
theorem
for a, b, c being Real st a < b & b < c holds
TriangularFS (a,b,c) is Lipschitzian
proof
 let a, b, c be Real;
 assume A1:a < b & b < c; then
 B5: c-b > b-b by XREAL_1:9;
 set a1 = 1/(b-a);
 set p1 = 1/(c-b);
 set b1 = - a/(b-a);
 set q1 = c/(c-b);
 c > a by A1,XXREAL_0:2;then
 B4: c-a > a-a by XREAL_1:9;
 B7: b-a > a-a by XREAL_1:9,A1;
   (q1-b1)/(a1+p1) = (c/(c-b)+ a/(b-a))/(1/(b-a)+1/(c-b))
  .= (( (c*(b-a)+a*(c-b)) )/((b-a)*(c-b)) )/(1/(b-a)+1/(c-b))
         by XCMPLX_1:116,B5,B7
  .= (( (c*b-c*a+a*c-a*b) )/((b-a)*(c-b)) )
     /(( (1*(b-a)+1*(c-b)) )/((b-a)*(c-b)) ) by XCMPLX_1:116,B5,B7
  .= ((c-a)*b) / (c-a) by XCMPLX_1:55,B5,B7
  .=b by XCMPLX_1:89,B4;then
 for x be Real holds TriangularFS (a,b,c).x
 = max(0,min(1, ( ((AffineMap (a1,b1))|(].-infty,(q1-b1)/(a1+p1).[)) +*
 ((AffineMap (-p1,q1))|([.(q1-b1)/(a1+p1),+infty.[)) ) .x )) by asymTT9,A1;
 hence thesis by asymTT5,B7,B5;
end;
