reserve a,a1,a2,b,c,d for Ordinal,
  n,m,k for Nat,
  x,y,z,t,X,Y,Z for set;
reserve f,g for Function;
reserve A,B,C for array;

theorem Th29:
  x in dom f & y in dom f implies Swap(f,x,y).x = f.y
  proof assume
A1: x in dom f & y in dom f; then
A2: Swap(f,x,y) = f+*(x,f.y)+*(y,f.x) by FUNCT_7:def 12;
A3: dom f = dom(f+*(x,f.y)) by FUNCT_7:30;
    now assume x <> y; then
      Swap(f,x,y).x = (f+*(x,f.y)).x by A2,FUNCT_7:32;
      hence thesis by A1,FUNCT_7:31;
    end;
    hence Swap(f,x,y).x = f.y by A3,A1,A2,FUNCT_7:31;
  end;
