:: ARYTM_0 semantic presentation

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Lm1: {} in {{} }
by TARSKI:def 1;

Lm2: now

let e be set ; :: thesis:
assume A1: one = [0,e] ; :: thesis:
{0} = 0 \/ {0}
.= succ 0 by ORDINAL1:def 1
.= {{0},{0,e}} by A1, ORDINAL2:def 4, TARSKI:def 5 ;
then {0} in {0} by TARSKI:def 2;
hence contradiction ; :: thesis:

end;

theorem Th1: :: ARYTM_0:1

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proof end;

theorem Th2: :: ARYTM_0:2

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for x being Element of REAL+ st x <> {} holds
[{} ,x] in REAL

proof end;

theorem Th3: :: ARYTM_0:3

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for y being set st [{} ,y] in REAL holds
y <> {}

proof end;

theorem Th4: :: ARYTM_0:4

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for x, y being Element of REAL+ holds x - y in REAL

proof end;

theorem Th5: :: ARYTM_0:5

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REAL+ misses [:{{} },REAL+ :]

proof end;

theorem Th6: :: ARYTM_0:6

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for x, y being Element of REAL+ st x - y = {} holds
x = y

proof end;

theorem Th7: :: ARYTM_0:7

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for a, b being set holds not one = [a,b]

proof end;

theorem Th8: :: ARYTM_0:8

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for x, y, z being Element of REAL+ st x <> {} & x *' y = x *' z holds
y = z

proof end;

definition

let x, y be Element of REAL ;
canceled;
func + x,y -> Element of REAL means :Def2: :: ARYTM_0:def 2
ex x', y' being Element of REAL+ st
( x = x' & y = y' & it = x' + y' ) if ( x in REAL+ & y in REAL+ )
ex x', y' being Element of REAL+ st
( x = x' & y = [0,y'] & it = x' - y' ) if ( x in REAL+ & y in [:{0},REAL+ :] )
ex x', y' being Element of REAL+ st
( x = [0,x'] & y = y' & it = y' - x' ) if ( y in REAL+ & x in [:{0},REAL+ :] )
otherwise ex x', y' being Element of REAL+ st
( x = [0,x'] & y = [0,y'] & it = [0,(x' + y')] );
existence

proof end;

proof end;

consistency by Th5, XBOOLE_0:3;
commutativity ;
func * x,y -> Element of REAL means :Def3: :: ARYTM_0:def 3
ex x', y' being Element of REAL+ st
( x = x' & y = y' & it = x' *' y' ) if ( x in REAL+ & y in REAL+ )
ex x', y' being Element of REAL+ st
( x = x' & y = [0,y'] & it = [0,(x' *' y')] ) if ( x in REAL+ & y in [:{0},REAL+ :] & x <> 0 )
ex x', y' being Element of REAL+ st
( x = [0,x'] & y = y' & it = [0,(y' *' x')] ) if ( y in REAL+ & x in [:{0},REAL+ :] & y <> 0 )
ex x', y' being Element of REAL+ st
( x = [0,x'] & y = [0,y'] & it = y' *' x' ) if ( x in [:{0},REAL+ :] & y in [:{0},REAL+ :] )
otherwise it = 0;
existence

proof end;

proof end;

consistency by Th5, XBOOLE_0:3;
commutativity ;

end;

:: deftheorem ARYTM_0:def 1 :

:: deftheorem Def2 defines + ARYTM_0:def 2 :

:: deftheorem Def3 defines * ARYTM_0:def 3 :

definition

let x be Element of REAL ;
func opp x -> Element of REAL means :Def4: :: ARYTM_0:def 4
+ x,it = 0;
existence

proof end;

proof end;

involutiveness ;
func inv x -> Element of REAL means :Def5: :: ARYTM_0:def 5
* x,it = one if x <> 0
otherwise it = 0;
existence

proof end;

proof end;

consistency ;
involutiveness

proof end;

end;

:: deftheorem Def4 defines opp ARYTM_0:def 4 :

:: deftheorem Def5 defines inv ARYTM_0:def 5 :

Lm3: for x, y, z being set st [x,y] = {z} holds
( z = {x} & x = y )

proof end;

theorem :: ARYTM_0:9

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canceled;

theorem Th10: :: ARYTM_0:10

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for a, b being Element of REAL holds not 0,one --> a,b in REAL

proof end;

definition

let x, y be Element of REAL ;
canceled;
func [*x,y*] -> Element of COMPLEX equals :Def7: :: ARYTM_0:def 7
x if y = 0
otherwise 0,one --> x,y;
consistency ;
coherence

proof end;

end;

:: deftheorem ARYTM_0:def 6 :

:: deftheorem Def7 defines [* ARYTM_0:def 7 :

theorem :: ARYTM_0:11

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for c being Element of COMPLEX ex r, s being Element of REAL st c = [*r,s*]

proof end;

theorem :: ARYTM_0:12

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for x1, x2, y1, y2 being Element of REAL st [*x1,x2*] = [*y1,y2*] holds
( x1 = y1 & x2 = y2 )

proof end;

set RR = [:{0},REAL+ :] \ {[0,0]};

reconsider o = 0 as Element of REAL by Th1, ARYTM_2:21;

theorem Th13: :: ARYTM_0:13

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for x, o being Element of REAL st o = 0 holds
+ x,o = x

proof end;

theorem Th14: :: ARYTM_0:14

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for x, o being Element of REAL st o = 0 holds
* x,o = 0

proof end;

theorem Th15: :: ARYTM_0:15

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for x, y, z being Element of REAL holds * x,(* y,z) = * (* x,y),z

proof end;

theorem Th16: :: ARYTM_0:16

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for x, y, z being Element of REAL holds * x,(+ y,z) = + (* x,y),(* x,z)

proof end;

theorem :: ARYTM_0:17

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for x, y being Element of REAL holds * (opp x),y = opp (* x,y)

proof end;

theorem Th18: :: ARYTM_0:18

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for x being Element of REAL holds * x,x in REAL+

proof end;

theorem :: ARYTM_0:19

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for x, y being Element of REAL st + (* x,x),(* y,y) = 0 holds
x = 0

proof end;

theorem Th20: :: ARYTM_0:20

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for x, y, z being Element of REAL st x <> 0 & * x,y = one & * x,z = one holds
y = z

proof end;

theorem Th21: :: ARYTM_0:21

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for x, y being Element of REAL st y = one holds
* x,y = x

proof end;

reconsider j = one as Element of REAL by Th1, ARYTM_2:21;

theorem :: ARYTM_0:22

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for x, y being Element of REAL st y <> 0 holds
* (* x,y),(inv y) = x

proof end;

theorem Th23: :: ARYTM_0:23

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for x, y being Element of REAL holds
( not * x,y = 0 or x = 0 or y = 0 )

proof end;

theorem :: ARYTM_0:24

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for x, y being Element of REAL holds inv (* x,y) = * (inv x),(inv y)

proof end;

theorem Th25: :: ARYTM_0:25

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for x, y, z being Element of REAL holds + x,(+ y,z) = + (+ x,y),z

proof end;

theorem :: ARYTM_0:26

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for x, y being Element of REAL st [*x,y*] in REAL holds
y = 0

proof end;

theorem :: ARYTM_0:27

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for x, y being Element of REAL holds opp (+ x,y) = + (opp x),(opp y)

proof end;