let a, b, c be Real; :: thesis: for f being Function of REAL,REAL st b > 0 & c > 0 & ( for x being Real holds f . x = max (0,(b - |.((b * (x - a)) / c).|)) ) holds
centroid (f,['(a - c),(a + c)']) = a
let f be Function of REAL,REAL; :: thesis: ( b > 0 & c > 0 & ( for x being Real holds f . x = max (0,(b - |.((b * (x - a)) / c).|)) ) implies centroid (f,['(a - c),(a + c)']) = a )
assume that
A1: b > 0 and
A2: c > 0 and
A3: for x being Real holds f . x = max (0,(b - |.((b * (x - a)) / c).|)) ; :: thesis: centroid (f,['(a - c),(a + c)']) = a
f | ['(a - c),(a + c)'] = ((AffineMap ((b / c),(b - ((a * b) / c)))) | [.(lower_bound ['(a - c),(a + c)']),(((b + ((a * b) / c)) - (b - ((a * b) / c))) / ((b / c) - (- (b / c)))).]) +* ((AffineMap ((- (b / c)),(b + ((a * b) / c)))) | [.(((b + ((a * b) / c)) - (b - ((a * b) / c))) / ((b / c) - (- (b / c)))),(upper_bound ['(a - c),(a + c)']).]) by Th16, A1, A2, A3;
hence centroid (f,['(a - c),(a + c)']) = a by Th15, A1, A2; :: thesis: verum