theorem Th15: :: FUZZY_7:29

for a, b, c being Real
for f being Function of REAL,REAL st b > 0 & c > 0 & 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)']).]) holds
centroid (f,['(a - c),(a + c)']) = a