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. ) )

A3: x in dom (max+ f) by A2, Def2;
A4: x in dom (max- f) by A2, Def3;
A5: (max+ f) . x = max (f . x),0. by A3, Def2;
A6: (max- f) . x = max (- (f . x)),0. by A4, Def3;
thus ( ( (max+ f) . x = f . x or (max+ f) . x = 0. ) & ( (max- f) . x = - (f . x) or (max- f) . x = 0. ) ) by A5, A6, XXREAL_0:16; :: thesis: verum

end;

suppose A7: 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. ) )

thus ( ( (max+ f) . x = f . x or (max+ f) . x = 0. ) & ( (max- f) . x = - (f . x) or (max- f) . x = 0. ) ) by A1, A7, FUNCT_1:def 4; :: thesis: verum

end;

end;