
theorem asymTT51:
for a,b,p,q be Real st a > 0 & p > 0 holds
ex r being Real st
 ( 0 < r &
 ( for x1, x2 being Real st
x1 in dom ((AffineMap (a,b)|].-infty,(q-b)/(a+p).[)
         +* (AffineMap (-p,q)|[.(q-b)/(a+p),+infty.[)) &
x2 in dom ((AffineMap (a,b)|].-infty,(q-b)/(a+p).[)
         +* (AffineMap (-p,q)|[.(q-b)/(a+p),+infty.[)) holds
  |. ((AffineMap (a,b)|].-infty,(q-b)/(a+p).[)
         +* (AffineMap (-p,q)|[.(q-b)/(a+p),+infty.[)) . x1
  - ((AffineMap (a,b)|].-infty,(q-b)/(a+p).[)
         +* (AffineMap (-p,q)|[.(q-b)/(a+p),+infty.[)) . x2.|
  <= r * |.(x1 - x2).| ) )
proof
  let a,b,p,q be Real;
  assume AP: a > 0 & p > 0;
  set f = ( (AffineMap (a,b)|].-infty,(q-b)/(a+p).[)
     +* (AffineMap (-p,q)|[.(q-b)/(a+p),+infty.[) );
  reconsider f as Function of REAL,REAL by asymTT10;
  f is Lipschitzian by asymTT50,AP;
  hence thesis;
end;
