let C be non empty set ; :: thesis: for f being PartFunc of C,ExtREAL
for x being Element of C st 0. < (max- f) . x holds
(max+ f) . x = 0.
let f be PartFunc of C,ExtREAL; :: thesis: for x being Element of C st 0. < (max- f) . x holds
(max+ f) . x = 0.
let x be Element of C; :: thesis: ( 0. < (max- f) . x implies (max+ f) . x = 0. )
A1: dom (max- f) = dom f by Def3;

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

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

assume A3: 0. < (max- f) . x ; :: thesis: (max+ f) . x = 0.
A4: x in dom (max- f) by A2, Def3;
A5: x in dom (max+ f) by A2, Def2;
(max- f) . x = max ((- (f . x)),0.) by A4, Def3;
then - (- (f . x)) < - 0. by A3, XXREAL_0:28;
then max ((f . x),0.) = 0. by XXREAL_0:def 10;
hence (max+ f) . x = 0. by A5, Def2; :: thesis: verum

end;

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

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

end;

end;