let C be non empty set ; :: thesis: for f being PartFunc of C,ExtREAL
for x being Element of C holds
( ( (max+ f) . x = f . x or (max+ f) . x = 0. ) & ( (max- f) . x = - (f . x) or (max- f) . x = 0. ) )
let f be PartFunc of C,ExtREAL; :: thesis: for x being Element of C holds
( ( (max+ f) . x = f . x or (max+ f) . x = 0. ) & ( (max- f) . x = - (f . x) or (max- f) . x = 0. ) )
let x be Element of C; :: thesis: ( ( (max+ f) . x = f . x or (max+ f) . x = 0. ) & ( (max- f) . x = - (f . x) or (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 or (max+ f) . x = 0. ) & ( (max- f) . x = - (f . x) or (max- f) . x = 0. ) )

then A3: x in dom (max+ f) by Def2;
A4: x in dom (max- f) by A2, Def3;
A5: (max+ f) . x = max ((f . x),0.) by A3, Def2;
(max- f) . x = max ((- (f . x)),0.) by A4, Def3;
hence ( ( (max+ f) . x = f . x or (max+ f) . x = 0. ) & ( (max- f) . x = - (f . x) or (max- f) . x = 0. ) ) by A5, XXREAL_0:16; :: thesis: verum

end;

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

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

end;

end;