
theorem:: asymTT1:
for a,b,p,q be Real, f be Function of REAL,REAL st
for x be Real holds
f.x = max(0,min(1, ( ((AffineMap (a,b))|(].-infty,(q-b)/(a-p).[)) +*
                     ((AffineMap (p,q))|([.(q-b)/(a-p),+infty.[)) ) .x ))
holds f is FuzzySet of REAL
proof
 let a,b,p,q be Real;
 let f be Function of REAL,REAL;
 assume
 A3: for x be Real holds
 f.x = max(0,min(1, ( ((AffineMap (a,b))|(].-infty,(q-b)/(a-p).[)) +*
                      ((AffineMap (p,q))|([.(q-b)/(a-p),+infty.[)) ) .x ));
 set g = ( ((AffineMap (a,b))|(].-infty,(q-b)/(a-p).[)) +*
                      ((AffineMap (p,q))|([.(q-b)/(a-p),+infty.[)) );
 reconsider g as Function of REAL,REAL by asymTT10;
 for x being Real holds f.x= max(0,min(1, g.x)) by A3;
 hence thesis by MM40;
end;
