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<span class="kw">let </span><font color="Maroon" title="c1">X</font> be  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">Y</font>, <font color="Olive" title="b2">Z</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a> <br/> for <font color="Olive" title="b3">F</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>,<font color="Olive" title="b1">Y</font><br/> for <font color="Olive" title="b4">G</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>,<font color="Olive" title="b2">Z</font> holds   <a href="supinf_2.html#K9" title="SUPINF_2:func.9">rng</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">F</font> <a href="supinf_2.html#K13" title="SUPINF_2:func.13">+</a> <font color="Olive" title="b4">G</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K9" title="SUPINF_2:func.9">rng</a> <font color="Olive" title="b3">F</font></span>)</span> <a href="supinf_2.html#K5" title="SUPINF_2:func.5">+</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K9" title="SUPINF_2:func.9">rng</a> <font color="Olive" title="b4">G</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">Y</font>, <font color="Maroon" title="c3">Z</font> be  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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 <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font><br/> for <font color="Olive" title="b2">G</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c3">Z</font> holds   <a href="supinf_2.html#K9" title="SUPINF_2:func.9">rng</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">F</font> <a href="supinf_2.html#K13" title="SUPINF_2:func.13">+</a> <font color="Olive" title="b2">G</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K9" title="SUPINF_2:func.9">rng</a> <font color="Olive" title="b1">F</font></span>)</span> <a href="supinf_2.html#K5" title="SUPINF_2:func.5">+</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K9" title="SUPINF_2:func.9">rng</a> <font color="Olive" title="b2">G</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c4">F</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">G</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c3">Z</font> holds   <a href="supinf_2.html#K9" title="SUPINF_2:func.9">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="supinf_2.html#K13" title="SUPINF_2:func.13">+</a> <font color="Olive" title="b1">G</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K9" title="SUPINF_2:func.9">rng</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="supinf_2.html#K5" title="SUPINF_2:func.5">+</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K9" title="SUPINF_2:func.9">rng</a> <font color="Olive" title="b1">G</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c5">G</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c3">Z</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="supinf_2.html#K9" title="SUPINF_2:func.9">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="supinf_2.html#K13" title="SUPINF_2:func.13">+</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K9" title="SUPINF_2:func.9">rng</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="supinf_2.html#K5" title="SUPINF_2:func.5">+</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K9" title="SUPINF_2:func.9">rng</a> <font color="Maroon" title="c5">G</font></span>)</span></span><br/>







<a NAME="E1:44"/><span class="lab"><font color="Green" title="E13">A1</font></span>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="supinf_2.html#K13" title="SUPINF_2:func.13">+</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K9" title="SUPINF_2:func.9">rng</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="supinf_2.html#K5" title="SUPINF_2:func.5">+</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K9" title="SUPINF_2:func.9">rng</a> <font color="Maroon" title="c5">G</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="supinf_2.html#K13" title="SUPINF_2:func.13">+</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K9" title="SUPINF_2:func.9">rng</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="supinf_2.html#K5" title="SUPINF_2:func.5">+</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K9" title="SUPINF_2:func.9">rng</a> <font color="Maroon" title="c5">G</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:44_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="supinf_2.html#K13" title="SUPINF_2:func.13">+</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K9" title="SUPINF_2:func.9">rng</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="supinf_2.html#K5" title="SUPINF_2:func.5">+</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K9" title="SUPINF_2:func.9">rng</a> <font color="Maroon" title="c5">G</font></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">x</font> = <font color="Maroon" title="c6">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">a</font> = <font color="Maroon" title="c4">F</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c7">x</font>, <font color="Maroon" title="c9">b</font> = <font color="Maroon" title="c5">G</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c7">x</font> as   <a href="supinf_1.html#NM1" title="SUPINF_1:NM.1">R_eal</a> ;<br/>
<a NAME="E4:44_1"/><span class="lab"><font color="Green" title="E14">A2</font></span>: 
<font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_2.html#K9" title="SUPINF_2:func.9">rng</a> <font color="Maroon" title="c4">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T6" title="FUNCT_2:th.6">FUNCT_2:6</a></span>;<br/>
<a NAME="E5:44_1"/><span class="lab"><font color="Green" title="E15">A3</font></span>: 
<font color="Maroon" title="c9">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_2.html#K9" title="SUPINF_2:func.9">rng</a> <font color="Maroon" title="c5">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T6" title="FUNCT_2:th.6">FUNCT_2:6</a></span>;<br/>
<a NAME="E6:44_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="supinf_2.html#K13" title="SUPINF_2:func.13">+</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">a</font> <a href="supinf_2.html#K2" title="SUPINF_2:func.2">+</a> <font color="Maroon" title="c9">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="supinf_2.html#D9" target="_self" title="SUPINF_2:def.9">Def9</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:44_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="supinf_2.html#K13" title="SUPINF_2:func.13">+</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K9" title="SUPINF_2:func.9">rng</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="supinf_2.html#K5" title="SUPINF_2:func.5">+</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K9" title="SUPINF_2:func.9">rng</a> <font color="Maroon" title="c5">G</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:44_1"><span class="lab"><font color="Green" title="E14">A2</font></span></a>, <a class="txt" href="#E5:44_1"><span class="lab"><font color="Green" title="E15">A3</font></span></a>, <a class="ref" href="supinf_2.html#D5" target="_self" title="SUPINF_2:def.5">Def5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:44"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="supinf_2.html#K13" title="SUPINF_2:func.13">+</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E3:44"/><span class="kw">then </span>
<font color="Maroon" title="c4">F</font> <a href="supinf_2.html#K13" title="SUPINF_2:func.13">+</a> <font color="Maroon" title="c5">G</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K9" title="SUPINF_2:func.9">rng</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="supinf_2.html#K5" title="SUPINF_2:func.5">+</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K9" title="SUPINF_2:func.9">rng</a> <font color="Maroon" title="c5">G</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:44"><span class="lab"><font color="Green" title="E13">A1</font></span></a>, <a class="ref" href="funct_2.html#T5" title="FUNCT_2:th.5">FUNCT_2:5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:44"/>
 <a href="supinf_2.html#K9" title="SUPINF_2:func.9">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="supinf_2.html#K13" title="SUPINF_2:func.13">+</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K9" title="SUPINF_2:func.9">rng</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="supinf_2.html#K5" title="SUPINF_2:func.5">+</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K9" title="SUPINF_2:func.9">rng</a> <font color="Maroon" title="c5">G</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D19" title="RELAT_1:def.19">RELAT_1:def 19</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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