
theorem
for a, b, c, d being Real st a < b & b < c & c < d holds
TrapezoidalFS (a,b,c,d) is Lipschitzian
proof
 let a, b, c ,d be Real;
 assume A1:a < b & b < c & c < d; then
 A20: a-a < b-a by XREAL_1:9;
 A30: c-c < d-c by XREAL_1:9,A1;
 set a1 = 1/(b-a);
 set b1 = - a/(b-a);
 set p1 = 1/(d-c);
 set q1 = d/(d-c);
 B5: (q1-b1)/(a1+p1) = (d/(d-c) + a/(b-a))/(1/(b-a) + 1/(d-c))
  .=( (d*(b-a)+a*(d-c)) /((d-c)*(b-a)))/(1/(b-a) + 1/(d-c))
        by XCMPLX_1:116,A20,A30
  .=((d*b-d*a+a*d-a*c)/((d-c)*(b-a)))/( (1*(b-a)+1*(d-c)) /((d-c)*(b-a)) )
        by XCMPLX_1:116,A20,A30
  .= (b*d-a*c)/(d-c+b-a) by XCMPLX_1:55,A20,A30;
 for x be Real holds
 TrapezoidalFS (a,b,c,d).x = max(0,min(1, (
 (AffineMap ( 1/(b-a),- a/(b-a) )|].-infty,(b*d-a*c)/(d-c+b-a).[) +*
 (AffineMap ( - 1/(d-c),d/(d-c) )|[.(b*d-a*c)/(d-c+b-a),+infty.[) ).x  ))
   by asymTT8,A1;
 hence thesis by asymTT5,A20,A30,B5;
end;
