reserve A for non empty closed_interval Subset of REAL;

theorem Lm221:
for a,b,c,d be Real, f be Function of REAL,REAL st
b > 0 & c > 0 & d > 0 & d < b &
( for x be Real holds f.x = min(d, max(0, b - |. b*(x-a)/c .|)) )
holds centroid (f,['a-c,a+c']) = a
proof
 let a,b,c,d be Real;
 let f be Function of REAL,REAL;
 assume A1: b > 0 & c > 0 & d > 0 & d < b;
 assume for x be Real holds f.x = min(d, max(0, b - |. b*(x-a)/c .|));
 hence centroid (f,['a-c,a+c'])
 = centroid (
 d (#) TrapezoidalFS (a-c,a+(d-b)/(b/c),a+(b-d)/(b/c),a+c),['a-c,a+c'])
 by Lm220,A1
 .= a by Lm223,A1;
end;
