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 & dom (max+ f) = dom f ) by Def2, Def3;

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.
( x in dom (max+ f) & x in dom (max- f) ) by A2, Def2, Def3;
then A4: ( (max+ f) . x = max (f . x),0. & (max- f) . x = max (- (f . x)),0. ) by Def2, Def3;
then 0. <= f . x by A3, XXREAL_0:def 10;
then - (f . x) <= - 0. by SUPINF_2:16;
hence (max- f) . x = 0. by A4, EXTREAL1:24, 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 4; :: thesis: verum

end;

end;