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<span class="kw">let </span><font color="Maroon" title="c1">n</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">f1</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>, 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="c1">n</font></span>)</span><br/>  for <font color="Olive" title="b2">g1</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>,<span class="p1">(<span class="default"><a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>  st <font color="Olive" title="b1">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g1</font> holds <br/> <a href="vfunct_1.html#K5" title="VFUNCT_1:func.5">-</a> <font color="Olive" title="b1">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="nfcont_4.html#K2" title="NFCONT_4:func.2">-</a> <font color="Olive" title="b2">g1</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">f1</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">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">g1</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>,<span class="p1">(<span class="default"><a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>  st <font color="Maroon" title="c2">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g1</font> holds <br/> <a href="vfunct_1.html#K5" title="VFUNCT_1:func.5">-</a> <font color="Maroon" title="c2">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="nfcont_4.html#K2" title="NFCONT_4:func.2">-</a> <font color="Olive" title="b1">g1</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">g1</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</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> <font color="Maroon" title="c1">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="c2">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g1</font> implies  <a href="vfunct_1.html#K5" title="VFUNCT_1:func.5">-</a> <font color="Maroon" title="c2">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="nfcont_4.html#K2" title="NFCONT_4:func.2">-</a> <font color="Maroon" title="c3">g1</font> )</span><br/>





<span class="kw">assume </span><a NAME="E1:11"/><span class="lab"><font color="Green" title="E11">AS</font></span>: 
<font color="Maroon" title="c2">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="vfunct_1.html#K5" title="VFUNCT_1:func.5">-</a> <font color="Maroon" title="c2">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="nfcont_4.html#K2" title="NFCONT_4:func.2">-</a> <font color="Maroon" title="c3">g1</font></span><br/>

<a NAME="E2:11"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="vfunct_1.html#K5" title="VFUNCT_1:func.5">-</a> <font color="Maroon" title="c2">f1</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="c2">f1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vfunct_1.html#D5" title="VFUNCT_1:def.5">VFUNCT_1:def 5</a></span>;<br/>
<a NAME="E3:11"/><span class="kw">then </span><span class="lab"><font color="Green" title="E12">A2</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="vfunct_1.html#K5" title="VFUNCT_1:func.5">-</a> <font color="Maroon" title="c2">f1</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="nfcont_4.html#K2" title="NFCONT_4:func.2">-</a> <font color="Maroon" title="c3">g1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:11"><span class="lab"><font color="Green" title="E11">AS</font></span></a>, <a class="ref" href="nfcont_4.html#D3" target="_self" title="NFCONT_4:def.3">Def5</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E13">B1</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">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"><a href="vfunct_1.html#K5" title="VFUNCT_1:func.5">-</a> <font color="Maroon" title="c2">f1</font></span>)</span> implies <span class="p1">(<span class="default"><a href="vfunct_1.html#K5" title="VFUNCT_1:func.5">-</a> <font color="Maroon" title="c2">f1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="nfcont_4.html#K2" title="NFCONT_4:func.2">-</a> <font color="Maroon" title="c3">g1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> )</span><br/><span class="kw">assume </span><a NAME="E1:11_1"/><span class="lab"><font color="Green" title="E14">A3</font></span>: 
<font color="Maroon" title="c4">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"><a href="vfunct_1.html#K5" title="VFUNCT_1:func.5">-</a> <font color="Maroon" title="c2">f1</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"><a href="vfunct_1.html#K5" title="VFUNCT_1:func.5">-</a> <font color="Maroon" title="c2">f1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="nfcont_4.html#K2" title="NFCONT_4:func.2">-</a> <font color="Maroon" title="c3">g1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font></span><br/><a NAME="E2:11_1"/>
<font color="Maroon" title="c2">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:11"><span class="lab"><font color="Green" title="E11">AS</font></span></a>, <a class="ref" href="real_ns1.html#D4" title="REAL_NS1:def.4">REAL_NS1:def 4</a></span>;<br/><a NAME="E3:11_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E15">A5</font></span>: 
 <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid.html#K6" title="EUCLID:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="real_ns1.html#T4" title="REAL_NS1:th.4">REAL_NS1:4</a></span>;<br/><a NAME="E4:11_1"/><span class="lab"><font color="Green" title="E16">A6</font></span>: 
<span class="p1">(<span class="default"><a href="vfunct_1.html#K5" title="VFUNCT_1:func.5">-</a> <font color="Maroon" title="c2">f1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:11_1"><span class="lab"><font color="Green" title="E14">A3</font></span></a>, <a class="ref" href="vfunct_1.html#D5" title="VFUNCT_1:def.5">VFUNCT_1:def 5</a></span>;<br/><a NAME="E5:11_1"/>
<span class="p1">(<span class="default"><a href="nfcont_4.html#K2" title="NFCONT_4:func.2">-</a> <font color="Maroon" title="c3">g1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid.html#K6" title="EUCLID:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:11"><span class="lab"><font color="Green" title="E12">A2</font></span></a>, <a class="txt" href="#E1:11_1"><span class="lab"><font color="Green" title="E14">A3</font></span></a>, <a class="ref" href="nfcont_4.html#D3" target="_self" title="NFCONT_4:def.3">Def5</a></span>;<br/><a NAME="E6:11_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="vfunct_1.html#K5" title="VFUNCT_1:func.5">-</a> <font color="Maroon" title="c2">f1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="nfcont_4.html#K2" title="NFCONT_4:func.2">-</a> <font color="Maroon" title="c3">g1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:11_1"><span class="lab"><font color="Green" title="E14">A3</font></span></a>, <a class="txt" href="#E3:11_1"><span class="lab"><font color="Green" title="E15">A5</font></span></a>, <a class="txt" href="#E4:11_1"><span class="lab"><font color="Green" title="E16">A6</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>;<br/><span class="kw">hence </span><a NAME="E7:11_1"/>
<span class="p1">(<span class="default"><a href="vfunct_1.html#K5" title="VFUNCT_1:func.5">-</a> <font color="Maroon" title="c2">f1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="nfcont_4.html#K2" title="NFCONT_4:func.2">-</a> <font color="Maroon" title="c3">g1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:11"><span class="lab"><font color="Green" title="E12">A2</font></span></a>, <a class="txt" href="#E1:11_1"><span class="lab"><font color="Green" title="E14">A3</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</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="E5:11"/>
 <a href="vfunct_1.html#K5" title="VFUNCT_1:func.5">-</a> <font color="Maroon" title="c2">f1</font> is   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</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> <font color="Maroon" title="c1">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/>
<span class="kw">hence </span><a NAME="E6:11"/>
 <a href="vfunct_1.html#K5" title="VFUNCT_1:func.5">-</a> <font color="Maroon" title="c2">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="nfcont_4.html#K2" title="NFCONT_4:func.2">-</a> <font color="Maroon" title="c3">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:11"><span class="lab"><font color="Green" title="E12">A2</font></span></a>, <a class="txt" href="#E4:11"><span class="lab"><font color="Green" title="E13">B1</font></span></a>, <a class="ref" href="partfun1.html#T5" title="PARTFUN1:th.5">PARTFUN1:5</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/>


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