let X, Y be ComplexNormSpace; :: thesis: for f, g being Point of (C_NormSpace_of_BoundedLinearOperators (X,Y))

for c being Complex holds

( ( ||.f.|| = 0 implies f = 0. (C_NormSpace_of_BoundedLinearOperators (X,Y)) ) & ( f = 0. (C_NormSpace_of_BoundedLinearOperators (X,Y)) implies ||.f.|| = 0 ) & ||.(c * f).|| = |.c.| * ||.f.|| & ||.(f + g).|| <= ||.f.|| + ||.g.|| )

let f, g be Point of (C_NormSpace_of_BoundedLinearOperators (X,Y)); :: thesis: for c being Complex holds

( ( ||.f.|| = 0 implies f = 0. (C_NormSpace_of_BoundedLinearOperators (X,Y)) ) & ( f = 0. (C_NormSpace_of_BoundedLinearOperators (X,Y)) implies ||.f.|| = 0 ) & ||.(c * f).|| = |.c.| * ||.f.|| & ||.(f + g).|| <= ||.f.|| + ||.g.|| )

let c be Complex; :: thesis: ( ( ||.f.|| = 0 implies f = 0. (C_NormSpace_of_BoundedLinearOperators (X,Y)) ) & ( f = 0. (C_NormSpace_of_BoundedLinearOperators (X,Y)) implies ||.f.|| = 0 ) & ||.(c * f).|| = |.c.| * ||.f.|| & ||.(f + g).|| <= ||.f.|| + ||.g.|| )

for c being Complex holds

( ( ||.f.|| = 0 implies f = 0. (C_NormSpace_of_BoundedLinearOperators (X,Y)) ) & ( f = 0. (C_NormSpace_of_BoundedLinearOperators (X,Y)) implies ||.f.|| = 0 ) & ||.(c * f).|| = |.c.| * ||.f.|| & ||.(f + g).|| <= ||.f.|| + ||.g.|| )

let f, g be Point of (C_NormSpace_of_BoundedLinearOperators (X,Y)); :: thesis: for c being Complex holds

( ( ||.f.|| = 0 implies f = 0. (C_NormSpace_of_BoundedLinearOperators (X,Y)) ) & ( f = 0. (C_NormSpace_of_BoundedLinearOperators (X,Y)) implies ||.f.|| = 0 ) & ||.(c * f).|| = |.c.| * ||.f.|| & ||.(f + g).|| <= ||.f.|| + ||.g.|| )

let c be Complex; :: thesis: ( ( ||.f.|| = 0 implies f = 0. (C_NormSpace_of_BoundedLinearOperators (X,Y)) ) & ( f = 0. (C_NormSpace_of_BoundedLinearOperators (X,Y)) implies ||.f.|| = 0 ) & ||.(c * f).|| = |.c.| * ||.f.|| & ||.(f + g).|| <= ||.f.|| + ||.g.|| )

A1: now :: thesis: ( f = 0. (C_NormSpace_of_BoundedLinearOperators (X,Y)) implies ||.f.|| = 0 )

A9:
||.(f + g).|| <= ||.f.|| + ||.g.||
assume A2:
f = 0. (C_NormSpace_of_BoundedLinearOperators (X,Y))
; :: thesis: ||.f.|| = 0

thus ||.f.|| = 0 :: thesis: verum

end;thus ||.f.|| = 0 :: thesis: verum

proof

reconsider g = f as Lipschitzian LinearOperator of X,Y by Def7;

set z = the carrier of X --> (0. Y);

reconsider z = the carrier of X --> (0. Y) as Function of the carrier of X, the carrier of Y ;

consider r0 being object such that

A3: r0 in PreNorms g by XBOOLE_0:def 1;

reconsider r0 = r0 as Real by A3;

A4: ( ( for s being Real st s in PreNorms g holds

s <= 0 ) implies upper_bound (PreNorms g) <= 0 ) by SEQ_4:45;

A5: ( not PreNorms g is empty & PreNorms g is bounded_above ) by Th26;

A6: z = g by A2, Th30;

then upper_bound (PreNorms g) = 0 by A7, A5, A3, A4, SEQ_4:def 1;

then (BoundedLinearOperatorsNorm (X,Y)) . f = 0 by Th29;

hence ||.f.|| = 0 ; :: thesis: verum

end;set z = the carrier of X --> (0. Y);

reconsider z = the carrier of X --> (0. Y) as Function of the carrier of X, the carrier of Y ;

consider r0 being object such that

A3: r0 in PreNorms g by XBOOLE_0:def 1;

reconsider r0 = r0 as Real by A3;

A4: ( ( for s being Real st s in PreNorms g holds

s <= 0 ) implies upper_bound (PreNorms g) <= 0 ) by SEQ_4:45;

A5: ( not PreNorms g is empty & PreNorms g is bounded_above ) by Th26;

A6: z = g by A2, Th30;

A7: now :: thesis: for r being Real st r in PreNorms g holds

( 0 <= r & r <= 0 )

then
0 <= r0
by A3;( 0 <= r & r <= 0 )

let r be Real; :: thesis: ( r in PreNorms g implies ( 0 <= r & r <= 0 ) )

assume r in PreNorms g ; :: thesis: ( 0 <= r & r <= 0 )

then consider t being VECTOR of X such that

A8: r = ||.(g . t).|| and

||.t.|| <= 1 ;

||.(g . t).|| = ||.(0. Y).|| by A6, FUNCOP_1:7

.= 0 by NORMSP_0:def 6 ;

hence ( 0 <= r & r <= 0 ) by A8; :: thesis: verum

end;assume r in PreNorms g ; :: thesis: ( 0 <= r & r <= 0 )

then consider t being VECTOR of X such that

A8: r = ||.(g . t).|| and

||.t.|| <= 1 ;

||.(g . t).|| = ||.(0. Y).|| by A6, FUNCOP_1:7

.= 0 by NORMSP_0:def 6 ;

hence ( 0 <= r & r <= 0 ) by A8; :: thesis: verum

then upper_bound (PreNorms g) = 0 by A7, A5, A3, A4, SEQ_4:def 1;

then (BoundedLinearOperatorsNorm (X,Y)) . f = 0 by Th29;

hence ||.f.|| = 0 ; :: thesis: verum

proof

A19:
||.(c * f).|| = |.c.| * ||.f.||
reconsider f1 = f, g1 = g, h1 = f + g as Lipschitzian LinearOperator of X,Y by Def7;

A10: ( ( for s being Real st s in PreNorms h1 holds

s <= ||.f.|| + ||.g.|| ) implies upper_bound (PreNorms h1) <= ||.f.|| + ||.g.|| ) by SEQ_4:45;

hence ||.(f + g).|| <= ||.f.|| + ||.g.|| by A18, A10; :: thesis: verum

end;A10: ( ( for s being Real st s in PreNorms h1 holds

s <= ||.f.|| + ||.g.|| ) implies upper_bound (PreNorms h1) <= ||.f.|| + ||.g.|| ) by SEQ_4:45;

A11: now :: thesis: for t being VECTOR of X st ||.t.|| <= 1 holds

||.(h1 . t).|| <= ||.f.|| + ||.g.||

||.(h1 . t).|| <= ||.f.|| + ||.g.||

let t be VECTOR of X; :: thesis: ( ||.t.|| <= 1 implies ||.(h1 . t).|| <= ||.f.|| + ||.g.|| )

assume A12: ||.t.|| <= 1 ; :: thesis: ||.(h1 . t).|| <= ||.f.|| + ||.g.||

0 <= ||.g.|| by Th32;

then A13: ||.g.|| * ||.t.|| <= ||.g.|| * 1 by A12, XREAL_1:64;

0 <= ||.f.|| by Th32;

then ||.f.|| * ||.t.|| <= ||.f.|| * 1 by A12, XREAL_1:64;

then A14: (||.f.|| * ||.t.||) + (||.g.|| * ||.t.||) <= (||.f.|| * 1) + (||.g.|| * 1) by A13, XREAL_1:7;

A15: ||.((f1 . t) + (g1 . t)).|| <= ||.(f1 . t).|| + ||.(g1 . t).|| by CLVECT_1:def 13;

A16: ||.(g1 . t).|| <= ||.g.|| * ||.t.|| by Th31;

||.(f1 . t).|| <= ||.f.|| * ||.t.|| by Th31;

then ||.(f1 . t).|| + ||.(g1 . t).|| <= (||.f.|| * ||.t.||) + (||.g.|| * ||.t.||) by A16, XREAL_1:7;

then A17: ||.(f1 . t).|| + ||.(g1 . t).|| <= ||.f.|| + ||.g.|| by A14, XXREAL_0:2;

||.(h1 . t).|| = ||.((f1 . t) + (g1 . t)).|| by Th34;

hence ||.(h1 . t).|| <= ||.f.|| + ||.g.|| by A15, A17, XXREAL_0:2; :: thesis: verum

end;assume A12: ||.t.|| <= 1 ; :: thesis: ||.(h1 . t).|| <= ||.f.|| + ||.g.||

0 <= ||.g.|| by Th32;

then A13: ||.g.|| * ||.t.|| <= ||.g.|| * 1 by A12, XREAL_1:64;

0 <= ||.f.|| by Th32;

then ||.f.|| * ||.t.|| <= ||.f.|| * 1 by A12, XREAL_1:64;

then A14: (||.f.|| * ||.t.||) + (||.g.|| * ||.t.||) <= (||.f.|| * 1) + (||.g.|| * 1) by A13, XREAL_1:7;

A15: ||.((f1 . t) + (g1 . t)).|| <= ||.(f1 . t).|| + ||.(g1 . t).|| by CLVECT_1:def 13;

A16: ||.(g1 . t).|| <= ||.g.|| * ||.t.|| by Th31;

||.(f1 . t).|| <= ||.f.|| * ||.t.|| by Th31;

then ||.(f1 . t).|| + ||.(g1 . t).|| <= (||.f.|| * ||.t.||) + (||.g.|| * ||.t.||) by A16, XREAL_1:7;

then A17: ||.(f1 . t).|| + ||.(g1 . t).|| <= ||.f.|| + ||.g.|| by A14, XXREAL_0:2;

||.(h1 . t).|| = ||.((f1 . t) + (g1 . t)).|| by Th34;

hence ||.(h1 . t).|| <= ||.f.|| + ||.g.|| by A15, A17, XXREAL_0:2; :: thesis: verum

A18: now :: thesis: for r being Real st r in PreNorms h1 holds

r <= ||.f.|| + ||.g.||

(BoundedLinearOperatorsNorm (X,Y)) . (f + g) = upper_bound (PreNorms h1)
by Th29;r <= ||.f.|| + ||.g.||

let r be Real; :: thesis: ( r in PreNorms h1 implies r <= ||.f.|| + ||.g.|| )

assume r in PreNorms h1 ; :: thesis: r <= ||.f.|| + ||.g.||

then ex t being VECTOR of X st

( r = ||.(h1 . t).|| & ||.t.|| <= 1 ) ;

hence r <= ||.f.|| + ||.g.|| by A11; :: thesis: verum

end;assume r in PreNorms h1 ; :: thesis: r <= ||.f.|| + ||.g.||

then ex t being VECTOR of X st

( r = ||.(h1 . t).|| & ||.t.|| <= 1 ) ;

hence r <= ||.f.|| + ||.g.|| by A11; :: thesis: verum

hence ||.(f + g).|| <= ||.f.|| + ||.g.|| by A18, A10; :: thesis: verum

proof

reconsider f1 = f, h1 = c * f as Lipschitzian LinearOperator of X,Y by Def7;

A20: ( ( for s being Real st s in PreNorms h1 holds

s <= |.c.| * ||.f.|| ) implies upper_bound (PreNorms h1) <= |.c.| * ||.f.|| ) by SEQ_4:45;

then ||.(c * f).|| <= |.c.| * ||.f.|| by A27, A20;

hence ||.(c * f).|| = |.c.| * ||.f.|| by A28, XXREAL_0:1; :: thesis: verum

end;A20: ( ( for s being Real st s in PreNorms h1 holds

s <= |.c.| * ||.f.|| ) implies upper_bound (PreNorms h1) <= |.c.| * ||.f.|| ) by SEQ_4:45;

A21: now :: thesis: for t being VECTOR of X st ||.t.|| <= 1 holds

||.(h1 . t).|| <= |.c.| * ||.f.||

||.(h1 . t).|| <= |.c.| * ||.f.||

A22:
0 <= ||.f.||
by Th32;

let t be VECTOR of X; :: thesis: ( ||.t.|| <= 1 implies ||.(h1 . t).|| <= |.c.| * ||.f.|| )

assume ||.t.|| <= 1 ; :: thesis: ||.(h1 . t).|| <= |.c.| * ||.f.||

then A23: ||.f.|| * ||.t.|| <= ||.f.|| * 1 by A22, XREAL_1:64;

||.(f1 . t).|| <= ||.f.|| * ||.t.|| by Th31;

then A24: ||.(f1 . t).|| <= ||.f.|| by A23, XXREAL_0:2;

A25: ||.(c * (f1 . t)).|| = |.c.| * ||.(f1 . t).|| by CLVECT_1:def 13;

A26: 0 <= |.c.| by COMPLEX1:46;

||.(h1 . t).|| = ||.(c * (f1 . t)).|| by Th35;

hence ||.(h1 . t).|| <= |.c.| * ||.f.|| by A25, A24, A26, XREAL_1:64; :: thesis: verum

end;let t be VECTOR of X; :: thesis: ( ||.t.|| <= 1 implies ||.(h1 . t).|| <= |.c.| * ||.f.|| )

assume ||.t.|| <= 1 ; :: thesis: ||.(h1 . t).|| <= |.c.| * ||.f.||

then A23: ||.f.|| * ||.t.|| <= ||.f.|| * 1 by A22, XREAL_1:64;

||.(f1 . t).|| <= ||.f.|| * ||.t.|| by Th31;

then A24: ||.(f1 . t).|| <= ||.f.|| by A23, XXREAL_0:2;

A25: ||.(c * (f1 . t)).|| = |.c.| * ||.(f1 . t).|| by CLVECT_1:def 13;

A26: 0 <= |.c.| by COMPLEX1:46;

||.(h1 . t).|| = ||.(c * (f1 . t)).|| by Th35;

hence ||.(h1 . t).|| <= |.c.| * ||.f.|| by A25, A24, A26, XREAL_1:64; :: thesis: verum

A27: now :: thesis: for r being Real st r in PreNorms h1 holds

r <= |.c.| * ||.f.||

r <= |.c.| * ||.f.||

let r be Real; :: thesis: ( r in PreNorms h1 implies r <= |.c.| * ||.f.|| )

assume r in PreNorms h1 ; :: thesis: r <= |.c.| * ||.f.||

then ex t being VECTOR of X st

( r = ||.(h1 . t).|| & ||.t.|| <= 1 ) ;

hence r <= |.c.| * ||.f.|| by A21; :: thesis: verum

end;assume r in PreNorms h1 ; :: thesis: r <= |.c.| * ||.f.||

then ex t being VECTOR of X st

( r = ||.(h1 . t).|| & ||.t.|| <= 1 ) ;

hence r <= |.c.| * ||.f.|| by A21; :: thesis: verum

A28: now :: thesis: ( ( c <> 0c & |.c.| * ||.f.|| <= ||.(c * f).|| ) or ( c = 0c & ||.(c * f).|| = |.c.| * ||.f.|| ) )end;

(BoundedLinearOperatorsNorm (X,Y)) . (c * f) = upper_bound (PreNorms h1)
by Th29;per cases
( c <> 0c or c = 0c )
;

end;

case A29:
c <> 0c
; :: thesis: |.c.| * ||.f.|| <= ||.(c * f).||

s <= (|.c.| ") * ||.(c * f).|| ) implies upper_bound (PreNorms f1) <= (|.c.| ") * ||.(c * f).|| ) by SEQ_4:45;

A39: 0 <= |.c.| by COMPLEX1:46;

(BoundedLinearOperatorsNorm (X,Y)) . f = upper_bound (PreNorms f1) by Th29;

then ||.f.|| <= (|.c.| ") * ||.(c * f).|| by A37, A38;

then |.c.| * ||.f.|| <= |.c.| * ((|.c.| ") * ||.(c * f).||) by A39, XREAL_1:64;

then A40: |.c.| * ||.f.|| <= (|.c.| * (|.c.| ")) * ||.(c * f).|| ;

|.c.| <> 0 by A29, COMPLEX1:47;

then |.c.| * ||.f.|| <= 1 * ||.(c * f).|| by A40, XCMPLX_0:def 7;

hence |.c.| * ||.f.|| <= ||.(c * f).|| ; :: thesis: verum

end;

A30: now :: thesis: for t being VECTOR of X st ||.t.|| <= 1 holds

||.(f1 . t).|| <= (|.c.| ") * ||.(c * f).||

||.(f1 . t).|| <= (|.c.| ") * ||.(c * f).||

A31:
0 <= ||.(c * f).||
by Th32;

let t be VECTOR of X; :: thesis: ( ||.t.|| <= 1 implies ||.(f1 . t).|| <= (|.c.| ") * ||.(c * f).|| )

assume ||.t.|| <= 1 ; :: thesis: ||.(f1 . t).|| <= (|.c.| ") * ||.(c * f).||

then A32: ||.(c * f).|| * ||.t.|| <= ||.(c * f).|| * 1 by A31, XREAL_1:64;

||.(h1 . t).|| <= ||.(c * f).|| * ||.t.|| by Th31;

then A33: ||.(h1 . t).|| <= ||.(c * f).|| by A32, XXREAL_0:2;

h1 . t = c * (f1 . t) by Th35;

then A34: (c ") * (h1 . t) = ((c ") * c) * (f1 . t) by CLVECT_1:def 4

.= 1r * (f1 . t) by A29, COMPLEX1:def 4, XCMPLX_0:def 7

.= f1 . t by CLVECT_1:def 5 ;

A35: |.(c ").| = |.c.| " by COMPLEX1:66;

A36: 0 <= |.(c ").| by COMPLEX1:46;

||.((c ") * (h1 . t)).|| = |.(c ").| * ||.(h1 . t).|| by CLVECT_1:def 13;

hence ||.(f1 . t).|| <= (|.c.| ") * ||.(c * f).|| by A34, A33, A36, A35, XREAL_1:64; :: thesis: verum

end;let t be VECTOR of X; :: thesis: ( ||.t.|| <= 1 implies ||.(f1 . t).|| <= (|.c.| ") * ||.(c * f).|| )

assume ||.t.|| <= 1 ; :: thesis: ||.(f1 . t).|| <= (|.c.| ") * ||.(c * f).||

then A32: ||.(c * f).|| * ||.t.|| <= ||.(c * f).|| * 1 by A31, XREAL_1:64;

||.(h1 . t).|| <= ||.(c * f).|| * ||.t.|| by Th31;

then A33: ||.(h1 . t).|| <= ||.(c * f).|| by A32, XXREAL_0:2;

h1 . t = c * (f1 . t) by Th35;

then A34: (c ") * (h1 . t) = ((c ") * c) * (f1 . t) by CLVECT_1:def 4

.= 1r * (f1 . t) by A29, COMPLEX1:def 4, XCMPLX_0:def 7

.= f1 . t by CLVECT_1:def 5 ;

A35: |.(c ").| = |.c.| " by COMPLEX1:66;

A36: 0 <= |.(c ").| by COMPLEX1:46;

||.((c ") * (h1 . t)).|| = |.(c ").| * ||.(h1 . t).|| by CLVECT_1:def 13;

hence ||.(f1 . t).|| <= (|.c.| ") * ||.(c * f).|| by A34, A33, A36, A35, XREAL_1:64; :: thesis: verum

A37: now :: thesis: for r being Real st r in PreNorms f1 holds

r <= (|.c.| ") * ||.(c * f).||

A38:
( ( for s being Real st s in PreNorms f1 holds r <= (|.c.| ") * ||.(c * f).||

let r be Real; :: thesis: ( r in PreNorms f1 implies r <= (|.c.| ") * ||.(c * f).|| )

assume r in PreNorms f1 ; :: thesis: r <= (|.c.| ") * ||.(c * f).||

then ex t being VECTOR of X st

( r = ||.(f1 . t).|| & ||.t.|| <= 1 ) ;

hence r <= (|.c.| ") * ||.(c * f).|| by A30; :: thesis: verum

end;assume r in PreNorms f1 ; :: thesis: r <= (|.c.| ") * ||.(c * f).||

then ex t being VECTOR of X st

( r = ||.(f1 . t).|| & ||.t.|| <= 1 ) ;

hence r <= (|.c.| ") * ||.(c * f).|| by A30; :: thesis: verum

s <= (|.c.| ") * ||.(c * f).|| ) implies upper_bound (PreNorms f1) <= (|.c.| ") * ||.(c * f).|| ) by SEQ_4:45;

A39: 0 <= |.c.| by COMPLEX1:46;

(BoundedLinearOperatorsNorm (X,Y)) . f = upper_bound (PreNorms f1) by Th29;

then ||.f.|| <= (|.c.| ") * ||.(c * f).|| by A37, A38;

then |.c.| * ||.f.|| <= |.c.| * ((|.c.| ") * ||.(c * f).||) by A39, XREAL_1:64;

then A40: |.c.| * ||.f.|| <= (|.c.| * (|.c.| ")) * ||.(c * f).|| ;

|.c.| <> 0 by A29, COMPLEX1:47;

then |.c.| * ||.f.|| <= 1 * ||.(c * f).|| by A40, XCMPLX_0:def 7;

hence |.c.| * ||.f.|| <= ||.(c * f).|| ; :: thesis: verum

case A41:
c = 0c
; :: thesis: ||.(c * f).|| = |.c.| * ||.f.||

reconsider fz = f as VECTOR of (C_VectorSpace_of_BoundedLinearOperators (X,Y)) ;

c * f = (Mult_ ((BoundedLinearOperators (X,Y)),(C_VectorSpace_of_LinearOperators (X,Y)))) . [c,f] by CLVECT_1:def 1

.= c * fz by CLVECT_1:def 1

.= 0. (C_VectorSpace_of_BoundedLinearOperators (X,Y)) by A41, CLVECT_1:1

.= 0. (C_NormSpace_of_BoundedLinearOperators (X,Y)) ;

hence ||.(c * f).|| = |.c.| * ||.f.|| by A41, Th33, COMPLEX1:44; :: thesis: verum

end;c * f = (Mult_ ((BoundedLinearOperators (X,Y)),(C_VectorSpace_of_LinearOperators (X,Y)))) . [c,f] by CLVECT_1:def 1

.= c * fz by CLVECT_1:def 1

.= 0. (C_VectorSpace_of_BoundedLinearOperators (X,Y)) by A41, CLVECT_1:1

.= 0. (C_NormSpace_of_BoundedLinearOperators (X,Y)) ;

hence ||.(c * f).|| = |.c.| * ||.f.|| by A41, Th33, COMPLEX1:44; :: thesis: verum

then ||.(c * f).|| <= |.c.| * ||.f.|| by A27, A20;

hence ||.(c * f).|| = |.c.| * ||.f.|| by A28, XXREAL_0:1; :: thesis: verum

now :: thesis: ( ||.f.|| = 0 implies f = 0. (C_NormSpace_of_BoundedLinearOperators (X,Y)) )

hence
( ( ||.f.|| = 0 implies f = 0. (C_NormSpace_of_BoundedLinearOperators (X,Y)) ) & ( f = 0. (C_NormSpace_of_BoundedLinearOperators (X,Y)) implies ||.f.|| = 0 ) & ||.(c * f).|| = |.c.| * ||.f.|| & ||.(f + g).|| <= ||.f.|| + ||.g.|| )
by A1, A19, A9; :: thesis: verumreconsider g = f as Lipschitzian LinearOperator of X,Y by Def7;

set z = the carrier of X --> (0. Y);

reconsider z = the carrier of X --> (0. Y) as Function of the carrier of X, the carrier of Y ;

assume A42: ||.f.|| = 0 ; :: thesis: f = 0. (C_NormSpace_of_BoundedLinearOperators (X,Y))

hence f = 0. (C_NormSpace_of_BoundedLinearOperators (X,Y)) by Th30; :: thesis: verum

end;set z = the carrier of X --> (0. Y);

reconsider z = the carrier of X --> (0. Y) as Function of the carrier of X, the carrier of Y ;

assume A42: ||.f.|| = 0 ; :: thesis: f = 0. (C_NormSpace_of_BoundedLinearOperators (X,Y))

now :: thesis: for t being VECTOR of X holds g . t = z . t

then
g = z
by FUNCT_2:63;let t be VECTOR of X; :: thesis: g . t = z . t

||.(g . t).|| <= ||.f.|| * ||.t.|| by Th31;

then ||.(g . t).|| = 0 by A42, CLVECT_1:105;

hence g . t = 0. Y by NORMSP_0:def 5

.= z . t by FUNCOP_1:7 ;

:: thesis: verum

end;||.(g . t).|| <= ||.f.|| * ||.t.|| by Th31;

then ||.(g . t).|| = 0 by A42, CLVECT_1:105;

hence g . t = 0. Y by NORMSP_0:def 5

.= z . t by FUNCOP_1:7 ;

:: thesis: verum

hence f = 0. (C_NormSpace_of_BoundedLinearOperators (X,Y)) by Th30; :: thesis: verum