 reserve a,b,c,x for Real;
 reserve C for non empty set;

theorem
  for a,b,c,d being Real st a < b & b < c & c < d holds
    TrapezoidalFS (a,b,c,d) is normalized
  proof
    let a,b,c,d be Real;
    set F = TrapezoidalFS (a,b,c,d);
    reconsider bb = c as Element of REAL by XREAL_0:def 1;
    assume
Z1: a < b & b < c & c < d;
s0: bb in [.c,d.] by Z1;
S1: F = 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 Z1,TPDef;
s2: dom AffineMap (-1/(d-c),d/(d-c)) = REAL by FUNCT_2:def 1;
    c + 0 < d by Z1; then
t1: d - c > 0 by XREAL_1:20;
    bb in [.c,d.] by Z1; then
    bb in dom (AffineMap (-1/(d-c),d/(d-c)) | [.c,d.]) by s2,RELAT_1:57; then
    F.bb = (AffineMap (-1/(d-c),d/(d-c)) | [.c,d.]).bb by FUNCT_4:13,S1
        .= (AffineMap (-1/(d-c),d/(d-c))).bb by FUNCT_1:49,s0
        .= 1 by Cb1,t1;
    hence thesis;
  end;
