reserve f,g,h for Function,
  A for set;
reserve F for Function,
  B,x,y,y1,y2,z for set;
reserve x,z for object;

theorem
  (A --> [x,y])~ = A --> [y,x]
proof
A1: dom ((A --> [x,y])~) = dom (A --> [x,y]) by Def1;
  then
A2: dom ((A --> [x,y])~) = A;
    now
    let z be object;
    assume
A4: z in dom ((A --> [x,y])~);
    then (A --> [x,y]).z = [x,y] by A2,Th7;
    hence ((A --> [x,y])~).z = [y,x] by A1,A4,Def1
      .= (A --> [y,x]).z by A2,A4,Th7;
  end;
  hence thesis by A2;
end;
