reserve A for non empty closed_interval Subset of REAL;

theorem
for a, b, c being Real st a < b & b < c holds
(for x be Real st x in [.b,c.] holds
TriangularFS (a,b,c).x = AffineMap (- (1 / (c - b)), c / (c - b)).x )
proof
 let a, b, c be Real;
 assume A1: a < b & b < c;
  let x be Real;
  assume B1: x in [.b,c.]; then
    B21: x in dom(AffineMap ((- (1 / (c - b))),(c / (c - b))) | [.b,c.])
      by FUNCT_2:def 1;
  thus TriangularFS (a,b,c).x
   = ((((AffineMap (0,0)) | (REAL \ ].a,c.[)) +*
  ((AffineMap ((1 / (b - a)),(- (a / (b - a))))) | [.a,b.])) +*
  ((AffineMap ((- (1 / (c - b))),(c / (c - b)))) | [.b,c.])).x
   by A1,FUZNUM_1:def 7
  .= ( AffineMap (- (1 / (c - b)), c / (c - b)) | [.b,c.] ).x
   by B21,FUNCT_4:13
  .= ( AffineMap (- (1 / (c - b)), c / (c - b))).x
   by B1,FUNCT_1:49;
end;
