:: ARYTM_2 semantic presentation

definition
func DEDEKIND_CUTS -> Subset-Family of RAT+ equals :: ARYTM_2:def 1
{ A where A is Subset of RAT+ : for r being Element of RAT+ st r in A holds
( ( for s being Element of RAT+ st s <=' r holds
s in A ) & ex s being Element of RAT+ st
( s in A & r < s ) )
}
\ {RAT+ };
coherence
{ A where A is Subset of RAT+ : for r being Element of RAT+ st r in A holds
( ( for s being Element of RAT+ st s <=' r holds
s in A ) & ex s being Element of RAT+ st
( s in A & r < s ) )
}
\ {RAT+ } is Subset-Family of RAT+
proof end;
end;

:: deftheorem Def1 defines DEDEKIND_CUTS ARYTM_2:def 1 :
DEDEKIND_CUTS = { A where A is Subset of RAT+ : for r being Element of RAT+ st r in A holds
( ( for s being Element of RAT+ st s <=' r holds
s in A ) & ex s being Element of RAT+ st
( s in A & r < s ) )
}
\ {RAT+ };

registration
cluster DEDEKIND_CUTS -> non empty ;
coherence
not DEDEKIND_CUTS is empty
proof end;
end;

definition
func REAL+ -> set equals :: ARYTM_2:def 2
(RAT+ \/ DEDEKIND_CUTS ) \ { { s where s is Element of RAT+ : s < t } where t is Element of RAT+ : t <> {} } ;
coherence
(RAT+ \/ DEDEKIND_CUTS ) \ { { s where s is Element of RAT+ : s < t } where t is Element of RAT+ : t <> {} } is set
;
end;

:: deftheorem Def2 defines REAL+ ARYTM_2:def 2 :
REAL+ = (RAT+ \/ DEDEKIND_CUTS ) \ { { s where s is Element of RAT+ : s < t } where t is Element of RAT+ : t <> {} } ;

set IR = { A where A is Subset of RAT+ : for r being Element of RAT+ st r in A holds
( ( for s being Element of RAT+ st s <=' r holds
s in A ) & ex s being Element of RAT+ st
( s in A & r < s ) )
}
;

set RA = { { s where s is Element of RAT+ : s < t } where t is Element of RAT+ : t <> {} } ;

Lemma29: for x, y being set holds not [x,y] in { A where A is Subset of RAT+ : for r being Element of RAT+ st r in A holds
( ( for s being Element of RAT+ st s <=' r holds
s in A ) & ex s being Element of RAT+ st
( s in A & r < s ) )
}

proof end;

Lemma34: RAT+ c= RAT+ \/ DEDEKIND_CUTS
by XBOOLE_1:7;

Lemma35: RAT+ misses { { s where s is Element of RAT+ : s < t } where t is Element of RAT+ : t <> {} }
proof end;

theorem Th1: :: ARYTM_2:1
RAT+ c= REAL+ by , , XBOOLE_1:86;

RAT+ \/ DEDEKIND_CUTS c= RAT+ \/ { A where A is Subset of RAT+ : for r being Element of RAT+ st r in A holds
( ( for s being Element of RAT+ st s <=' r holds
s in A ) & ex s being Element of RAT+ st
( s in A & r < s ) )
}

by XBOOLE_1:9;

then Lemma41: REAL+ c= RAT+ \/ { A where A is Subset of RAT+ : for r being Element of RAT+ st r in A holds
( ( for s being Element of RAT+ st s <=' r holds
s in A ) & ex s being Element of RAT+ st
( s in A & r < s ) )
}

by XBOOLE_1:1;

REAL+ = RAT+ \/ (({ A where A is Subset of RAT+ : for r being Element of RAT+ st r in A holds
( ( for s being Element of RAT+ st s <=' r holds
s in A ) & ex s being Element of RAT+ st
( s in A & r < s ) )
}
\ {RAT+ }
)
\ { { s where s is Element of RAT+ : s < t } where t is Element of RAT+ : t <> {} }
)

by , XBOOLE_1:87;

then Lemma42: DEDEKIND_CUTS \ { { s where s is Element of RAT+ : s < t } where t is Element of RAT+ : t <> {} } c= REAL+
by XBOOLE_1:7;

theorem Th2: :: ARYTM_2:2
omega c= REAL+ by , ARYTM_3:34, XBOOLE_1:1;

registration
cluster REAL+ -> non empty ;
coherence
not REAL+ is empty
by , ORDINAL2:19;
end;

definition
let x be Element of REAL+ ;
func DEDEKIND_CUT c1 -> Element of DEDEKIND_CUTS means :Def3: :: ARYTM_2:def 3
ex r being Element of RAT+ st
( IR = r & it = { s where s is Element of RAT+ : s < r } ) if IR in RAT+
otherwise it = IR;
existence
( ( x in RAT+ implies ex b1 being Element of DEDEKIND_CUTS ex r being Element of RAT+ st
( x = r & b1 = { s where s is Element of RAT+ : s < r } ) ) & ( not x in RAT+ implies ex b1 being Element of DEDEKIND_CUTS st b1 = x ) )
proof end;
uniqueness
for b1, b2 being Element of DEDEKIND_CUTS holds
( ( x in RAT+ & ex r being Element of RAT+ st
( x = r & b1 = { s where s is Element of RAT+ : s < r } ) & ex r being Element of RAT+ st
( x = r & b2 = { s where s is Element of RAT+ : s < r } ) implies b1 = b2 ) & ( not x in RAT+ & b1 = x & b2 = x implies b1 = b2 ) )
;
consistency
for b1 being Element of DEDEKIND_CUTS holds verum
;
end;

:: deftheorem Def3 defines DEDEKIND_CUT ARYTM_2:def 3 :
for x being Element of REAL+
for b2 being Element of DEDEKIND_CUTS holds
( ( x in RAT+ implies ( b2 = DEDEKIND_CUT x iff ex r being Element of RAT+ st
( x = r & b2 = { s where s is Element of RAT+ : s < r } ) ) ) & ( not x in RAT+ implies ( b2 = DEDEKIND_CUT x iff b2 = x ) ) );

theorem Th3: :: ARYTM_2:3
for y being set holds not [{} ,y] in REAL+
proof end;

Lemma47: for s, t being Element of RAT+ st ( for r being Element of RAT+ holds
( r < s iff r < t ) ) holds
s = t
proof end;

definition
let x be Element of DEDEKIND_CUTS ;
func GLUED c1 -> Element of REAL+ means :Def4: :: ARYTM_2:def 4
ex r being Element of RAT+ st
( it = r & ( for s being Element of RAT+ holds
( s in IR iff s < r ) ) ) if ex r being Element of RAT+ st
for s being Element of RAT+ holds
( s in IR iff s < r )
otherwise it = IR;
uniqueness
for b1, b2 being Element of REAL+ holds
( ( ex r being Element of RAT+ st
for s being Element of RAT+ holds
( s in x iff s < r ) & ex r being Element of RAT+ st
( b1 = r & ( for s being Element of RAT+ holds
( s in x iff s < r ) ) ) & ex r being Element of RAT+ st
( b2 = r & ( for s being Element of RAT+ holds
( s in x iff s < r ) ) ) implies b1 = b2 ) & ( ( for r being Element of RAT+ holds
not for s being Element of RAT+ holds
( s in x iff s < r ) ) & b1 = x & b2 = x implies b1 = b2 ) )
proof end;
existence
( ( ex r being Element of RAT+ st
for s being Element of RAT+ holds
( s in x iff s < r ) implies ex b1 being Element of REAL+ ex r being Element of RAT+ st
( b1 = r & ( for s being Element of RAT+ holds
( s in x iff s < r ) ) ) ) & ( ( for r being Element of RAT+ holds
not for s being Element of RAT+ holds
( s in x iff s < r ) ) implies ex b1 being Element of REAL+ st b1 = x ) )
proof end;
consistency
for b1 being Element of REAL+ holds verum
;
end;

:: deftheorem Def4 defines GLUED ARYTM_2:def 4 :
for x being Element of DEDEKIND_CUTS
for b2 being Element of REAL+ holds
( ( ex r being Element of RAT+ st
for s being Element of RAT+ holds
( s in x iff s < r ) implies ( b2 = GLUED x iff ex r being Element of RAT+ st
( b2 = r & ( for s being Element of RAT+ holds
( s in x iff s < r ) ) ) ) ) & ( ( for r being Element of RAT+ holds
not for s being Element of RAT+ holds
( s in x iff s < r ) ) implies ( b2 = GLUED x iff b2 = x ) ) );

Lemma51: for r being Element of RAT+
for B being set st B in DEDEKIND_CUTS & r in B holds
ex s being Element of RAT+ st
( s in B & r < s )
proof end;

Lemma53: {} in DEDEKIND_CUTS
proof end;

Lemma54: DEDEKIND_CUTS /\ RAT+ = {{} }
proof end;

Lemma55: for x being Element of REAL+ holds
( DEDEKIND_CUT x = {} iff x = {} )
proof end;

Lemma56: for A being Element of DEDEKIND_CUTS holds
( GLUED A = {} iff A = {} )
proof end;

Lemma57: for A being Element of DEDEKIND_CUTS holds DEDEKIND_CUT (GLUED A) = A
proof end;

Lemma58: for x, y being set st x in { A where A is Subset of RAT+ : for r being Element of RAT+ st r in A holds
( ( for s being Element of RAT+ st s <=' r holds
s in A ) & ex s being Element of RAT+ st
( s in A & r < s ) )
}
& y in { A where A is Subset of RAT+ : for r being Element of RAT+ st r in A holds
( ( for s being Element of RAT+ st s <=' r holds
s in A ) & ex s being Element of RAT+ st
( s in A & r < s ) )
}
& not x c= y holds
y c= x
proof end;

definition
let x be Element of REAL+ , y be Element of REAL+ ;
pred c1 <=' c2 means :Def5: :: ARYTM_2:def 5
ex x', y' being Element of RAT+ st
( IR = x' & RA = y' & x' <=' y' ) if ( IR in RAT+ & RA in RAT+ )
IR in RA if ( IR in RAT+ & not RA in RAT+ )
not RA in IR if ( not IR in RAT+ & RA in RAT+ )
otherwise IR c= RA;
consistency
( ( x in RAT+ & y in RAT+ & x in RAT+ & not y in RAT+ implies ( ex x', y' being Element of RAT+ st
( x = x' & y = y' & x' <=' y' ) iff x in y ) ) & ( x in RAT+ & y in RAT+ & not x in RAT+ & y in RAT+ implies ( ex x', y' being Element of RAT+ st
( x = x' & y = y' & x' <=' y' ) iff not y in x ) ) & ( x in RAT+ & not y in RAT+ & not x in RAT+ & y in RAT+ implies ( x in y iff not y in x ) ) )
;
connectedness
for x, y being Element of REAL+ st ( ( x in RAT+ & y in RAT+ & ( for x', y' being Element of RAT+ holds
( not x = x' or not y = y' or not x' <=' y' ) ) ) or ( x in RAT+ & not y in RAT+ & not x in y ) or ( not x in RAT+ & y in RAT+ & y in x ) or ( ( not x in RAT+ or not y in RAT+ ) & ( not x in RAT+ or y in RAT+ ) & ( x in RAT+ or not y in RAT+ ) & not x c= y ) ) holds
( ( y in RAT+ & x in RAT+ implies ex x', y' being Element of RAT+ st
( y = x' & x = y' & x' <=' y' ) ) & ( y in RAT+ & not x in RAT+ implies y in x ) & ( not y in RAT+ & x in RAT+ implies not x in y ) & ( ( not y in RAT+ or not x in RAT+ ) & ( not y in RAT+ or x in RAT+ ) & ( y in RAT+ or not x in RAT+ ) implies y c= x ) )
proof end;
end;

:: deftheorem Def5 defines <=' ARYTM_2:def 5 :
for x, y being Element of REAL+ holds
( ( x in RAT+ & y in RAT+ implies ( x <=' y iff ex x', y' being Element of RAT+ st
( x = x' & y = y' & x' <=' y' ) ) ) & ( x in RAT+ & not y in RAT+ implies ( x <=' y iff x in y ) ) & ( not x in RAT+ & y in RAT+ implies ( x <=' y iff not y in x ) ) & ( ( not x in RAT+ or not y in RAT+ ) & ( not x in RAT+ or y in RAT+ ) & ( x in RAT+ or not y in RAT+ ) implies ( x <=' y iff x c= y ) ) );

notation
let x be Element of REAL+ , y be Element of REAL+ ;
antonym c2 < c1 for c1 <=' c2;
end;

Lemma62: for x', y' being Element of RAT+
for x, y being Element of REAL+ st x = x' & y = y' holds
( x <=' y iff x' <=' y' )
proof end;

Lemma63: for B being set st B in DEDEKIND_CUTS & B <> {} holds
ex r being Element of RAT+ st
( r in B & r <> {} )
proof end;

Lemma66: for r, s being Element of RAT+
for B being set st B in DEDEKIND_CUTS & r in B & s <=' r holds
s in B
proof end;

Lemma67: for B being set st B in DEDEKIND_CUTS & B <> {} holds
{} in B
proof end;

Lemma68: for r being Element of RAT+
for B being set st B in DEDEKIND_CUTS \ { { s where s is Element of RAT+ : s < t } where t is Element of RAT+ : t <> {} } & not r in B & B <> {} holds
ex s being Element of RAT+ st
( not s in B & s < r )
proof end;

Lemma69: for x being Element of REAL+ st DEDEKIND_CUT x in { { s where s is Element of RAT+ : s < t } where t is Element of RAT+ : t <> {} } holds
x in RAT+
proof end;

Lemma70: for x, y being Element of REAL+ st DEDEKIND_CUT x c= DEDEKIND_CUT y holds
x <=' y
proof end;

Lemma94: for x, y being Element of REAL+ st x <=' y & y <=' x holds
x = y
proof end;

Lemma95: for x, y being Element of REAL+ st DEDEKIND_CUT x = DEDEKIND_CUT y holds
x = y
proof end;

Lemma96: for x being Element of REAL+ holds GLUED (DEDEKIND_CUT x) = x
proof end;

definition
let A be Element of DEDEKIND_CUTS , B be Element of DEDEKIND_CUTS ;
func c1 + c2 -> Element of DEDEKIND_CUTS equals :Def6: :: ARYTM_2:def 6
{ (r + s) where r is Element of RAT+ , s is Element of RAT+ : ( r in IR & s in RA ) } ;
coherence
{ (r + s) where r is Element of RAT+ , s is Element of RAT+ : ( r in A & s in B ) } is Element of DEDEKIND_CUTS
proof end;
commutativity
for b1, A, B being Element of DEDEKIND_CUTS st b1 = { (r + s) where r is Element of RAT+ , s is Element of RAT+ : ( r in A & s in B ) } holds
b1 = { (r + s) where r is Element of RAT+ , s is Element of RAT+ : ( r in B & s in A ) }
proof end;
end;

:: deftheorem Def6 defines + ARYTM_2:def 6 :
for A, B being Element of DEDEKIND_CUTS holds A + B = { (r + s) where r is Element of RAT+ , s is Element of RAT+ : ( r in A & s in B ) } ;

Lemma105: for B being set st B in DEDEKIND_CUTS holds
ex r being Element of RAT+ st not r in B
proof end;

definition
let A be Element of DEDEKIND_CUTS , B be Element of DEDEKIND_CUTS ;
func c1 *' c2 -> Element of DEDEKIND_CUTS equals :: ARYTM_2:def 7
{ (r *' s) where r is Element of RAT+ , s is Element of RAT+ : ( r in IR & s in RA ) } ;
coherence
{ (r *' s) where r is Element of RAT+ , s is Element of RAT+ : ( r in A & s in B ) } is Element of DEDEKIND_CUTS
proof end;
commutativity
for b1, A, B being Element of DEDEKIND_CUTS st b1 = { (r *' s) where r is Element of RAT+ , s is Element of RAT+ : ( r in A & s in B ) } holds
b1 = { (r *' s) where r is Element of RAT+ , s is Element of RAT+ : ( r in B & s in A ) }
proof end;
end;

:: deftheorem Def7 defines *' ARYTM_2:def 7 :
for A, B being Element of DEDEKIND_CUTS holds A *' B = { (r *' s) where r is Element of RAT+ , s is Element of RAT+ : ( r in A & s in B ) } ;

definition
let x be Element of REAL+ , y be Element of REAL+ ;
func c1 + c2 -> Element of REAL+ equals :Def8: :: ARYTM_2:def 8
IR if RA = {}
RA if IR = {}
otherwise GLUED ((DEDEKIND_CUT IR) + (DEDEKIND_CUT RA));
coherence
( ( y = {} implies x is Element of REAL+ ) & ( x = {} implies y is Element of REAL+ ) & ( not y = {} & not x = {} implies GLUED ((DEDEKIND_CUT x) + (DEDEKIND_CUT y)) is Element of REAL+ ) )
;
consistency
for b1 being Element of REAL+ st y = {} & x = {} holds
( b1 = x iff b1 = y )
;
commutativity
for b1, x, y being Element of REAL+ st ( y = {} implies b1 = x ) & ( x = {} implies b1 = y ) & ( not y = {} & not x = {} implies b1 = GLUED ((DEDEKIND_CUT x) + (DEDEKIND_CUT y)) ) holds
( ( x = {} implies b1 = y ) & ( y = {} implies b1 = x ) & ( not x = {} & not y = {} implies b1 = GLUED ((DEDEKIND_CUT y) + (DEDEKIND_CUT x)) ) )
;
func c1 *' c2 -> Element of REAL+ equals :: ARYTM_2:def 9
GLUED ((DEDEKIND_CUT IR) *' (DEDEKIND_CUT RA));
coherence
GLUED ((DEDEKIND_CUT x) *' (DEDEKIND_CUT y)) is Element of REAL+
;
commutativity
for b1, x, y being Element of REAL+ st b1 = GLUED ((DEDEKIND_CUT x) *' (DEDEKIND_CUT y)) holds
b1 = GLUED ((DEDEKIND_CUT y) *' (DEDEKIND_CUT x))
;
end;

:: deftheorem Def8 defines + ARYTM_2:def 8 :
for x, y being Element of REAL+ holds
( ( y = {} implies x + y = x ) & ( x = {} implies x + y = y ) & ( not y = {} & not x = {} implies x + y = GLUED ((DEDEKIND_CUT x) + (DEDEKIND_CUT y)) ) );

:: deftheorem Def9 defines *' ARYTM_2:def 9 :
for x, y being Element of REAL+ holds x *' y = GLUED ((DEDEKIND_CUT x) *' (DEDEKIND_CUT y));

theorem Th4: :: ARYTM_2:4
for x, y being Element of REAL+ st x = {} holds
x *' y = {}
proof end;

theorem Th5: :: ARYTM_2:5
canceled;

theorem Th6: :: ARYTM_2:6
for x, y being Element of REAL+ st x + y = {} holds
x = {}
proof end;

Lemma109: for A, B, C being Element of DEDEKIND_CUTS holds A + (B + C) c= (A + B) + C
proof end;

Lemma110: for A, B, C being Element of DEDEKIND_CUTS holds A + (B + C) = (A + B) + C
proof end;

Lemma111: for A, B being Element of DEDEKIND_CUTS holds
( not A + B = {} or A = {} or B = {} )
proof end;

theorem Th7: :: ARYTM_2:7
for x, y, z being Element of REAL+ holds x + (y + z) = (x + y) + z
proof end;

theorem Th8: :: ARYTM_2:8
{ A where A is Subset of RAT+ : for r being Element of RAT+ st r in A holds
( ( for s being Element of RAT+ st s <=' r holds
s in A ) & ex s being Element of RAT+ st
( s in A & r < s ) )
}
is c=-linear
proof end;

Lemma113: for e being set st e in REAL+ holds
e <> RAT+
proof end;

Lemma114: for r, s being Element of RAT+
for B being set st B in { A where A is Subset of RAT+ : for r being Element of RAT+ st r in A holds
( ( for s being Element of RAT+ st s <=' r holds
s in A ) & ex s being Element of RAT+ st
( s in A & r < s ) )
}
& r in B & s <=' r holds
s in B
proof end;

Lemma115: for y, x being Element of REAL+ st y < x holds
ex z being Element of REAL+ st
( z in RAT+ & z < x & y < z )
proof end;

Lemma117: for x, y, z being Element of REAL+ st x <=' y & y <=' z holds
x <=' z
proof end;

theorem Th9: :: ARYTM_2:9
for X, Y being Subset of REAL+ st ex x being Element of REAL+ st x in X & ex x being Element of REAL+ st x in Y & ( for x, y being Element of REAL+ st x in X & y in Y holds
x <=' y ) holds
ex z being Element of REAL+ st
for x, y being Element of REAL+ st x in X & y in Y holds
( x <=' z & z <=' y )
proof end;

Lemma146: one = 1
;

Lemma147: {} = {}
;

Lemma148: for A, B being Element of DEDEKIND_CUTS st A + B = A & A <> {} holds
B = {}
proof end;

Lemma153: for x, y being Element of REAL+ st x + y = x holds
y = {}
proof end;

Lemma154: for A, B being Element of DEDEKIND_CUTS st A <> {} & A c= B & A <> B holds
ex C being Element of DEDEKIND_CUTS st A + C = B
proof end;

Lemma161: for x, y being Element of REAL+ st x <=' y holds
DEDEKIND_CUT x c= DEDEKIND_CUT y
proof end;

theorem Th10: :: ARYTM_2:10
for x, y being Element of REAL+ st x <=' y holds
ex z being Element of REAL+ st x + z = y
proof end;

theorem Th11: :: ARYTM_2:11
for x, y being Element of REAL+ ex z being Element of REAL+ st
( x + z = y or y + z = x )
proof end;

theorem Th12: :: ARYTM_2:12
for x, y, z being Element of REAL+ st x + y = x + z holds
y = z
proof end;

Lemma165: for A, B, C being Element of DEDEKIND_CUTS holds A *' (B *' C) c= (A *' B) *' C
proof end;

Lemma166: for A, B, C being Element of DEDEKIND_CUTS holds A *' (B *' C) = (A *' B) *' C
proof end;

theorem Th13: :: ARYTM_2:13
for x, y, z being Element of REAL+ holds x *' (y *' z) = (x *' y) *' z
proof end;

Lemma167: for x, y being Element of REAL+ holds
( not x *' y = {} or x = {} or y = {} )
proof end;

Lemma168: for A, B, C being Element of DEDEKIND_CUTS holds A *' (B + C) = (A *' B) + (A *' C)
proof end;

theorem Th14: :: ARYTM_2:14
for x, y, z being Element of REAL+ holds x *' (y + z) = (x *' y) + (x *' z)
proof end;

one in RAT+
;

then reconsider rone = one as Element of REAL+ by ;

Lemma174: for B being set st B in { A where A is Subset of RAT+ : for r being Element of RAT+ st r in A holds
( ( for s being Element of RAT+ st s <=' r holds
s in A ) & ex s being Element of RAT+ st
( s in A & r < s ) )
}
& B <> {} holds
ex r being Element of RAT+ st
( r in B & r <> {} )
proof end;

Lemma175: for A being Element of DEDEKIND_CUTS st A <> {} holds
ex B being Element of DEDEKIND_CUTS st A *' B = DEDEKIND_CUT rone
proof end;

theorem Th15: :: ARYTM_2:15
for x being Element of REAL+ st x <> {} holds
ex y being Element of REAL+ st x *' y = one
proof end;

Lemma209: for A, B being Element of DEDEKIND_CUTS st A = { r where r is Element of RAT+ : r < one } holds
A *' B = B
proof end;

theorem Th16: :: ARYTM_2:16
for x, y being Element of REAL+ st x = one holds
x *' y = y
proof end;

Lemma210: for i, j being Element of omega
for x', y' being Element of RAT+ st i = x' & j = y' holds
i +^ j = x' + y'
proof end;

Lemma215: for z', x', y' being Element of RAT+ st z' < x' + y' & x' <> {} & y' <> {} holds
ex r, s being Element of RAT+ st
( z' = r + s & r < x' & s < y' )
proof end;

Lemma216: for x, y being Element of REAL+ st x in RAT+ & y in RAT+ holds
ex x', y' being Element of RAT+ st
( x = x' & y = y' & x + y = x' + y' )
proof end;

theorem Th17: :: ARYTM_2:17
for x, y being Element of REAL+ st x in omega & y in omega holds
y + x in omega
proof end;

theorem Th18: :: ARYTM_2:18
for A being Subset of REAL+ st {} in A & ( for x, y being Element of REAL+ st x in A & y = one holds
x + y in A ) holds
omega c= A
proof end;

theorem Th19: :: ARYTM_2:19
for x being Element of REAL+ st x in omega holds
for y being Element of REAL+ holds
( y in x iff ( y in omega & y <> x & y <=' x ) )
proof end;

theorem Th20: :: ARYTM_2:20
for x, y, z being Element of REAL+ st x = y + z holds
z <=' x
proof end;

theorem Th21: :: ARYTM_2:21
( {} in REAL+ & one in REAL+ )
proof end;

theorem Th22: :: ARYTM_2:22
for x, y being Element of REAL+ st x in RAT+ & y in RAT+ holds
ex x', y' being Element of RAT+ st
( x = x' & y = y' & x *' y = x' *' y' )
proof end;