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 Def2;

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, Def2;
A5: x in dom (max- f) by A2, Def3;
A6: (max+ f) . x = max (f . x),0. by A4, Def2;
A7: ( not f . x <= 0. or not 0. <= 0. ) by A3, A6, XXREAL_0:28;
A8: max (- (f . x)),0. = 0. by A7, XXREAL_0:def 10;
thus (max- f) . x = 0. by A5, A8, Def3; :: thesis: verum

end;

suppose A9: not x in dom f ; :: thesis: ( 0. < (max+ f) . x implies (max- f) . x = 0. )

thus ( 0. < (max+ f) . x implies (max- f) . x = 0. ) by A1, A9, FUNCT_1:def 4; :: thesis: verum

end;

end;