reserve X for non empty set;
reserve e,e1,e2,e19,e29 for Equivalence_Relation of X,
  x,y,x9,y9 for set;
reserve A for non empty set,
  L for lower-bounded LATTICE;

theorem Th15:
  for d being distance_function of A,L st d is onto holds alpha d is one-to-one
proof
  let d be distance_function of A,L;
  set f = alpha d;
  assume d is onto;
  then
A1: rng d = the carrier of L by FUNCT_2:def 3;
  for a,b being Element of L st f.a = f.b holds a = b
  proof
    let a,b be Element of L;
    assume
A2: f.a = f.b;
    consider z1 be object such that
A3: z1 in [:A,A:] and
A4: d.z1 = a by A1,FUNCT_2:11;
    consider x1,y1 being object such that
A5: x1 in A & y1 in A and
A6: z1 = [x1,y1] by A3,ZFMISC_1:def 2;
    reconsider x1,y1 as Element of A by A5;
    consider z2 be object such that
A7: z2 in [:A,A:] and
A8: d.z2 = b by A1,FUNCT_2:11;
    consider x2,y2 being object such that
A9: x2 in A & y2 in A and
A10: z2 = [x2,y2] by A7,ZFMISC_1:def 2;
    reconsider x2,y2 as Element of A by A9;
    consider E1 being Equivalence_Relation of A such that
A11: E1 = f.a and
A12: for x,y be Element of A holds [x,y] in E1 iff d.(x,y) <= a by Def8;
    consider E2 being Equivalence_Relation of A such that
A13: E2 = f.b and
A14: for x,y be Element of A holds [x,y] in E2 iff d.(x,y) <= b by Def8;
A15: d.(x2,y2) = b by A8,A10;
    then [x2,y2] in E2 by A14;
    then
A16: b <= a by A2,A15,A11,A12,A13;
A17: d.(x1,y1) = a by A4,A6;
    then [x1,y1] in E1 by A12;
    then a <= b by A2,A17,A11,A13,A14;
    hence thesis by A16,ORDERS_2:2;
  end;
  hence thesis by WAYBEL_1:def 1;
end;
