reserve U1,U2,U3 for Universal_Algebra,
  n,m for Nat,
  x,y,z for object,
  A,B for non empty set,
  h1 for FinSequence of [:A,B:];
reserve h1 for homogeneous quasi_total non empty PartFunc of
    (the carrier of U1)*,the carrier of U1,
  h2 for homogeneous quasi_total non empty PartFunc of
    (the carrier of U2)*,the carrier of U2;

theorem
  for A,B be non empty set holds Inv (A,B) is one-to-one
proof
  let A,B be non empty set;
  let x,y be object;
  assume that
A1: x in dom Inv(A,B) & y in dom Inv(A,B) and
A2: Inv(A,B).x = Inv(A,B).y;
  reconsider x1 = x,y1 = y as Element of [:A,B:] by A1,FUNCT_2:def 1;
  Inv(A,B).x1 = [x1`2,x1`1] & Inv(A,B).y1 = [y1`2,y1`1] by Def6;
  then x1`1 =y1`1 & x1`2 = y1`2 by A2,XTUPLE_0:1;
  hence thesis by DOMAIN_1:2;
end;
