theorem
  f is one-to-one & x in dom f & y in dom f & f/.x = f/.y implies x = y
proof
  assume that
A1: f is one-to-one and
A2: x in dom f and
A3: y in dom f;
  assume f/.x = f/.y;
  then f.x = f/.y by A2,PARTFUN1:def 6
    .= f.y by A3,PARTFUN1:def 6;
  hence thesis by A1,A2,A3;
end;
