 reserve A for non empty Subset of REAL;
 reserve A for non empty closed_interval Subset of REAL;

theorem
  for a,b,p,q be Real, f be Function of REAL,REAL st a <> p &
  f = (AffineMap (a,b)|].-infty,(q-b)/(a-p).]) +*
    (AffineMap (p,q)|[.(q-b)/(a-p),+infty.[) holds
  f is Lipschitzian
proof
 let a,b,p,q be Real, f be Function of REAL,REAL;
 assume A1: a <> p &
 f = (AffineMap (a,b)|].-infty,(q-b)/(a-p).]) +*
   (AffineMap (p,q)|[.(q-b)/(a-p),+infty.[); then
  f = (AffineMap (a,b)|].-infty,(q-b)/(a-p).[) +*
   (AffineMap (p,q)|[.(q-b)/(a-p),+infty.[) by FUZZY_6:35;
 hence thesis by FUZZY_6:28,A1;
end;
