let C be non empty set ; :: thesis: for f being PartFunc of C,ExtREAL
for x being Element of C st (max- f) . x = - (f . x) holds
(max+ f) . x = 0.
let f be PartFunc of C,ExtREAL; :: thesis: for x being Element of C st (max- f) . x = - (f . x) holds
(max+ f) . x = 0.
let x be Element of C; :: thesis: ( (max- f) . x = - (f . x) implies (max+ f) . x = 0. )
A1: dom (max+ f) = dom f by Def2;

per cases ( x in dom f or not x in dom f ) ;

suppose A2: x in dom f ; :: thesis: ( (max- f) . x = - (f . x) implies (max+ f) . x = 0. )

assume A3: (max- f) . x = - (f . x) ; :: thesis: (max+ f) . x = 0.
A4: x in dom (max+ f) by A2, Def2;
A5: x in dom (max- f) by A2, Def3;
A6: (max+ f) . x = max ((f . x),0.) by A4, Def2;
(max- f) . x = max ((- (f . x)),0.) by A5, Def3;
then - (- (f . x)) <= - 0. by A3, XXREAL_0:def 10;
hence (max+ f) . x = 0. by A6, XXREAL_0:def 10; :: thesis: verum

end;

suppose not x in dom f ; :: thesis: ( (max- f) . x = - (f . x) implies (max+ f) . x = 0. )

hence ( (max- f) . x = - (f . x) implies (max+ f) . x = 0. ) by A1, FUNCT_1:def 2; :: thesis: verum

end;

end;