theorem
  x in dom A & y in dom A & A is permutation of B implies
  Swap(A,x,y) is permutation of B & A is permutation of Swap(B,x,y)
  proof set X = dom A;
    assume
A1: x in X & y in X & A is permutation of B; then
    X = dom B by Th37; then
    Swap(A,x,y) is permutation of A &
    B is permutation of Swap(B,x,y) by A1,Th43;
    hence thesis by A1,Th40;
  end;
