 reserve A for non empty Subset of REAL;
 reserve A for non empty closed_interval Subset of REAL;

theorem
  for a, b, c, d, r be Real st
    a < b & b < c & c < d holds
  TrapezoidalFS (a,b,c,d)
    = (AffineMap (0,0) | (REAL \ ].a,d.[)) +*
      TrapezoidalFS (a,b,c,d) | [.a,d.]
proof
 let a, b, c, d, r be Real;
 assume A1: a < b & b < c & c < d;
 set f1 = (AffineMap (0,0)) | (REAL \ ].a,d.[);
 set f2 = AffineMap ((1 / (b - a)),(- (a / (b - a)))) | [.a,b.];
 set f3 = (AffineMap (0,1)) | [.b,c.];
 set f4 = (AffineMap ((- (1 / (d - c))),(d / (d - c)))) | [.c,d.];
 thus TrapezoidalFS (a,b,c,d)
 = ((((AffineMap (0,0)) | (REAL \ ].a,d.[)) +*
 ((AffineMap ((1 / (b - a)),(- (a / (b - a))))) | [.a,b.])) +*
 ((AffineMap (0,1)) | [.b,c.])) +*
 ((AffineMap ((- (1 / (d - c))),(d / (d - c)))) | [.c,d.])
 by FUZNUM_1:def 8,A1
 .= f1 +* (f2 +* f3) +* f4 by FUNCT_4:14
 .= f1 +* ((f2 +* f3) +* f4) by FUNCT_4:14
 .= (AffineMap (0,0) | (REAL \ ].a,d.[)) +* TrapezoidalFS (a,b,c,d) | [.a,d.]
  by LmSymTrape2,A1;
end;
