:: ARYTM_0 semantic presentation
Lemma1:
{} in {{} }
by TARSKI:def 1;
Lemma2:
1 = succ 0
;
theorem Th1: :: ARYTM_0:1
theorem Th2: :: ARYTM_0:2
theorem Th3: :: ARYTM_0:3
theorem Th4: :: ARYTM_0:4
theorem Th5: :: ARYTM_0:5
theorem Th6: :: ARYTM_0:6
theorem Th7: :: ARYTM_0:7
for b
1, b
2 being
set holds
not 1
= [b1,b2]
theorem Th8: :: ARYTM_0:8
definition
let c
1, c
2 be
Element of
REAL ;
canceled;func + c
1,c
2 -> Element of
REAL means :
Def2:
:: ARYTM_0:def 2
ex b
1, b
2 being
Element of
REAL+ st
( a
1 = b
1 & a
2 = b
2 & a
3 = b
1 + b
2 )
if ( a
1 in REAL+ & a
2 in REAL+ )
ex b
1, b
2 being
Element of
REAL+ st
( a
1 = b
1 & a
2 = [0,b2] & a
3 = b
1 - b
2 )
if ( a
1 in REAL+ & a
2 in [:{0},REAL+ :] )
ex b
1, b
2 being
Element of
REAL+ st
( a
1 = [0,b1] & a
2 = b
2 & a
3 = b
2 - b
1 )
if ( a
2 in REAL+ & a
1 in [:{0},REAL+ :] )
otherwise ex b
1, b
2 being
Element of
REAL+ st
( a
1 = [0,b1] & a
2 = [0,b2] & a
3 = [0,(b1 + b2)] );
existence
( not ( c1 in REAL+ & c2 in REAL+ & ( for b1 being Element of REAL
for b2, b3 being Element of REAL+ holds
not ( c1 = b2 & c2 = b3 & b1 = b2 + b3 ) ) ) & not ( c1 in REAL+ & c2 in [:{0},REAL+ :] & ( for b1 being Element of REAL
for b2, b3 being Element of REAL+ holds
not ( c1 = b2 & c2 = [0,b3] & b1 = b2 - b3 ) ) ) & not ( c2 in REAL+ & c1 in [:{0},REAL+ :] & ( for b1 being Element of REAL
for b2, b3 being Element of REAL+ holds
not ( c1 = [0,b2] & c2 = b3 & b1 = b3 - b2 ) ) ) & not ( not ( c1 in REAL+ & c2 in REAL+ ) & not ( c1 in REAL+ & c2 in [:{0},REAL+ :] ) & not ( c2 in REAL+ & c1 in [:{0},REAL+ :] ) & ( for b1 being Element of REAL
for b2, b3 being Element of REAL+ holds
not ( c1 = [0,b2] & c2 = [0,b3] & b1 = [0,(b2 + b3)] ) ) ) )
uniqueness
for b1, b2 being Element of REAL holds
( ( c1 in REAL+ & c2 in REAL+ & ex b3, b4 being Element of REAL+ st
( c1 = b3 & c2 = b4 & b1 = b3 + b4 ) & ex b3, b4 being Element of REAL+ st
( c1 = b3 & c2 = b4 & b2 = b3 + b4 ) implies b1 = b2 ) & ( c1 in REAL+ & c2 in [:{0},REAL+ :] & ex b3, b4 being Element of REAL+ st
( c1 = b3 & c2 = [0,b4] & b1 = b3 - b4 ) & ex b3, b4 being Element of REAL+ st
( c1 = b3 & c2 = [0,b4] & b2 = b3 - b4 ) implies b1 = b2 ) & ( c2 in REAL+ & c1 in [:{0},REAL+ :] & ex b3, b4 being Element of REAL+ st
( c1 = [0,b3] & c2 = b4 & b1 = b4 - b3 ) & ex b3, b4 being Element of REAL+ st
( c1 = [0,b3] & c2 = b4 & b2 = b4 - b3 ) implies b1 = b2 ) & ( not ( c1 in REAL+ & c2 in REAL+ ) & not ( c1 in REAL+ & c2 in [:{0},REAL+ :] ) & not ( c2 in REAL+ & c1 in [:{0},REAL+ :] ) & ex b3, b4 being Element of REAL+ st
( c1 = [0,b3] & c2 = [0,b4] & b1 = [0,(b3 + b4)] ) & ex b3, b4 being Element of REAL+ st
( c1 = [0,b3] & c2 = [0,b4] & b2 = [0,(b3 + b4)] ) implies b1 = b2 ) )
consistency
for b1 being Element of REAL holds
( ( c1 in REAL+ & c2 in REAL+ & c1 in REAL+ & c2 in [:{0},REAL+ :] implies ( not ( ex b2, b3 being Element of REAL+ st
( c1 = b2 & c2 = b3 & b1 = b2 + b3 ) & ( for b2, b3 being Element of REAL+ holds
not ( c1 = b2 & c2 = [0,b3] & b1 = b2 - b3 ) ) ) & not ( ex b2, b3 being Element of REAL+ st
( c1 = b2 & c2 = [0,b3] & b1 = b2 - b3 ) & ( for b2, b3 being Element of REAL+ holds
not ( c1 = b2 & c2 = b3 & b1 = b2 + b3 ) ) ) ) ) & ( c1 in REAL+ & c2 in REAL+ & c2 in REAL+ & c1 in [:{0},REAL+ :] implies ( not ( ex b2, b3 being Element of REAL+ st
( c1 = b2 & c2 = b3 & b1 = b2 + b3 ) & ( for b2, b3 being Element of REAL+ holds
not ( c1 = [0,b2] & c2 = b3 & b1 = b3 - b2 ) ) ) & not ( ex b2, b3 being Element of REAL+ st
( c1 = [0,b2] & c2 = b3 & b1 = b3 - b2 ) & ( for b2, b3 being Element of REAL+ holds
not ( c1 = b2 & c2 = b3 & b1 = b2 + b3 ) ) ) ) ) & ( c1 in REAL+ & c2 in [:{0},REAL+ :] & c2 in REAL+ & c1 in [:{0},REAL+ :] implies ( not ( ex b2, b3 being Element of REAL+ st
( c1 = b2 & c2 = [0,b3] & b1 = b2 - b3 ) & ( for b2, b3 being Element of REAL+ holds
not ( c1 = [0,b2] & c2 = b3 & b1 = b3 - b2 ) ) ) & not ( ex b2, b3 being Element of REAL+ st
( c1 = [0,b2] & c2 = b3 & b1 = b3 - b2 ) & ( for b2, b3 being Element of REAL+ holds
not ( c1 = b2 & c2 = [0,b3] & b1 = b2 - b3 ) ) ) ) ) )
by Th5, XBOOLE_0:3;
commutativity
for b1, b2, b3 being Element of REAL holds
( not ( b2 in REAL+ & b3 in REAL+ & ( for b4, b5 being Element of REAL+ holds
not ( b2 = b4 & b3 = b5 & b1 = b4 + b5 ) ) ) & not ( b2 in REAL+ & b3 in [:{0},REAL+ :] & ( for b4, b5 being Element of REAL+ holds
not ( b2 = b4 & b3 = [0,b5] & b1 = b4 - b5 ) ) ) & not ( b3 in REAL+ & b2 in [:{0},REAL+ :] & ( for b4, b5 being Element of REAL+ holds
not ( b2 = [0,b4] & b3 = b5 & b1 = b5 - b4 ) ) ) & not ( not ( b2 in REAL+ & b3 in REAL+ ) & not ( b2 in REAL+ & b3 in [:{0},REAL+ :] ) & not ( b3 in REAL+ & b2 in [:{0},REAL+ :] ) & ( for b4, b5 being Element of REAL+ holds
not ( b2 = [0,b4] & b3 = [0,b5] & b1 = [0,(b4 + b5)] ) ) ) implies ( not ( b3 in REAL+ & b2 in REAL+ & ( for b4, b5 being Element of REAL+ holds
not ( b3 = b4 & b2 = b5 & b1 = b4 + b5 ) ) ) & not ( b3 in REAL+ & b2 in [:{0},REAL+ :] & ( for b4, b5 being Element of REAL+ holds
not ( b3 = b4 & b2 = [0,b5] & b1 = b4 - b5 ) ) ) & not ( b2 in REAL+ & b3 in [:{0},REAL+ :] & ( for b4, b5 being Element of REAL+ holds
not ( b3 = [0,b4] & b2 = b5 & b1 = b5 - b4 ) ) ) & not ( not ( b3 in REAL+ & b2 in REAL+ ) & not ( b3 in REAL+ & b2 in [:{0},REAL+ :] ) & not ( b2 in REAL+ & b3 in [:{0},REAL+ :] ) & ( for b4, b5 being Element of REAL+ holds
not ( b3 = [0,b4] & b2 = [0,b5] & b1 = [0,(b4 + b5)] ) ) ) ) )
;
func * c
1,c
2 -> Element of
REAL means :
Def3:
:: ARYTM_0:def 3
ex b
1, b
2 being
Element of
REAL+ st
( a
1 = b
1 & a
2 = b
2 & a
3 = b
1 *' b
2 )
if ( a
1 in REAL+ & a
2 in REAL+ )
ex b
1, b
2 being
Element of
REAL+ st
( a
1 = b
1 & a
2 = [0,b2] & a
3 = [0,(b1 *' b2)] )
if ( a
1 in REAL+ & a
2 in [:{0},REAL+ :] & a
1 <> 0 )
ex b
1, b
2 being
Element of
REAL+ st
( a
1 = [0,b1] & a
2 = b
2 & a
3 = [0,(b2 *' b1)] )
if ( a
2 in REAL+ & a
1 in [:{0},REAL+ :] & a
2 <> 0 )
ex b
1, b
2 being
Element of
REAL+ st
( a
1 = [0,b1] & a
2 = [0,b2] & a
3 = b
2 *' b
1 )
if ( a
1 in [:{0},REAL+ :] & a
2 in [:{0},REAL+ :] )
otherwise a
3 = 0;
existence
( not ( c1 in REAL+ & c2 in REAL+ & ( for b1 being Element of REAL
for b2, b3 being Element of REAL+ holds
not ( c1 = b2 & c2 = b3 & b1 = b2 *' b3 ) ) ) & not ( c1 in REAL+ & c2 in [:{0},REAL+ :] & c1 <> 0 & ( for b1 being Element of REAL
for b2, b3 being Element of REAL+ holds
not ( c1 = b2 & c2 = [0,b3] & b1 = [0,(b2 *' b3)] ) ) ) & not ( c2 in REAL+ & c1 in [:{0},REAL+ :] & c2 <> 0 & ( for b1 being Element of REAL
for b2, b3 being Element of REAL+ holds
not ( c1 = [0,b2] & c2 = b3 & b1 = [0,(b3 *' b2)] ) ) ) & not ( c1 in [:{0},REAL+ :] & c2 in [:{0},REAL+ :] & ( for b1 being Element of REAL
for b2, b3 being Element of REAL+ holds
not ( c1 = [0,b2] & c2 = [0,b3] & b1 = b3 *' b2 ) ) ) & not ( not ( c1 in REAL+ & c2 in REAL+ ) & not ( c1 in REAL+ & c2 in [:{0},REAL+ :] & c1 <> 0 ) & not ( c2 in REAL+ & c1 in [:{0},REAL+ :] & c2 <> 0 ) & not ( c1 in [:{0},REAL+ :] & c2 in [:{0},REAL+ :] ) & ( for b1 being Element of REAL holds
not b1 = 0 ) ) )
uniqueness
for b1, b2 being Element of REAL holds
( ( c1 in REAL+ & c2 in REAL+ & ex b3, b4 being Element of REAL+ st
( c1 = b3 & c2 = b4 & b1 = b3 *' b4 ) & ex b3, b4 being Element of REAL+ st
( c1 = b3 & c2 = b4 & b2 = b3 *' b4 ) implies b1 = b2 ) & ( c1 in REAL+ & c2 in [:{0},REAL+ :] & c1 <> 0 & ex b3, b4 being Element of REAL+ st
( c1 = b3 & c2 = [0,b4] & b1 = [0,(b3 *' b4)] ) & ex b3, b4 being Element of REAL+ st
( c1 = b3 & c2 = [0,b4] & b2 = [0,(b3 *' b4)] ) implies b1 = b2 ) & ( c2 in REAL+ & c1 in [:{0},REAL+ :] & c2 <> 0 & ex b3, b4 being Element of REAL+ st
( c1 = [0,b3] & c2 = b4 & b1 = [0,(b4 *' b3)] ) & ex b3, b4 being Element of REAL+ st
( c1 = [0,b3] & c2 = b4 & b2 = [0,(b4 *' b3)] ) implies b1 = b2 ) & ( c1 in [:{0},REAL+ :] & c2 in [:{0},REAL+ :] & ex b3, b4 being Element of REAL+ st
( c1 = [0,b3] & c2 = [0,b4] & b1 = b4 *' b3 ) & ex b3, b4 being Element of REAL+ st
( c1 = [0,b3] & c2 = [0,b4] & b2 = b4 *' b3 ) implies b1 = b2 ) & ( not ( c1 in REAL+ & c2 in REAL+ ) & not ( c1 in REAL+ & c2 in [:{0},REAL+ :] & c1 <> 0 ) & not ( c2 in REAL+ & c1 in [:{0},REAL+ :] & c2 <> 0 ) & not ( c1 in [:{0},REAL+ :] & c2 in [:{0},REAL+ :] ) & b1 = 0 & b2 = 0 implies b1 = b2 ) )
consistency
for b1 being Element of REAL holds
( ( c1 in REAL+ & c2 in REAL+ & c1 in REAL+ & c2 in [:{0},REAL+ :] & c1 <> 0 implies ( not ( ex b2, b3 being Element of REAL+ st
( c1 = b2 & c2 = b3 & b1 = b2 *' b3 ) & ( for b2, b3 being Element of REAL+ holds
not ( c1 = b2 & c2 = [0,b3] & b1 = [0,(b2 *' b3)] ) ) ) & not ( ex b2, b3 being Element of REAL+ st
( c1 = b2 & c2 = [0,b3] & b1 = [0,(b2 *' b3)] ) & ( for b2, b3 being Element of REAL+ holds
not ( c1 = b2 & c2 = b3 & b1 = b2 *' b3 ) ) ) ) ) & ( c1 in REAL+ & c2 in REAL+ & c2 in REAL+ & c1 in [:{0},REAL+ :] & c2 <> 0 implies ( not ( ex b2, b3 being Element of REAL+ st
( c1 = b2 & c2 = b3 & b1 = b2 *' b3 ) & ( for b2, b3 being Element of REAL+ holds
not ( c1 = [0,b2] & c2 = b3 & b1 = [0,(b3 *' b2)] ) ) ) & not ( ex b2, b3 being Element of REAL+ st
( c1 = [0,b2] & c2 = b3 & b1 = [0,(b3 *' b2)] ) & ( for b2, b3 being Element of REAL+ holds
not ( c1 = b2 & c2 = b3 & b1 = b2 *' b3 ) ) ) ) ) & ( c1 in REAL+ & c2 in REAL+ & c1 in [:{0},REAL+ :] & c2 in [:{0},REAL+ :] implies ( not ( ex b2, b3 being Element of REAL+ st
( c1 = b2 & c2 = b3 & b1 = b2 *' b3 ) & ( for b2, b3 being Element of REAL+ holds
not ( c1 = [0,b2] & c2 = [0,b3] & b1 = b3 *' b2 ) ) ) & not ( ex b2, b3 being Element of REAL+ st
( c1 = [0,b2] & c2 = [0,b3] & b1 = b3 *' b2 ) & ( for b2, b3 being Element of REAL+ holds
not ( c1 = b2 & c2 = b3 & b1 = b2 *' b3 ) ) ) ) ) & ( c1 in REAL+ & c2 in [:{0},REAL+ :] & c1 <> 0 & c2 in REAL+ & c1 in [:{0},REAL+ :] & c2 <> 0 implies ( not ( ex b2, b3 being Element of REAL+ st
( c1 = b2 & c2 = [0,b3] & b1 = [0,(b2 *' b3)] ) & ( for b2, b3 being Element of REAL+ holds
not ( c1 = [0,b2] & c2 = b3 & b1 = [0,(b3 *' b2)] ) ) ) & not ( ex b2, b3 being Element of REAL+ st
( c1 = [0,b2] & c2 = b3 & b1 = [0,(b3 *' b2)] ) & ( for b2, b3 being Element of REAL+ holds
not ( c1 = b2 & c2 = [0,b3] & b1 = [0,(b2 *' b3)] ) ) ) ) ) & ( c1 in REAL+ & c2 in [:{0},REAL+ :] & c1 <> 0 & c1 in [:{0},REAL+ :] & c2 in [:{0},REAL+ :] implies ( not ( ex b2, b3 being Element of REAL+ st
( c1 = b2 & c2 = [0,b3] & b1 = [0,(b2 *' b3)] ) & ( for b2, b3 being Element of REAL+ holds
not ( c1 = [0,b2] & c2 = [0,b3] & b1 = b3 *' b2 ) ) ) & not ( ex b2, b3 being Element of REAL+ st
( c1 = [0,b2] & c2 = [0,b3] & b1 = b3 *' b2 ) & ( for b2, b3 being Element of REAL+ holds
not ( c1 = b2 & c2 = [0,b3] & b1 = [0,(b2 *' b3)] ) ) ) ) ) & ( c2 in REAL+ & c1 in [:{0},REAL+ :] & c2 <> 0 & c1 in [:{0},REAL+ :] & c2 in [:{0},REAL+ :] implies ( not ( ex b2, b3 being Element of REAL+ st
( c1 = [0,b2] & c2 = b3 & b1 = [0,(b3 *' b2)] ) & ( for b2, b3 being Element of REAL+ holds
not ( c1 = [0,b2] & c2 = [0,b3] & b1 = b3 *' b2 ) ) ) & not ( ex b2, b3 being Element of REAL+ st
( c1 = [0,b2] & c2 = [0,b3] & b1 = b3 *' b2 ) & ( for b2, b3 being Element of REAL+ holds
not ( c1 = [0,b2] & c2 = b3 & b1 = [0,(b3 *' b2)] ) ) ) ) ) )
by Th5, XBOOLE_0:3;
commutativity
for b1, b2, b3 being Element of REAL holds
( not ( b2 in REAL+ & b3 in REAL+ & ( for b4, b5 being Element of REAL+ holds
not ( b2 = b4 & b3 = b5 & b1 = b4 *' b5 ) ) ) & not ( b2 in REAL+ & b3 in [:{0},REAL+ :] & b2 <> 0 & ( for b4, b5 being Element of REAL+ holds
not ( b2 = b4 & b3 = [0,b5] & b1 = [0,(b4 *' b5)] ) ) ) & not ( b3 in REAL+ & b2 in [:{0},REAL+ :] & b3 <> 0 & ( for b4, b5 being Element of REAL+ holds
not ( b2 = [0,b4] & b3 = b5 & b1 = [0,(b5 *' b4)] ) ) ) & not ( b2 in [:{0},REAL+ :] & b3 in [:{0},REAL+ :] & ( for b4, b5 being Element of REAL+ holds
not ( b2 = [0,b4] & b3 = [0,b5] & b1 = b5 *' b4 ) ) ) & ( not ( b2 in REAL+ & b3 in REAL+ ) & not ( b2 in REAL+ & b3 in [:{0},REAL+ :] & b2 <> 0 ) & not ( b3 in REAL+ & b2 in [:{0},REAL+ :] & b3 <> 0 ) & not ( b2 in [:{0},REAL+ :] & b3 in [:{0},REAL+ :] ) implies b1 = 0 ) implies ( not ( b3 in REAL+ & b2 in REAL+ & ( for b4, b5 being Element of REAL+ holds
not ( b3 = b4 & b2 = b5 & b1 = b4 *' b5 ) ) ) & not ( b3 in REAL+ & b2 in [:{0},REAL+ :] & b3 <> 0 & ( for b4, b5 being Element of REAL+ holds
not ( b3 = b4 & b2 = [0,b5] & b1 = [0,(b4 *' b5)] ) ) ) & not ( b2 in REAL+ & b3 in [:{0},REAL+ :] & b2 <> 0 & ( for b4, b5 being Element of REAL+ holds
not ( b3 = [0,b4] & b2 = b5 & b1 = [0,(b5 *' b4)] ) ) ) & not ( b3 in [:{0},REAL+ :] & b2 in [:{0},REAL+ :] & ( for b4, b5 being Element of REAL+ holds
not ( b3 = [0,b4] & b2 = [0,b5] & b1 = b5 *' b4 ) ) ) & ( not ( b3 in REAL+ & b2 in REAL+ ) & not ( b3 in REAL+ & b2 in [:{0},REAL+ :] & b3 <> 0 ) & not ( b2 in REAL+ & b3 in [:{0},REAL+ :] & b2 <> 0 ) & not ( b3 in [:{0},REAL+ :] & b2 in [:{0},REAL+ :] ) implies b1 = 0 ) ) )
;
end;
:: deftheorem Def1 ARYTM_0:def 1 :
canceled;
:: deftheorem Def2 defines + ARYTM_0:def 2 :
for b
1, b
2, b
3 being
Element of
REAL holds
( ( b
1 in REAL+ & b
2 in REAL+ implies ( b
3 = + b
1,b
2 iff ex b
4, b
5 being
Element of
REAL+ st
( b
1 = b
4 & b
2 = b
5 & b
3 = b
4 + b
5 ) ) ) & ( b
1 in REAL+ & b
2 in [:{0},REAL+ :] implies ( b
3 = + b
1,b
2 iff ex b
4, b
5 being
Element of
REAL+ st
( b
1 = b
4 & b
2 = [0,b5] & b
3 = b
4 - b
5 ) ) ) & ( b
2 in REAL+ & b
1 in [:{0},REAL+ :] implies ( b
3 = + b
1,b
2 iff ex b
4, b
5 being
Element of
REAL+ st
( b
1 = [0,b4] & b
2 = b
5 & b
3 = b
5 - b
4 ) ) ) & ( not ( b
1 in REAL+ & b
2 in REAL+ ) & not ( b
1 in REAL+ & b
2 in [:{0},REAL+ :] ) & not ( b
2 in REAL+ & b
1 in [:{0},REAL+ :] ) implies ( b
3 = + b
1,b
2 iff ex b
4, b
5 being
Element of
REAL+ st
( b
1 = [0,b4] & b
2 = [0,b5] & b
3 = [0,(b4 + b5)] ) ) ) );
:: deftheorem Def3 defines * ARYTM_0:def 3 :
for b
1, b
2, b
3 being
Element of
REAL holds
( ( b
1 in REAL+ & b
2 in REAL+ implies ( b
3 = * b
1,b
2 iff ex b
4, b
5 being
Element of
REAL+ st
( b
1 = b
4 & b
2 = b
5 & b
3 = b
4 *' b
5 ) ) ) & ( b
1 in REAL+ & b
2 in [:{0},REAL+ :] & b
1 <> 0 implies ( b
3 = * b
1,b
2 iff ex b
4, b
5 being
Element of
REAL+ st
( b
1 = b
4 & b
2 = [0,b5] & b
3 = [0,(b4 *' b5)] ) ) ) & ( b
2 in REAL+ & b
1 in [:{0},REAL+ :] & b
2 <> 0 implies ( b
3 = * b
1,b
2 iff ex b
4, b
5 being
Element of
REAL+ st
( b
1 = [0,b4] & b
2 = b
5 & b
3 = [0,(b5 *' b4)] ) ) ) & ( b
1 in [:{0},REAL+ :] & b
2 in [:{0},REAL+ :] implies ( b
3 = * b
1,b
2 iff ex b
4, b
5 being
Element of
REAL+ st
( b
1 = [0,b4] & b
2 = [0,b5] & b
3 = b
5 *' b
4 ) ) ) & ( not ( b
1 in REAL+ & b
2 in REAL+ ) & not ( b
1 in REAL+ & b
2 in [:{0},REAL+ :] & b
1 <> 0 ) & not ( b
2 in REAL+ & b
1 in [:{0},REAL+ :] & b
2 <> 0 ) & not ( b
1 in [:{0},REAL+ :] & b
2 in [:{0},REAL+ :] ) implies ( b
3 = * b
1,b
2 iff b
3 = 0 ) ) );
definition
let c
1 be
Element of
REAL ;
func opp c
1 -> Element of
REAL means :
Def4:
:: ARYTM_0:def 4
+ a
1,a
2 = 0;
existence
ex b1 being Element of REAL st + c1,b1 = 0
uniqueness
for b1, b2 being Element of REAL holds
( + c1,b1 = 0 & + c1,b2 = 0 implies b1 = b2 )
involutiveness
for b1, b2 being Element of REAL holds
( + b2,b1 = 0 implies + b1,b2 = 0 )
;
func inv c
1 -> Element of
REAL means :
Def5:
:: ARYTM_0:def 5
* a
1,a
2 = 1
if a
1 <> 0
otherwise a
2 = 0;
existence
( not ( c1 <> 0 & ( for b1 being Element of REAL holds
not * c1,b1 = 1 ) ) & not ( not c1 <> 0 & ( for b1 being Element of REAL holds
not b1 = 0 ) ) )
uniqueness
for b1, b2 being Element of REAL holds
( ( c1 <> 0 & * c1,b1 = 1 & * c1,b2 = 1 implies b1 = b2 ) & ( not c1 <> 0 & b1 = 0 & b2 = 0 implies b1 = b2 ) )
consistency
for b1 being Element of REAL holds
verum
;
involutiveness
for b1, b2 being Element of REAL holds
( ( b2 <> 0 implies * b2,b1 = 1 ) & ( not b2 <> 0 implies b1 = 0 ) implies ( ( b1 <> 0 implies * b1,b2 = 1 ) & ( not b1 <> 0 implies b2 = 0 ) ) )
end;
:: deftheorem Def4 defines opp ARYTM_0:def 4 :
:: deftheorem Def5 defines inv ARYTM_0:def 5 :
for b
1, b
2 being
Element of
REAL holds
( ( b
1 <> 0 implies ( b
2 = inv b
1 iff
* b
1,b
2 = 1 ) ) & ( not b
1 <> 0 implies ( b
2 = inv b
1 iff b
2 = 0 ) ) );
Lemma15:
for b1, b2, b3 being set holds
( [b1,b2] = {b3} implies ( b3 = {b1} & b1 = b2 ) )
theorem Th9: :: ARYTM_0:9
canceled;
theorem Th10: :: ARYTM_0:10
:: deftheorem Def6 ARYTM_0:def 6 :
canceled;
:: deftheorem Def7 defines [* ARYTM_0:def 7 :
theorem Th11: :: ARYTM_0:11
theorem Th12: :: ARYTM_0:12
set c1 = [:{0},REAL+ :] \ {[0,0]};
reconsider c2 = 0 as Element of REAL by Th1, ARYTM_2:21;
theorem Th13: :: ARYTM_0:13
for b
1, b
2 being
Element of
REAL holds
( b
2 = 0 implies
+ b
1,b
2 = b
1 )
theorem Th14: :: ARYTM_0:14
theorem Th15: :: ARYTM_0:15
theorem Th16: :: ARYTM_0:16
theorem Th17: :: ARYTM_0:17
theorem Th18: :: ARYTM_0:18
theorem Th19: :: ARYTM_0:19
for b
1, b
2 being
Element of
REAL holds
(
+ (* b1,b1),
(* b2,b2) = 0 implies b
1 = 0 )
theorem Th20: :: ARYTM_0:20
for b
1, b
2, b
3 being
Element of
REAL holds
( b
1 <> 0 &
* b
1,b
2 = 1 &
* b
1,b
3 = 1 implies b
2 = b
3 )
theorem Th21: :: ARYTM_0:21
for b
1, b
2 being
Element of
REAL holds
( b
2 = 1 implies
* b
1,b
2 = b
1 )
reconsider c3 = 1 as Element of REAL by Th1, ARYTM_2:21;
theorem Th22: :: ARYTM_0:22
theorem Th23: :: ARYTM_0:23
for b
1, b
2 being
Element of
REAL holds
not (
* b
1,b
2 = 0 & not b
1 = 0 & not b
2 = 0 )
theorem Th24: :: ARYTM_0:24
theorem Th25: :: ARYTM_0:25
theorem Th26: :: ARYTM_0:26
theorem Th27: :: ARYTM_0:27