span>)</span> = 
<font color="Maroon" title="c4">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">x1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c8">y1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:13"><span class="lab"><font color="Green" title="E15">A1</font></span></a>, <a class="ref" href="real_ns1.html#T2" title="REAL_NS1:th.2">REAL_NS1:2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">x1</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="c4">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">y1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:13"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="ref" href="grcat_1.html#D8" title="GRCAT_1:def.8">GRCAT_1:def 8</a></span>
;<br/><span class="kw">hence </span><a NAME="E3:13_2"/>
<font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c6">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">y</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:13"><span class="lab"><font color="Green" title="E15">A1</font></span></a>, <a class="ref" href="real_ns1.html#T2" title="REAL_NS1:th.2">REAL_NS1:2</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>
<span class="kw">hence </span><a NAME="E5:13"/>
<font color="Maroon" title="c3">f</font> is  <a href="pdiff_6.html#V1" title="PDIFF_6:attr.1">additive</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pdiff_6.html#D1" target="_self" title="PDIFF_6:def.1">Def1</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>
