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 rng (Inv (A,B)) = [:B,A:]
proof
  let A,B be non empty set;
  thus rng Inv(A,B) c= [:B,A:];
  let x be object;
A1: dom Inv (A,B) = [:A,B:] by FUNCT_2:def 1;
  assume x in [:B,A:];
  then reconsider y = x as Element of [:B,A:];
  Inv(A,B).[y`2,y`1] = [[y`2,y`1]`2,[y`2,y`1]`1] by Def6
    .= [y`1,[y`2,y`1]`1]
    .= [y`1,y`2]
    .= y by MCART_1:21;
  hence thesis by A1,FUNCT_1:def 3;
end;
