let NORM1, NORM2 be Function of (),REAL; :: thesis: ( ( for x being object st x in BoundedLinearOperators (X,Y) holds
NORM1 . x = upper_bound (PreNorms (modetrans (x,X,Y))) ) & ( for x being object st x in BoundedLinearOperators (X,Y) holds
NORM2 . x = upper_bound (PreNorms (modetrans (x,X,Y))) ) implies NORM1 = NORM2 )

assume that
A2: for x being object st x in BoundedLinearOperators (X,Y) holds
NORM1 . x = upper_bound (PreNorms (modetrans (x,X,Y))) and
A3: for x being object st x in BoundedLinearOperators (X,Y) holds
NORM2 . x = upper_bound (PreNorms (modetrans (x,X,Y))) ; :: thesis: NORM1 = NORM2
A4: for z being object st z in BoundedLinearOperators (X,Y) holds
NORM1 . z = NORM2 . z
proof
let z be object ; :: thesis: ( z in BoundedLinearOperators (X,Y) implies NORM1 . z = NORM2 . z )
assume A5: z in BoundedLinearOperators (X,Y) ; :: thesis: NORM1 . z = NORM2 . z
NORM1 . z = upper_bound (PreNorms (modetrans (z,X,Y))) by A2, A5;
hence NORM1 . z = NORM2 . z by A3, A5; :: thesis: verum
end;
A6: dom NORM2 = BoundedLinearOperators (X,Y) by FUNCT_2:def 1;
dom NORM1 = BoundedLinearOperators (X,Y) by FUNCT_2:def 1;
hence NORM1 = NORM2 by A6, A4; :: thesis: verum