reserve f, g, h for Function;
reserve x, y, z, u, X for set,
  A for non empty set,
  n for Element of NAT,
  f for Function of X, X;
reserve f for c=-monotone Function of bool X, bool X,
  S for Subset of X;
reserve X, Y for non empty set,
  f for Function of X, Y,
  g for Function of Y, X;

theorem Th11: ::: EULER_1:12
  f is bijective implies X,Y are_equipotent
proof
  assume
A1: f is bijective;
  take h = f;
A2: h is one-to-one onto by A1,FUNCT_2:def 4;
  then
A3: rng h = Y by FUNCT_2:def 3;
  hereby
    let x be object;
    assume
A4: x in X;
    reconsider y = h.x as object;
    take y;
    thus y in Y by A3,A4,FUNCT_2:4;
    x in dom h by A4,FUNCT_2:def 1;
    hence [x,y] in h by FUNCT_1:1;
  end;
  hereby
    let y be object;
    assume y in Y;
    then consider x being object such that
A5: x in dom h & y = h.x by A3,FUNCT_1:def 3;
    reconsider x as object;
    take x;
    thus x in X & [x,y] in h by A5,FUNCT_1:1;
  end;
  let x,y,z,u be object;
  assume that
A6: [x,y] in h and
A7: [z,u] in h;
A8: z in dom h & u = h.z by A7,FUNCT_1:1;
  x in dom h & y = h.x by A6,FUNCT_1:1;
  hence thesis by A2,A8;
end;
