:: REAL_NS1 semantic presentation
Lemma1:
for b1 being Nat holds
ex b2 being BinOp of REAL b1 st
( ( for b3, b4 being Element of REAL b1 holds b2 . b3,b4 = b3 + b4 ) & b2 is commutative & b2 is associative )
Lemma2:
for b1 being Nat holds
ex b2 being Function of [:REAL ,(REAL b1):], REAL b1 st
for b3 being Element of REAL
for b4 being Element of REAL b1 holds b2 . b3,b4 = b3 * b4
:: deftheorem Def1 defines Euclid_add REAL_NS1:def 1 :
definition
let c
1 be
Nat;
func Euclid_mult c
1 -> Function of
[:REAL ,(REAL a1):],
REAL a
1 means :
Def2:
:: REAL_NS1:def 2
for b
1 being
Element of
REAL for b
2 being
Element of
REAL a
1 holds a
2 . b
1,b
2 = b
1 * b
2;
existence
ex b1 being Function of [:REAL ,(REAL c1):], REAL c1 st
for b2 being Element of REAL
for b3 being Element of REAL c1 holds b1 . b2,b3 = b2 * b3
by Lemma2;
uniqueness
for b1, b2 being Function of [:REAL ,(REAL c1):], REAL c1 holds
( ( for b3 being Element of REAL
for b4 being Element of REAL c1 holds b1 . b3,b4 = b3 * b4 ) & ( for b3 being Element of REAL
for b4 being Element of REAL c1 holds b2 . b3,b4 = b3 * b4 ) implies b1 = b2 )
end;
:: deftheorem Def2 defines Euclid_mult REAL_NS1:def 2 :
:: deftheorem Def3 defines Euclid_norm REAL_NS1:def 3 :
:: deftheorem Def4 defines REAL-NS REAL_NS1:def 4 :
theorem Th1: :: REAL_NS1:1
theorem Th2: :: REAL_NS1:2
theorem Th3: :: REAL_NS1:3
theorem Th4: :: REAL_NS1:4
theorem Th5: :: REAL_NS1:5
theorem Th6: :: REAL_NS1:6
theorem Th7: :: REAL_NS1:7
theorem Th8: :: REAL_NS1:8
theorem Th9: :: REAL_NS1:9
theorem Th10: :: REAL_NS1:10
theorem Th11: :: REAL_NS1:11
theorem Th12: :: REAL_NS1:12
Lemma19:
for b1 being Nat holds
REAL-NS b1 is RealBanachSpace
definition
let c
1 be
Nat;
func Euclid_scalar c
1 -> Function of
[:(REAL a1),(REAL a1):],
REAL means :
Def5:
:: REAL_NS1:def 5
for b
1, b
2 being
Element of
REAL a
1 holds a
2 . b
1,b
2 = Sum (mlt b1,b2);
existence
ex b1 being Function of [:(REAL c1),(REAL c1):], REAL st
for b2, b3 being Element of REAL c1 holds b1 . b2,b3 = Sum (mlt b2,b3)
uniqueness
for b1, b2 being Function of [:(REAL c1),(REAL c1):], REAL holds
( ( for b3, b4 being Element of REAL c1 holds b1 . b3,b4 = Sum (mlt b3,b4) ) & ( for b3, b4 being Element of REAL c1 holds b2 . b3,b4 = Sum (mlt b3,b4) ) implies b1 = b2 )
end;
:: deftheorem Def5 defines Euclid_scalar REAL_NS1:def 5 :
:: deftheorem Def6 defines REAL-US REAL_NS1:def 6 :
theorem Th13: :: REAL_NS1:13
theorem Th14: :: REAL_NS1:14
theorem Th15: :: REAL_NS1:15
theorem Th16: :: REAL_NS1:16
theorem Th17: :: REAL_NS1:17
theorem Th18: :: REAL_NS1:18
Lemma26:
for b1 being Nat holds REAL-US b1 is Hilbert