:: VFUNCT_2 semantic presentation  Show TPTP formulae Show IDV graph for whole article:: Showing IDV graph ... (Click the Palm Trees again to close it)

definition
let M be non empty set ;
let V be ComplexNormSpace;
let f1, f2 be PartFunc of M,the carrier of V;
deffunc H1( Element of M) -> Element of the carrier of V = (f1 /. $1) + (f2 /. $1);
deffunc H2( Element of M) -> 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 M,the carrier of V means :Def1: :: VFUNCT_2:def 1
( dom it = (dom f1) /\ (dom f2) & ( for c being Element of M st c in dom it holds
it /. c = (f1 /. c) + (f2 /. c) ) );
existence
ex b1 being PartFunc of M,the carrier of V st
( dom b1 = (dom f1) /\ (dom f2) & ( for c being Element of M st c in dom b1 holds
b1 /. c = (f1 /. c) + (f2 /. c) ) )
proof end;
uniqueness
for b1, b2 being PartFunc of M,the carrier of V st dom b1 = (dom f1) /\ (dom f2) & ( for c being Element of M st c in dom b1 holds
b1 /. c = (f1 /. c) + (f2 /. c) ) & dom b2 = (dom f1) /\ (dom f2) & ( for c being Element of M st c in dom b2 holds
b2 /. c = (f1 /. c) + (f2 /. c) ) holds
b1 = b2
proof end;
func f1 - f2 -> PartFunc of M,the carrier of V means :Def2: :: VFUNCT_2:def 2
( dom it = (dom f1) /\ (dom f2) & ( for c being Element of M st c in dom it holds
it /. c = (f1 /. c) - (f2 /. c) ) );
existence
ex b1 being PartFunc of M,the carrier of V st
( dom b1 = (dom f1) /\ (dom f2) & ( for c being Element of M st c in dom b1 holds
b1 /. c = (f1 /. c) - (f2 /. c) ) )
proof end;
uniqueness
for b1, b2 being PartFunc of M,the carrier of V st dom b1 = (dom f1) /\ (dom f2) & ( for c being Element of M st c in dom b1 holds
b1 /. c = (f1 /. c) - (f2 /. c) ) & dom b2 = (dom f1) /\ (dom f2) & ( for c being Element of M st c in dom b2 holds
b2 /. c = (f1 /. c) - (f2 /. c) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def1 defines + VFUNCT_2:def 1 :
for M being non empty set
for V being ComplexNormSpace
for f1, f2, b5 being PartFunc of M,the carrier of V holds
( b5 = f1 + f2 iff ( dom b5 = (dom f1) /\ (dom f2) & ( for c being Element of M st c in dom b5 holds
b5 /. c = (f1 /. c) + (f2 /. c) ) ) );

:: deftheorem Def2 defines - VFUNCT_2:def 2 :
for M being non empty set
for V being ComplexNormSpace
for f1, f2, b5 being PartFunc of M,the carrier of V holds
( b5 = f1 - f2 iff ( dom b5 = (dom f1) /\ (dom f2) & ( for c being Element of M st c in dom b5 holds
b5 /. c = (f1 /. c) - (f2 /. c) ) ) );

definition
let M be non empty set ;
let V be ComplexNormSpace;
let f1 be PartFunc of M, COMPLEX ;
let f2 be PartFunc of M,the carrier of V;
deffunc H1( Element of M) -> 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 M,the carrier of V means :Def3: :: VFUNCT_2:def 3
( dom it = (dom f1) /\ (dom f2) & ( for c being Element of M st c in dom it holds
it /. c = (f1 /. c) * (f2 /. c) ) );
existence
ex b1 being PartFunc of M,the carrier of V st
( dom b1 = (dom f1) /\ (dom f2) & ( for c being Element of M st c in dom b1 holds
b1 /. c = (f1 /. c) * (f2 /. c) ) )
proof end;
uniqueness
for b1, b2 being PartFunc of M,the carrier of V st dom b1 = (dom f1) /\ (dom f2) & ( for c being Element of M st c in dom b1 holds
b1 /. c = (f1 /. c) * (f2 /. c) ) & dom b2 = (dom f1) /\ (dom f2) & ( for c being Element of M st c in dom b2 holds
b2 /. c = (f1 /. c) * (f2 /. c) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def3 defines (#) VFUNCT_2:def 3 :
for M being non empty set
for V being ComplexNormSpace
for f1 being PartFunc of M, COMPLEX
for f2, b5 being PartFunc of M,the carrier of V holds
( b5 = f1 (#) f2 iff ( dom b5 = (dom f1) /\ (dom f2) & ( for c being Element of M st c in dom b5 holds
b5 /. c = (f1 /. c) * (f2 /. c) ) ) );

definition
let X be non empty set ;
let V be ComplexNormSpace;
let f be PartFunc of X,the carrier of V;
let z be Complex;
deffunc H1( Element of X) -> Element of the carrier of V = z * (f /. $1);
defpred S1[ set ] means $1 in dom f;
set M = dom f;
func z (#) f -> PartFunc of X,the carrier of V means :Def4: :: VFUNCT_2:def 4
( dom it = dom f & ( for x being Element of X st x in dom it holds
it /. x = z * (f /. x) ) );
existence
ex b1 being PartFunc of X,the carrier of V st
( dom b1 = dom f & ( for x being Element of X st x in dom b1 holds
b1 /. x = z * (f /. x) ) )
proof end;
uniqueness
for b1, b2 being PartFunc of X,the carrier of V st dom b1 = dom f & ( for x being Element of X st x in dom b1 holds
b1 /. x = z * (f /. x) ) & dom b2 = dom f & ( for x being Element of X st x in dom b2 holds
b2 /. x = z * (f /. x) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def4 defines (#) VFUNCT_2:def 4 :
for X being non empty set
for V being ComplexNormSpace
for f being PartFunc of X,the carrier of V
for z being Complex
for b5 being PartFunc of X,the carrier of V holds
( b5 = z (#) f iff ( dom b5 = dom f & ( for x being Element of X st x in dom b5 holds
b5 /. x = z * (f /. x) ) ) );

definition
let X be non empty set ;
let V be ComplexNormSpace;
let f be PartFunc of X,the carrier of V;
deffunc H1( Element of X) -> Element of REAL = ||.(f /. $1).||;
deffunc H2( Element of X) -> Element of the carrier of V = - (f /. $1);
defpred S1[ set ] means $1 in dom f;
set M = dom f;
func ||.f.|| -> PartFunc of X, REAL means :Def5: :: VFUNCT_2:def 5
( dom it = dom f & ( for x being Element of X st x in dom it holds
it . x = ||.(f /. x).|| ) );
existence
ex b1 being PartFunc of X, REAL st
( dom b1 = dom f & ( for x being Element of X st x in dom b1 holds
b1 . x = ||.(f /. x).|| ) )
proof end;
uniqueness
for b1, b2 being PartFunc of X, REAL st dom b1 = dom f & ( for x being Element of X st x in dom b1 holds
b1 . x = ||.(f /. x).|| ) & dom b2 = dom f & ( for x being Element of X st x in dom b2 holds
b2 . x = ||.(f /. x).|| ) holds
b1 = b2
proof end;
func - f -> PartFunc of X,the carrier of V means :Def6: :: VFUNCT_2:def 6
( dom it = dom f & ( for x being Element of X st x in dom it holds
it /. x = - (f /. x) ) );
existence
ex b1 being PartFunc of X,the carrier of V st
( dom b1 = dom f & ( for x being Element of X st x in dom b1 holds
b1 /. x = - (f /. x) ) )
proof end;
uniqueness
for b1, b2 being PartFunc of X,the carrier of V st dom b1 = dom f & ( for x being Element of X st x in dom b1 holds
b1 /. x = - (f /. x) ) & dom b2 = dom f & ( for x being Element of X st x in dom b2 holds
b2 /. x = - (f /. x) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def5 defines ||. VFUNCT_2:def 5 :
for X being non empty set
for V being ComplexNormSpace
for f being PartFunc of X,the carrier of V
for b4 being PartFunc of X, REAL holds
( b4 = ||.f.|| iff ( dom b4 = dom f & ( for x being Element of X st x in dom b4 holds
b4 . x = ||.(f /. x).|| ) ) );

:: deftheorem Def6 defines - VFUNCT_2:def 6 :
for X being non empty set
for V being ComplexNormSpace
for f, b4 being PartFunc of X,the carrier of V holds
( b4 = - f iff ( dom b4 = dom f & ( for x being Element of X st x in dom b4 holds
b4 /. x = - (f /. x) ) ) );

theorem :: VFUNCT_2:1  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for M being non empty set
for V being ComplexNormSpace
for f1 being PartFunc of M, COMPLEX
for f2 being PartFunc of M,the carrier of V holds (dom (f1 (#) f2)) \ ((f1 (#) f2) " {(0. V)}) = ((dom f1) \ (f1 " {0})) /\ ((dom f2) \ (f2 " {(0. V)}))
proof end;

theorem :: VFUNCT_2:2  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for M being non empty set
for V being ComplexNormSpace
for f being PartFunc of M,the carrier of V holds
( ||.f.|| " {0} = f " {(0. V)} & (- f) " {(0. V)} = f " {(0. V)} )
proof end;

theorem :: VFUNCT_2:3  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for M being non empty set
for V being ComplexNormSpace
for f being PartFunc of M,the carrier of V
for z being Complex st z <> 0c holds
(z (#) f) " {(0. V)} = f " {(0. V)}
proof end;

theorem :: VFUNCT_2:4  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for M being non empty set
for V being ComplexNormSpace
for f1, f2 being PartFunc of M,the carrier of V holds f1 + f2 = f2 + f1
proof end;

theorem :: VFUNCT_2:5  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for M being non empty set
for V being ComplexNormSpace
for f1, f2, f3 being PartFunc of M,the carrier of V holds (f1 + f2) + f3 = f1 + (f2 + f3)
proof end;

theorem :: VFUNCT_2:6  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for M being non empty set
for V being ComplexNormSpace
for f1, f2 being PartFunc of M, COMPLEX
for f3 being PartFunc of M,the carrier of V holds (f1 (#) f2) (#) f3 = f1 (#) (f2 (#) f3)
proof end;

theorem :: VFUNCT_2:7  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for M being non empty set
for V being ComplexNormSpace
for f3 being PartFunc of M,the carrier of V
for f1, f2 being PartFunc of M, COMPLEX holds (f1 + f2) (#) f3 = (f1 (#) f3) + (f2 (#) f3)
proof end;

theorem :: VFUNCT_2:8  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for M being non empty set
for V being ComplexNormSpace
for f1, f2 being PartFunc of M,the carrier of V
for f3 being PartFunc of M, COMPLEX holds f3 (#) (f1 + f2) = (f3 (#) f1) + (f3 (#) f2)
proof end;

theorem :: VFUNCT_2:9  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for M being non empty set
for V being ComplexNormSpace
for f2 being PartFunc of M,the carrier of V
for z being Complex
for f1 being PartFunc of M, COMPLEX holds z (#) (f1 (#) f2) = (z (#) f1) (#) f2
proof end;

theorem :: VFUNCT_2:10  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for M being non empty set
for V being ComplexNormSpace
for f2 being PartFunc of M,the carrier of V
for z being Complex
for f1 being PartFunc of M, COMPLEX holds z (#) (f1 (#) f2) = f1 (#) (z (#) f2)
proof end;

theorem :: VFUNCT_2:11  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for M being non empty set
for V being ComplexNormSpace
for f3 being PartFunc of M,the carrier of V
for f1, f2 being PartFunc of M, COMPLEX holds (f1 - f2) (#) f3 = (f1 (#) f3) - (f2 (#) f3)
proof end;

theorem :: VFUNCT_2:12  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for M being non empty set
for V being ComplexNormSpace
for f1, f2 being PartFunc of M,the carrier of V
for f3 being PartFunc of M, COMPLEX holds (f3 (#) f1) - (f3 (#) f2) = f3 (#) (f1 - f2)
proof end;

theorem :: VFUNCT_2:13  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for M being non empty set
for V being ComplexNormSpace
for f1, f2 being PartFunc of M,the carrier of V
for z being Complex holds z (#) (f1 + f2) = (z (#) f1) + (z (#) f2)
proof end;

theorem Th14: :: VFUNCT_2:14  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for M being non empty set
for V being ComplexNormSpace
for f being PartFunc of M,the carrier of V
for z1, z2 being Complex holds (z1 * z2) (#) f = z1 (#) (z2 (#) f)
proof end;

theorem :: VFUNCT_2:15  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for M being non empty set
for V being ComplexNormSpace
for f1, f2 being PartFunc of M,the carrier of V
for z being Complex holds z (#) (f1 - f2) = (z (#) f1) - (z (#) f2)
proof end;

theorem :: VFUNCT_2:16  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for M being non empty set
for V being ComplexNormSpace
for f1, f2 being PartFunc of M,the carrier of V holds f1 - f2 = (- 1r ) (#) (f2 - f1)
proof end;

theorem :: VFUNCT_2:17  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for M being non empty set
for V being ComplexNormSpace
for f1, f2, f3 being PartFunc of M,the carrier of V holds f1 - (f2 + f3) = (f1 - f2) - f3
proof end;

theorem Th18: :: VFUNCT_2:18  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for M being non empty set
for V being ComplexNormSpace
for f being PartFunc of M,the carrier of V holds 1r (#) f = f
proof end;

theorem :: VFUNCT_2:19  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for M being non empty set
for V being ComplexNormSpace
for f1, f2, f3 being PartFunc of M,the carrier of V holds f1 - (f2 - f3) = (f1 - f2) + f3
proof end;

theorem :: VFUNCT_2:20  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for M being non empty set
for V being ComplexNormSpace
for f1, f2, f3 being PartFunc of M,the carrier of V holds f1 + (f2 - f3) = (f1 + f2) - f3
proof end;

theorem :: VFUNCT_2:21  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for M being non empty set
for V being ComplexNormSpace
for f2 being PartFunc of M,the carrier of V
for f1 being PartFunc of M, COMPLEX holds ||.(f1 (#) f2).|| = |.f1.| (#) ||.f2.||
proof end;

theorem :: VFUNCT_2:22  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for M being non empty set
for V being ComplexNormSpace
for f being PartFunc of M,the carrier of V
for z being Complex holds ||.(z (#) f).|| = |.z.| (#) ||.f.||
proof end;

theorem Th23: :: VFUNCT_2:23  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for M being non empty set
for V being ComplexNormSpace
for f being PartFunc of M,the carrier of V holds - f = (- 1r ) (#) f
proof end;

theorem Th24: :: VFUNCT_2:24  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for M being non empty set
for V being ComplexNormSpace
for f being PartFunc of M,the carrier of V holds - (- f) = f
proof end;

theorem Th25: :: VFUNCT_2:25  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for M being non empty set
for V being ComplexNormSpace
for f1, f2 being PartFunc of M,the carrier of V holds f1 - f2 = f1 + (- f2)
proof end;

theorem :: VFUNCT_2:26  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for M being non empty set
for V being ComplexNormSpace
for f1, f2 being PartFunc of M,the carrier of V holds f1 - (- f2) = f1 + f2
proof end;

theorem Th27: :: VFUNCT_2:27  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for M being non empty set
for V being ComplexNormSpace
for f1, f2 being PartFunc of M,the carrier of V
for X being set holds
( (f1 + f2) | X = (f1 | X) + (f2 | X) & (f1 + f2) | X = (f1 | X) + f2 & (f1 + f2) | X = f1 + (f2 | X) )
proof end;

theorem :: VFUNCT_2:28  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for M being non empty set
for V being ComplexNormSpace
for f2 being PartFunc of M,the carrier of V
for X being set
for f1 being PartFunc of M, COMPLEX holds
( (f1 (#) f2) | X = (f1 | X) (#) (f2 | X) & (f1 (#) f2) | X = (f1 | X) (#) f2 & (f1 (#) f2) | X = f1 (#) (f2 | X) )
proof end;

theorem Th29: :: VFUNCT_2:29  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for M being non empty set
for V being ComplexNormSpace
for f being PartFunc of M,the carrier of V
for X being set holds
( (- f) | X = - (f | X) & ||.f.|| | X = ||.(f | X).|| )
proof end;

theorem :: VFUNCT_2:30  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for M being non empty set
for V being ComplexNormSpace
for f1, f2 being PartFunc of M,the carrier of V
for X being set holds
( (f1 - f2) | X = (f1 | X) - (f2 | X) & (f1 - f2) | X = (f1 | X) - f2 & (f1 - f2) | X = f1 - (f2 | X) )
proof end;

theorem :: VFUNCT_2:31  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for M being non empty set
for V being ComplexNormSpace
for f being PartFunc of M,the carrier of V
for z being Complex
for X being set holds (z (#) f) | X = z (#) (f | X)
proof end;

theorem Th32: :: VFUNCT_2:32  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for M being non empty set
for V being ComplexNormSpace
for f1, f2 being PartFunc of M,the carrier of V holds
( ( f1 is total & f2 is total implies f1 + f2 is total ) & ( f1 + f2 is total implies ( f1 is total & f2 is total ) ) & ( f1 is total & f2 is total implies f1 - f2 is total ) & ( f1 - f2 is total implies ( f1 is total & f2 is total ) ) )
proof end;

theorem Th33: :: VFUNCT_2:33  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for M being non empty set
for V being ComplexNormSpace
for f2 being PartFunc of M,the carrier of V
for f1 being PartFunc of M, COMPLEX holds
( ( f1 is total & f2 is total ) iff f1 (#) f2 is total )
proof end;

theorem Th34: :: VFUNCT_2:34  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for M being non empty set
for V being ComplexNormSpace
for f being PartFunc of M,the carrier of V
for z being Complex holds
( f is total iff z (#) f is total )
proof end;

theorem Th35: :: VFUNCT_2:35  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for M being non empty set
for V being ComplexNormSpace
for f being PartFunc of M,the carrier of V holds
( f is total iff - f is total )
proof end;

theorem Th36: :: VFUNCT_2:36  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for M being non empty set
for V being ComplexNormSpace
for f being PartFunc of M,the carrier of V holds
( f is total iff ||.f.|| is total )
proof end;

theorem :: VFUNCT_2:37  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for M being non empty set
for V being ComplexNormSpace
for f1, f2 being PartFunc of M,the carrier of V
for x being Element of M st f1 is total & f2 is total holds
( (f1 + f2) /. x = (f1 /. x) + (f2 /. x) & (f1 - f2) /. x = (f1 /. x) - (f2 /. x) )
proof end;

theorem :: VFUNCT_2:38  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for M being non empty set
for V being ComplexNormSpace
for f2 being PartFunc of M,the carrier of V
for f1 being PartFunc of M, COMPLEX
for x being Element of M st f1 is total & f2 is total holds
(f1 (#) f2) /. x = (f1 /. x) * (f2 /. x)
proof end;

theorem :: VFUNCT_2:39  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for M being non empty set
for V being ComplexNormSpace
for f being PartFunc of M,the carrier of V
for z being Complex
for x being Element of M st f is total holds
(z (#) f) /. x = z * (f /. x)
proof end;

theorem :: VFUNCT_2:40  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for M being non empty set
for V being ComplexNormSpace
for f being PartFunc of M,the carrier of V
for x being Element of M st f is total holds
( (- f) /. x = - (f /. x) & ||.f.|| . x = ||.(f /. x).|| )
proof end;

definition
let M be non empty set ;
let V be ComplexNormSpace;
let f be PartFunc of M,the carrier of V;
let Y be set ;
pred f is_bounded_on Y means :Def7: :: VFUNCT_2:def 7
ex r being Real st
for x being Element of M st x in Y /\ (dom f) holds
||.(f /. x).|| <= r;
end;

:: deftheorem Def7 defines is_bounded_on VFUNCT_2:def 7 :
for M being non empty set
for V being ComplexNormSpace
for f being PartFunc of M,the carrier of V
for Y being set holds
( f is_bounded_on Y iff ex r being Real st
for x being Element of M st x in Y /\ (dom f) holds
||.(f /. x).|| <= r );

theorem :: VFUNCT_2:41  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for M being non empty set
for V being ComplexNormSpace
for f being PartFunc of M,the carrier of V
for Y, X being set st Y c= X & f is_bounded_on X holds
f is_bounded_on Y
proof end;

theorem :: VFUNCT_2:42  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for M being non empty set
for V being ComplexNormSpace
for f being PartFunc of M,the carrier of V
for X being set st X misses dom f holds
f is_bounded_on X
proof end;

theorem :: VFUNCT_2:43  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for M being non empty set
for V being ComplexNormSpace
for f being PartFunc of M,the carrier of V
for Y being set holds 0c (#) f is_bounded_on Y
proof end;

theorem Th44: :: VFUNCT_2:44  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for M being non empty set
for V being ComplexNormSpace
for f being PartFunc of M,the carrier of V
for z being Complex
for Y being set st f is_bounded_on Y holds
z (#) f is_bounded_on Y
proof end;

theorem Th45: :: VFUNCT_2:45  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for M being non empty set
for V being ComplexNormSpace
for f being PartFunc of M,the carrier of V
for Y being set st f is_bounded_on Y holds
( ||.f.|| is_bounded_on Y & - f is_bounded_on Y )
proof end;

theorem Th46: :: VFUNCT_2:46  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for M being non empty set
for V being ComplexNormSpace
for f1, f2 being PartFunc of M,the carrier of V
for X, Y being set st f1 is_bounded_on X & f2 is_bounded_on Y holds
f1 + f2 is_bounded_on X /\ Y
proof end;

theorem :: VFUNCT_2:47  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for M being non empty set
for V being ComplexNormSpace
for f2 being PartFunc of M,the carrier of V
for X, Y being set
for f1 being PartFunc of M, COMPLEX st f1 is_bounded_on X & f2 is_bounded_on Y holds
f1 (#) f2 is_bounded_on X /\ Y
proof end;

theorem :: VFUNCT_2:48  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for M being non empty set
for V being ComplexNormSpace
for f1, f2 being PartFunc of M,the carrier of V
for X, Y being set st f1 is_bounded_on X & f2 is_bounded_on Y holds
f1 - f2 is_bounded_on X /\ Y
proof end;

theorem :: VFUNCT_2:49  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for M being non empty set
for V being ComplexNormSpace
for f being PartFunc of M,the carrier of V
for X, Y being set st f is_bounded_on X & f is_bounded_on Y holds
f is_bounded_on X \/ Y
proof end;

theorem :: VFUNCT_2:50  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for M being non empty set
for V being ComplexNormSpace
for f1, f2 being PartFunc of M,the carrier of V
for X, Y being set st f1 is_constant_on X & f2 is_constant_on Y holds
( f1 + f2 is_constant_on X /\ Y & f1 - f2 is_constant_on X /\ Y )
proof end;

theorem :: VFUNCT_2:51  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for M being non empty set
for V being ComplexNormSpace
for f2 being PartFunc of M,the carrier of V
for X, Y being set
for f1 being PartFunc of M, COMPLEX st f1 is_constant_on X & f2 is_constant_on Y holds
f1 (#) f2 is_constant_on X /\ Y
proof end;

theorem Th52: :: VFUNCT_2:52  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for M being non empty set
for V being ComplexNormSpace
for f being PartFunc of M,the carrier of V
for z being Complex
for Y being set st f is_constant_on Y holds
z (#) f is_constant_on Y
proof end;

theorem Th53: :: VFUNCT_2:53  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for M being non empty set
for V being ComplexNormSpace
for f being PartFunc of M,the carrier of V
for Y being set st f is_constant_on Y holds
( ||.f.|| is_constant_on Y & - f is_constant_on Y )
proof end;

theorem Th54: :: VFUNCT_2:54  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for M being non empty set
for V being ComplexNormSpace
for f being PartFunc of M,the carrier of V
for Y being set st f is_constant_on Y holds
f is_bounded_on Y
proof end;

theorem :: VFUNCT_2:55  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for M being non empty set
for V being ComplexNormSpace
for f being PartFunc of M,the carrier of V
for Y being set st f is_constant_on Y holds
( ( for z being Complex holds z (#) f is_bounded_on Y ) & - f is_bounded_on Y & ||.f.|| is_bounded_on Y )
proof end;

theorem :: VFUNCT_2:56  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for M being non empty set
for V being ComplexNormSpace
for f1, f2 being PartFunc of M,the carrier of V
for X, Y being set st f1 is_bounded_on X & f2 is_constant_on Y holds
f1 + f2 is_bounded_on X /\ Y
proof end;

theorem :: VFUNCT_2:57  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for M being non empty set
for V being ComplexNormSpace
for f1, f2 being PartFunc of M,the carrier of V
for X, Y being set st f1 is_bounded_on X & f2 is_constant_on Y holds
( f1 - f2 is_bounded_on X /\ Y & f2 - f1 is_bounded_on X /\ Y )
proof end;