begin
definition
let C be non
empty set ;
let V be
RealNormSpace;
let f1,
f2 be
PartFunc of ,;
deffunc H1(
Element of
C)
-> Element of the
carrier of
V =
(f1 /. $1) + (f2 /. $1);
deffunc H2(
Element of
C)
-> Element of the
carrier of
V =
(f1 /. $1) - (f2 /. $1);
defpred S1[
set ]
means $1
in (dom f1) /\ (dom f2);
set X =
(dom f1) /\ (dom f2);
func f1 + f2 -> PartFunc of ,
means :
Def1:
(
dom it = (dom f1) /\ (dom f2) & ( for
c being
Element of
C st
c in dom it holds
it /. c = (f1 /. c) + (f2 /. c) ) );
existence
ex b1 being PartFunc of , st
( dom b1 = (dom f1) /\ (dom f2) & ( for c being Element of C st c in dom b1 holds
b1 /. c = (f1 /. c) + (f2 /. c) ) )
uniqueness
for b1, b2 being PartFunc of , st dom b1 = (dom f1) /\ (dom f2) & ( for c being Element of C st c in dom b1 holds
b1 /. c = (f1 /. c) + (f2 /. c) ) & dom b2 = (dom f1) /\ (dom f2) & ( for c being Element of C st c in dom b2 holds
b2 /. c = (f1 /. c) + (f2 /. c) ) holds
b1 = b2
func f1 - f2 -> PartFunc of ,
means :
Def2:
(
dom it = (dom f1) /\ (dom f2) & ( for
c being
Element of
C st
c in dom it holds
it /. c = (f1 /. c) - (f2 /. c) ) );
existence
ex b1 being PartFunc of , st
( dom b1 = (dom f1) /\ (dom f2) & ( for c being Element of C st c in dom b1 holds
b1 /. c = (f1 /. c) - (f2 /. c) ) )
uniqueness
for b1, b2 being PartFunc of , st dom b1 = (dom f1) /\ (dom f2) & ( for c being Element of C st c in dom b1 holds
b1 /. c = (f1 /. c) - (f2 /. c) ) & dom b2 = (dom f1) /\ (dom f2) & ( for c being Element of C st c in dom b2 holds
b2 /. c = (f1 /. c) - (f2 /. c) ) holds
b1 = b2
end;
:: deftheorem Def1 defines + VFUNCT_1:def 1 :
:: deftheorem Def2 defines - VFUNCT_1:def 2 :
:: deftheorem Def3 defines (#) VFUNCT_1:def 3 :
:: deftheorem Def4 defines (#) VFUNCT_1:def 4 :
:: deftheorem Def5 defines ||. VFUNCT_1:def 5 :
:: deftheorem Def6 defines - VFUNCT_1:def 6 :
theorem
canceled;
theorem
canceled;
theorem
canceled;
theorem
canceled;
theorem
canceled;
theorem
canceled;
theorem
theorem
theorem
theorem
theorem
theorem
theorem
theorem
theorem
theorem
theorem
theorem
theorem
theorem Th20:
theorem
theorem
theorem
theorem Th24:
theorem
theorem
theorem
theorem
theorem Th29:
theorem Th30:
theorem Th31:
theorem
theorem Th33:
theorem
theorem Th35:
theorem
theorem
theorem Th38:
theorem Th39:
theorem Th40:
theorem Th41:
theorem Th42:
theorem
theorem
theorem
theorem
:: deftheorem Def7 defines is_bounded_on VFUNCT_1:def 7 :
theorem
canceled;
theorem
theorem
theorem
theorem Th51:
theorem Th52:
theorem Th53:
theorem
theorem
theorem
theorem
theorem
theorem Th59:
theorem Th60:
theorem Th61:
theorem
theorem
theorem