
theorem Th4:
  for X, Y being non empty TopSpace, x being Point of X, f being
  Function of [:Y, X | {x}:], Y st f = pr1(the carrier of Y, {x}) holds f is
  one-to-one
proof
  let X, Y be non empty TopSpace, x be Point of X, f be Function of [:Y, X | {
  x}:], Y;
  set Z = {x};
  assume
A1: f = pr1(the carrier of Y, Z);
  let z, y be object such that
A2: z in dom f and
A3: y in dom f and
A4: f.z = f.y;
A5: dom f = [:the carrier of Y, Z:] by A1,FUNCT_3:def 4;
  then consider x1, x2 being object such that
A6: x1 in the carrier of Y and
A7: x2 in Z and
A8: z = [x1, x2] by A2,ZFMISC_1:def 2;
  consider y1, y2 being object such that
A9: y1 in the carrier of Y and
A10: y2 in Z and
A11: y = [y1, y2] by A5,A3,ZFMISC_1:def 2;
A12: x2 = x by A7,TARSKI:def 1
    .= y2 by A10,TARSKI:def 1;
  x1 = f.(x1, x2) by A1,A6,A7,FUNCT_3:def 4
    .= f.(y1, y2) by A4,A8,A11
    .= y1 by A1,A9,A10,FUNCT_3:def 4;
  hence thesis by A8,A11,A12;
end;
