let X be non empty set ; for Y being RealNormSpace
for f, g, h being Point of (R_NormSpace_of_BoundedFunctions (X,Y))
for f9, g9, h9 being bounded Function of X, the carrier of Y st f9 = f & g9 = g & h9 = h holds
( h = f - g iff for x being Element of X holds h9 . x = (f9 . x) - (g9 . x) )
let Y be RealNormSpace; for f, g, h being Point of (R_NormSpace_of_BoundedFunctions (X,Y))
for f9, g9, h9 being bounded Function of X, the carrier of Y st f9 = f & g9 = g & h9 = h holds
( h = f - g iff for x being Element of X holds h9 . x = (f9 . x) - (g9 . x) )
let f, g, h be Point of (R_NormSpace_of_BoundedFunctions (X,Y)); for f9, g9, h9 being bounded Function of X, the carrier of Y st f9 = f & g9 = g & h9 = h holds
( h = f - g iff for x being Element of X holds h9 . x = (f9 . x) - (g9 . x) )
let f9, g9, h9 be bounded Function of X, the carrier of Y; ( f9 = f & g9 = g & h9 = h implies ( h = f - g iff for x being Element of X holds h9 . x = (f9 . x) - (g9 . x) ) )
assume A1:
( f9 = f & g9 = g & h9 = h )
; ( h = f - g iff for x being Element of X holds h9 . x = (f9 . x) - (g9 . x) )
A2:
now ( ( for x being Element of X holds h9 . x = (f9 . x) - (g9 . x) ) implies f - g = h )end;
now ( h = f - g implies for x being Element of X holds h9 . x = (f9 . x) - (g9 . x) )end;
hence
( h = f - g iff for x being Element of X holds h9 . x = (f9 . x) - (g9 . x) )
by A2; verum